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Generalized widths and reverse Urysohn inequalities
up vote 4 down vote favorite
This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of
minimum widths and constant widths. Some interesting ideas came up in those discussions which I would like to bring together with some ideas I had and some things I'm not clear on yet.
Let the $\mu$-width of a three-dimensional body $K$ be a function $$w_\mu(\vartheta)=\int_{S^2} h(\vartheta(\mathbf{u})) d\mu(\mathbf{u})\text,$$ where $\vartheta\in SO(3)$ is a rotation, $h(\mathbf
{u})=\max_{\mathbf{x}\in K} \mathbf{u}\cdot\mathbf{x}$ is the support height function, and $\mu$ is a (signed?) measure on $S^2$ with $\int \mathbf{u} d\mu(\mathbf{u})=0$ (to ensure translation
invariance). The standard width corresponds to a measure concentrated at two opposite points. The mean width corresponds to the uniform measure. The main properties I want to talk about are the
minimum $\mu$-width (the minimum of $w_\mu(\vartheta)$) and bodies of constant $\mu$-width. The mean $\mu$-width is not interesting because it reduces to the mean standard width (assuming $\mu(S^2)\
neq 0$).
The minimum standard width is the "mailslot width", the smallest mailslot through which the body can pass. If $\mu$ is concentrated uniformly on the equator (this is related to the spherical Radon
transform) then the minimum width is the "loop width", giving the smallest length of string loop through which the body can pass (this follows from the fact that the mean width of a planar body is
proportional to its perimeter). If $\mu$ is concentrated with equal weight at the vertices of a regular tetrahedron, the minimum "tetrahedral width" gives the linear size of the smallest regular
tetrahedron that contains the body. Bodies of constant tetrahdral width are the rotors of a tetrahedral cavity (see "Bodies of constant width?").
Let us define the "harmonic support" (h.s.) of a function or measure on $S^2$ as the set of integers $n>1$ such that the projection of the function to the space of spherical harmonics of degree $n$
does not vanish. Also let $\mathcal{K}_I$ be the space of convex bodies such that the harmonic support of their height function is a subset of $I$. If $I$ and $J$ are disjoint and their union is $\
{n>1\}$, I call $\mathcal K_I$ and $\mathcal K_J$ complementary spaces. Then the space of bodies of constant $\mu$-width is the complementary space to $\mathcal{K}_{h.s.(\mu)}$. Thus, it follows that
bodies of constant width and bodies of constant loop width are the same.
Urysohn's inequality says that the ratio $vol/\bar{w}^3$, where $\bar{w}$ is the mean width is maximized by balls. In general, the ball also maximizes the ratio $vol/w_\mu^3$ among bodies of constant
$\mu$-width $w_\mu$. I am interested in the complementary space, and whether $vol/w_\mu^3$, where $w_\mu$ is the minimum $\mu$-width, is minimized by balls among bodies in $\mathcal K_{h.s.(\mu)}$.
Clearly, this holds for the standard width: among all centrally-symmetric bodies of a given volume, balls maximize the mailslot width (not true if central symmetry is not assumed). However, based on
some experiments I made, I find that this is not true in general as a global statement. Still, I believe that balls are local minima. This is because it is pretty easy to show that if $K\in\mathcal
K_{h.s.(\mu)}$ is not a ball, then for some $\alpha_0>0$ the body $K_\alpha=(1-\alpha)B+\alpha K$ obtains a greater ratio than that of the ball for all $0<\alpha<\alpha_0$. (See "Local minimum from
directional derivatives in the space of convex bodies"). My question is, can you find a counterexample of my claim that $B$ is a local minimum of $vol/w_\mu^3$ among bodies in $\mathcal K_{h.s.(\mu)}
$; or can you see a way of proving it?
Dear Yoav, Do you have some idea on how $\alpha_0$ depends on $K$? I guess you need some sort of uniform estimate. Is $K=B$ the only "problem"? By this I mean that as $K$ approaches a ball the
estimate of $\alpha_0$ will get worse and worse because $f(K_\alpha)$ will be constant as a function of $\alpha$, but is the ball the only convex body for which you see this happening? I struggled
with a problem like this and at the end settled for the directional derivative result ... The problem was that there were lots and lots of directions for which the directional derivative was zero.
– alvarezpaiva Feb 12 '12 at 12:15
Dear Prof Alvarez, thank you for your comment. I am also leaning toward settling for the partial result, but not there yet. The directional derivative is strictly positive in my case for all $K\neq
B$. In fact, because $\alpha_0(K)$ is continuous (I think), there is a positive minimum $\alpha_0(\epsilon)$ over all $K$ with $d(K,B)=\epsilon$ by compactness. However, not all $K'$ with $d(K',B)\
le\epsilon$ can be written as $K_\alpha$ where $d(K,B)=\epsilon$. – Yoav Kallus Feb 12 '12 at 20:43
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1 Answer
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This is hardly a direct answer to your question, but a new paper—at least tangentially relevant—by HaiLin Jin and Qi Guo addresses the question of how assymetric can a constant-width
body be. In "Asymmetry of Convex Bodies of Constant Width" (Discrete & Computational Geometry Vol. 47, No. 2, Mar. 2012, 415-423), they establish tight bounds on the "Minkowski measure
up vote 1 of asymmetry for convex bodies." In particular, they extend the known result that Reuleaux triangles are most assymetric in $\mathbb{R}^2$ to showing that Meissner's tetrahedron is most
down vote assymetric in $\mathbb{R}^3$. An image of Meissner's tetrahedron appeared in the earlier MO question, "Are there smooth bodies of constant width?"
Thanks for beautifying my OP. Though your answer doesn't directly address my question, it does suggest other questions we may ask about generalized widths. E.g., we may ask how
1 asymmetric can a body of constant tetrahedral width be. This is much simpler then the question for constant standard width, because the corresponding space only has 13 d.o.f. (fixing
dilation, translation, and rotation). Even simpler if we discuss rotors of a octahedral cavity (only 8 d.o.f.). – Yoav Kallus Jan 27 '12 at 22:25
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Not the answer you're looking for? Browse other questions tagged convexity convex-geometry isoperimetry mg.metric-geometry harmonic-analysis or ask your own question. | {"url":"http://mathoverflow.net/questions/86797/generalized-widths-and-reverse-urysohn-inequalities","timestamp":"2014-04-19T07:14:53Z","content_type":null,"content_length":"61325","record_id":"<urn:uuid:461a748e-2c3f-4291-ba6f-234ed348ebd6>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00337-ip-10-147-4-33.ec2.internal.warc.gz"} |
Two set theory questions
February 5th 2010, 01:50 AM #1
Junior Member
Nov 2009
Two set theory questions
1) Prove that there is no function f with domain ω such that f (n+) ∈ f (n) for all n ∈ ω. [Hint: apply the Axiom of Foundation to ran f .] Deduce that, for any set x, it is false that x ∈ x.
2) Prove, using the Principle of Induction and the fact that each n ∈ ω is a transitive set, that n ∈ n is false for every natural number n.
Presumably, the idea in the first question is to show that if it were the case that there were such a function, then the range of f would contradict the axiom of Foundation, but I'm not entirely
sure how to do the details.
Any help would be greatly appreciated. Many thanks.
1) Prove that there is no function f with domain ω such that f (n+) ∈ f (n) for all n ∈ ω. [Hint: apply the Axiom of Foundation to ran f .] Deduce that, for any set x, it is false that x ∈ x.
2) Prove, using the Principle of Induction and the fact that each n ∈ ω is a transitive set, that n ∈ n is false for every natural number n.
Presumably, the idea in the first question is to show that if it were the case that there were such a function, then the range of f would contradict the axiom of Foundation, but I'm not entirely
sure how to do the details.
Any help would be greatly appreciated. Many thanks.
Rewrite the question and PREVIEW it to check it comes right. The way you wrote it can't be understood properly.
Which part can't be understood?
I see what's happened- the symbols are displayed on my screen, but clearly not on everyone's... I thought I couldn't see what the problem was...
In normal language, the questions are:
1) Prove that there is no function f with domain ω (omega) such that f (n+) is a member of f (n) for all n in ω (omega). Deduce that, for any set x, it is false that x is a member of itself.
(where n+ is the sucessor of n)
2) Prove, using the Principle of Induction and the fact that each n in ω (omega) is a transitive set, that the statement "n is a member of itself" is false for every natural number n. (i.e. prove
without using the Axiom of Foundation)
Sorry for the confusion.
Do you still want help with this?
What steps have you taken toward proofs of these?
February 5th 2010, 05:52 AM #2
Oct 2009
February 5th 2010, 08:26 AM #3
Junior Member
Nov 2009
February 5th 2010, 09:26 AM #4
Oct 2009
February 5th 2010, 09:38 AM #5
Junior Member
Nov 2009
February 9th 2010, 11:47 AM #6
Senior Member
Feb 2010 | {"url":"http://mathhelpforum.com/discrete-math/127281-two-set-theory-questions.html","timestamp":"2014-04-20T19:34:04Z","content_type":null,"content_length":"45237","record_id":"<urn:uuid:cf527d81-e00d-4fd3-9080-ba36d30d65f6>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00112-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ibanez Destroyer Information Exchange by TheDestroyerGuy - Home
Ibanez Destroyer Information Exchange
1985 DT-4550 BK-FE
TheDestroyerGuy is a fan site devoted to the free exchange of information relating to the various models of the Ibanez Destroyer electric guitar.
This site was last updated on 20 April 2014. Please refresh your browser cache often to ensure you get the latest changes.
Stomp on a pedal and jump to that section of this page:
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Click HERE to claim your free* t-shirt.
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Miscellaneous section of the site. This capture contains the contents of the the Ibanez picture disk advertised in 1985. Listen to the interview here.
We have removed the +/- images. Simply click on any section header image and that section will collapse/expand.
We have added new navigation images on the Home page.
We have updated the pick-up listing to include the newly released 2014 DT-420.
We have added a new page for the recently released 2014 DT-420 and we have also updated the Gallery page to include it, as well.
The new 2014 DT-420 TCH has been released. Get 'em while they're hot!
Wow! Lots of new Destroyers but the model numbers get a little bit confusing, so let's start at the beginning. The DT-520FM-CRS and the DT-520-BK were released first. Both have chrome hardware.
Aside from having different colored finishes, the difference between these two models is that the "FM" model has a Flame Maple cap on the body.
Recently, two variants of the DT-520 were released in the US, exclusively to the Guitar Center/Musician's Friend chains: the DT1GFM and the DT2G. These are versions of the DT-520FM and DT-520 that
have gold hardware rather than chrome. The DT1GFM comes in either a Cherry Red Sunburst or a Vintage Sunburst finish. The DT2G comes only in a Candy Apple finish. The major difference between the
DT1GFM/DT2G and the DT-520FM/DT-520 is that the tone control on the DT1GFM/DT2G has a push/pull operation that is a coil tap function.
If you look on the Ibanez (Japan) web site, you'll see what appears to be the DT-520FM-CRS, DT-520-BK, and the DT2G-CA except the model numbers shown are DT-520FMGB, DT-520GB, and DT-520GGB,
respectively. After asking an authoritative source, it seems the "GB" designation denotes "gigbag", which would make the model numbers align for the DT-520FM and DT-520.
It was our understanding that the DT2G was exclusive to Guitar Center/Musician's Friend yet the Ibanez (Japan) web site is clearly showing it as a model DT-520G.
A note on years: unless otherwise indicated, the years provided in the timelines below are the years each model first appeared
in Ibanez marketing materials
(i.e.: catalogs, flyers, brochures, etc). Actual dates of manufacture may precede these catalog appearance dates.
Three model numbers were reused: DT-200 (
), DT-400 (
), and DT-420 (
2012 sample
). The second incarnation of the
(2002) is a reissue of the second incarnation of the
(1996). The 2014
appears to be the same guitar as the 2013
but with a new finish and different pick-ups.
Known Models
1975: Mahogany Model 2459^[1]
1976: Model 2459
1980: DT-50, DT-400 (first incarnation)
1982: DT-300
1983: DT-100, DT-150, DT-155, DT-200 (first incarnation), DT-500, DT-555
1984: DT-355, DT-450
1985: DT-250, DT-330, DT-350, DT-380, DT-4550
1986: DG-350^[2], DG-351, DG-555^[2]
1992: DT-420^[3] (first incarnation), MED-320^[3]
1996: DT-400 (second incarnation)
2001: DTX-120
2002: DT-420^[3] (second incarnation)
2004: DT-200 (second incarnation)
2009: DTT-700
2011: GDTM21 Mikro^[3]
2013: DT-520 / DT1GFM / DT2G^[3]
2014: DT-420^[3] (third incarnation)
Known Samples
What is a "sample"?
is a pre-production edition of a guitar, usually a one-off, that is used as a demonstrator within Hoshino-Gakki to determine whether that particular design will become a production model. Model
samples may vary by hardware configuration or by something as trivial as having a different colored finish. Samples may also be used as demonstrators at trade shows to advertise an upcoming
product prior to its release.
2012: DT-420GFM, DT-520FM
Speculative Models (urban legend)
Unknown Models: We need your help identifying these models
1983: Mystery 6-String
1985: Mystery 6-String #2
Similar Models (not Destroyers)
Known Models
1976: Model 2459B
1981: DT-700^[3]
1983: DT-600
1984: DT-650^[3], DT-670, DT-870
1985: DB-700, DT-6750
1986: DB-800
2011: DTB-100^[3]
2013: MDB3^[3]
[1] the Model 2459 prototype was created for the 1975 Summer NAMM in Chicago, IL, USA (Ibanez - The Untold Story, Specht, Wright, and Donahue, 2005, Hoshino (U.S.A.) Inc, ISBN: 0-976-4277-0-2, pg
73, inset).
[2] note the model name changes in 1986 from DT-350 to DG-350 and from DT-555 to DG-555.
[3] the following models did not appear in any known marketing materials so the years provided are from actual dates of manufacture: 1981 DT-700, 1984 DT-650, 1992 DT-420, 1992 MED-320, 2002
DT-420, 2011 GDTM21 Mikro, 2011 DTB-100, 2013 MDB3, 2013 DT-520 / DT1GFM / DT2G, and the 2014 DT-420. | {"url":"http://www.thedestroyerguy.com/","timestamp":"2014-04-20T18:36:35Z","content_type":null,"content_length":"35069","record_id":"<urn:uuid:d09305b7-19a0-4d0b-b9ad-6410774fc1b1>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00391-ip-10-147-4-33.ec2.internal.warc.gz"} |
New monads
From HaskellWiki
(Difference between revisions)
(categorise) m (NewMonads moved to New monads)
← Older edit Newer edit →
Revision as of 15:55, 7 October 2006
1 MonadBase
It seems that the liftIO function from MonadIO can be generalized to access whatever the base of a transformer stack happens to be. So there is no need for a liftSTM, liftST, etc.
View NewMonads/MonadBase.
2 MonadLib
This is by Iavor S. Diatchki and can be found at http://www.cse.ogi.edu/~diatchki/monadLib/
It is a new version of the mtl package with transformers: ReaderT WriterT StateT ExceptT SearchT ContT
It also defines BaseM which is like MonadBase above.
3 LazyWriterT
This came up on the mailing list: Why is WriterT never lazy? The answer is it does not use lazy patterns with "~". So here is a more useful NewMonads/LazyWriterT that add two "~" to the definition of
(>>=) and renames WriterT to LazyWriterT.
4 MonadRandom
A simple monad transformer to allow computations in the transformed monad to generate random values.
View NewMonads/MonadRandom.
5 MonadSupply
Here is a simple monad/monad transformer for computations which consume values from a (finite or infinite) supply. Note that due to pattern matching, running out of supply in a non-MonadZero monad
will cause an error.
View NewMonads/MonadSupply.
6 MonadUndo
Here is a modified state monad transformer for keeping track of undo/redo states automatically.
View NewMonads/MonadUndo.
7 MonadUnique
This is a simple (trivial) monad transformer for supplying unique integer values to an algorithm.
View NewMonads/MonadUnique.
8 MonadSTO
Here's an extension of the ST monad in which the references are ordered and showable (they list their creation index).
View NewMonads/MonadSTO. | {"url":"http://www.haskell.org/haskellwiki/index.php?title=New_monads&diff=6564&oldid=6087","timestamp":"2014-04-18T00:45:25Z","content_type":null,"content_length":"18374","record_id":"<urn:uuid:6d297613-b531-4e1c-8e1c-766e0734fab1>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00536-ip-10-147-4-33.ec2.internal.warc.gz"} |
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A force is applied to a 3.61kg- radio-controlled model car parallel to the x-axis as it moves along a straight track. The x-component of the force varies with the x-coordinate of the car as shown in
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What is the work done by the force when the car moves from x=0 to x=3? I have tried to find the area of the graph using the area of trapezoid .5(b1+b2)*2 and got F=4 and distance is 3 = W=12 J
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Use the work-energy theorem to find the speed of the car at x=3.0m
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What is the work done by the force on the car from to 3m to 4m ? ANSWER IS 0 JOULES
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International Tables for Crystallography, Volume B, 3rd Edition, Reciprocal Space
ISBN: 978-1-4020-8205-4
696 pages
October 2008
International Tables for Crystallography
is the definitive resource and reference work for crystallography and structural science.
Each of the eight volumes in the series contains articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the
structure and properties of materials. Emphasis is given to symmetry, diffraction methods and techniques of crystal-structure determination, and the physical and chemical properties of crystals. The
data are accompanied by discussions of theory, practical explanations and examples, all of which are useful for teaching.
Volume B provides the reader with competent and useful accounts of the numerous aspects of reciprocal space in crystallographic research. Following an introductory chapter, the volume is divided into
five parts:
Part 1: Presents an account of structure factor formalisms, extensive treatment of the theory, algorithms and crystallographic applications of Fourier methods, and fundamental as well as advanced
treatments of symmetry in reciprocal space.
Part 2: Discusses crystallographic statistics, the theory of direct methods, Patterson techniques, isomorphous replacement and anomalous scattering, and treatments of the role of electron microscopy
and diffraction in crystal- structure determination.
Part 3: Includes applications of reciprocal space to molecular geometry and `best'-plane calculations, and contains a treatment of the principles of molecular graphics and modeling and their
Part 4:Contains treatments of various diffuse-scattering phenomena arising from crystal dynamics, disorder and low dimensionality (liquid crystals), and an exposition of the underlying theories and/
or experimental evidence. Polymer crystallography and reciprocal-space images of aperiodic crystals are also treated.
Part 5: Discusses introductory treatments of the theory of the interaction of radiation with matter (dynamical theory) as applied to X-ray, electron and neutron diffraction techniques.
Substantially revised and updated to take into account recent developments, the third edition of volume B includes contributions from seven new authors and a new chapter on extensions of the Ewald
method for Coulomb interactions in crystals. There are three new sections on electron diffraction and electron microscopy in structure determination.
Volume B is a vital addition to the library of scientists engaged in crystal-structure determination, crystallographic computing, crystal physics and other fields of crystallographic research.
Graduate students specializing in crystallography will find much material suitable for self-study and a rich source of references to the relevant literature.
See More
Preface (
U. Shmueli
Preface to the Second Edition (U. Shmueli).
Preface to the Third Edition (U. Shmueli).
1.1 Reciprocal Space in Crystallography (U. Shmueli).
1.1.1 Introduction.
1.1.2 Reciprocal Lattice in Crystallography.
1.1.3 Fundamental Relationships.
1.1.4 Tensor-Algebraic Formulation.
1.1.5 Transformations.
1.1.6 Some Analytical Aspects of the Reciprocal Space.
1.2 The Structure Factor (P. Coppens).
1.2.1 Introduction.
1.2.2 General Scattering Expression for X-Rays.
1.2.3 Scattering by a Crystal: Definition of a Structure Factor.
1.2.4 The Isolated-Atom Approximation in X-Ray Diffraction.
1.2.5 Scattering of Thermal Neutrons.
1.2.6 Effect of Bonding on the Atomic Electron Density within the Spherical-Atom Approximation: The Kappa Formalism.
1.2.7 Beyond the Spherical-Atom Description: The Atom-Centred Spherical Harmonic Expansion.
1.2.8 Fourier Transform of Orbital Products.
1.2.9 The Atomic Temperature Factor.
1.2.10 The Vibrational Probability Distribution and its Fourier Transform in the Harmonic Approximation.
1.2.11 Rigid-Body Analysis.
1.2.12 Treatment of Anharmonicity.
1.2.13 The Generalized Structure Factor.
1.2.14 Conclusion.
1.3 Fourier Transforms in Cystallography: Theory, Algorithms and Applications (G. Bricogne).
1.3.1 General Introduction.
1.3.2 The Mathematical Theory of the Fourier Transformation.
1.3.3 Numerical Computation of the Discrete Fourier Transform.
1.3.4 Crystallographic Applications of Fourier Transforms.
1.4 Symmetry in Reciprocal Space (U. Shmueli).
1.4.1 Introduction.
1.4.2 Effects of Symmetry on the Fourier Image of the Crystal.
1.4.3 Structure-Factor Tables.
1.4.4 Symmetry in Reciprocal Space: Space-Group Tables.
1.5 Crystallographic Viewpoints in the Classification of Space-Group Representations (M.I. Aroyo and H. Wondratschek).
1.5.1 List of Abbreviations and Symbols.
1.5.2 Introduction.
1.5.3 Basic Concepts.
1.5.4 Conventions in the Classification of Space-Group Irreps.
1.5.5 Examples and Discussion.
1.5.6 Conclusions.
Appendix A1.5.1. Reciprocal-space groups (G)*
2.1 Statistical Properties of the Weighted Reciprocal Lattice (U. Shmueli and A.J.C. Wilson).
2.1.1 Introduction.
2.1.2 The Average Intensity of General Reflections.
2.1.3 The Average Intensity of Zones and Rows.
2.1.4 Probability Density Distributions - Mathematical Preliminaries.
2.1.5 Ideal Probability Density Distributions.
2.1.6 Distributions of Sums, Averages and Ratios.
2.1.7 Non-Ideal Distributions: The Correction-Factor Approach.
2.1.8 Non-Ideal Distributions: The Fourier Method.
2.2 Direct Methods (C. Giacovazzo).
2.2.1 List of Symbols and Abbreviations.
2.2.2 Introduction.
2.2.3 Origin Specification.
2.2.4 Normalized Structure Factors.
2.2.5 Phase-Determining Formulae.
2.2.6 Direct Methods in Real and Reciprocal Space: Sayre's Equation.
2.2.7 Scheme of Procedure for Phase Determination: The Small-Molecule Case.
2.2.8 Other Mutlisolution Methods Applied to Small Molecules.
2.2.9 Some References to Direct-Methods Packages: The Small-Molecule Case.
2.2.10 Direct Methods in Macromolecular Crystallography.
2.3 Patterson and Molecular Replacement Techniques, and the Use of Noncrystallographic Symmetry in Phasing (L. Tong, M.G. Rossmann and E. Arnold).
2.3.1 Introduction.
2.3.2 Interpretation of Patterson Maps.
2.3.3 Isomorphous Replacement Difference Pattersons.
2.3.4 Anomalous Dispersion.
2.3.5 Noncrystallographic Symmetry.
2.3.6 Rotation Functions.
2.3.7 Translation Functions.
2.3.8 Molecular Replacement.
2.3.9 Conclusions.
2.4 Isomorphous Replacement and Anomalous Scattering (M. Vijayan and S. Ramaseshan).
2.4.1 Introduction.
2.4.2 Isomorphous Replacement Method.
2.4.3 Anomalous-Scattering Method.
2.4.4 Isomorphous Replacement and Anomalous Scattering in Protein Crystallography.
2.4.5 Anomalous Scattering of Neutrons and Synchrotron Radiation. The Multiwavelength Method.
2.5 Electron Diffraction and Electron Microscopy in Structure Determination (J.M. Cowley, J.C.H. Spence, M. Tanaka, B.K. Vainshtein, B.B. Zvyagin, P.A. Penczek and D.L. Dorset).
2.5.1 Foreword (J.M. Cowley and J.C.H. Spence).
2.5.2 Electron Diffraction and Electron Microscopy (J.M. Crowley).
2.5.3 Point-Group and Space-Group Determination by Convergent-Beam Electron Diffraction (M. Tanaka).
2.5.4 Electron-Diffraction Structure Analysis (B.K. Vainshtein and B.B. Zvyagin).
2.5.5 Image Reconstruction (B.K. Vainshtein).
2.5.6 Three-Dimensional Reconstruction (B.K. Vainshtein and P.A. Penczek).
2.5.7 Single-Particle Reconstruction (P.A. Penczek).
2.5.8 Direct Phase Determination in Electron Crystallography (D.L. Dorset).
3.1 Distances, Angles and Their Standard Uncertainties (D.E. Sands).
3.1.1 Introduction.
3.1.2 Scalar Product.
3.1.3 Length of a Vector.
3.1.4 Angle Between Two Vectors.
3.1.5 Vector Product.
3.1.6 Permutation Tensors.
3.1.7 Components of Vector Product.
3.1.8 Some Vector Relationships.
3.1.9 Planes.
3.1.10 Variance-Covariance Matrices.
3.1.11 Mean Values.
3.1.12 Computation.
3.2 The Least-Squares Plane (R.E. Marsh and V. Schomaker).
3.2.1 Introduction.
3.2.2 Least-Squares Plane Based on Uncorrelated, Isotropic Weights.
3.2.3 The Proper Least-Squares Plane, with Gaussian Weights.
3.3 Molecular Modelling and Graphics (R. Diamond and L.M.D. Cranswick).
3.3.1 Graphics (R. Diamond).
3.3.2 Molecular Modelling, Problems and Approaches (R. Diamond).
3.3.3 Implementations (R. Diamond).
3.3.4 Graphics Software for the Display of Small and Medium-Sized Molecules (L.M.D. Cranswick).
3.4 Accelerated Convergence Treatment of R^—n Lattice Sums (D.E. Williams).
3.4.1 Introduction.
3.4.2 Definition and Behaviour of the Direct-Space Sum.
3.4.3 Preliminary Description of the Method.
3.4.4 Preliminary Derivation to Obtain a Formula which Accelerates the Convergence of an R^—n Sum Over Lattice Points X(d).
3.4.5 Extension of the Method to a Composite Lattice.
3.4.6 The Case of n = 1 (Coulombic Lattice Energy).
3.4.7 The Cases of n = 2 and n = 3.
3.4.8 Derivation of the Accelerated Convergence Formula Via the Patterson Function.
3.4.9 Evaluation of the Incomplete Gamma Function.
3.4.10 Summation over the Asymmetric Unit and Elimination of Intramolecular Energy Terms.
3.4.11 Reference Formulae for Particular Values of n.
3.4.12 Numerical Illustrations.
3.5 Extensions of the Ewald Methods for Coulomb Interactions in Crystals (T.A. Darden).
3.5.1 Introduction.
3.5.2 Lattice Sums of Point Charges.
3.5.3 Generalization to Gaussian- and Hermite-Based Continous Charge Distributions.
3.5.4 Computational Efficiency.
4.1 Thermal Diffuse Scattering of X-Rays and Neutrons (B.T.M. Willis).
4.1.1 Introduction.
4.1.2 Dynamics of Three-Dimensional Crystals.
4.1.3 Scattering of X-Rays by Thermal Vibrations.
4.1.4 Scattering Neutrons by Thermal Vibrations.
4.1.5 Phonon Dispersion Relations.
4.1.6 Measurement of Elastic Constants.
4.2 Disorder Diffuse Scattering of X-Rays and Neutrons (F. Frey, H. Boysen and H. Jagodzinski).
4.2.1 Introduction.
4.2.2 Basic Scattering Theory.
4.2.3 Qualitative Treatment of Structural Disorder.
4.2.4 General Guidelines for Analysing a Disorder Problem.
4.2.5 Quantitative Interpretation.
4.2.6 Disorder Diffuse Scattering from Aperiodic Crystals.
4.2.7 Computer Simulations and Modelling.
4.2.8 Experimental Techniques and Data Evaluation.
4.3 Diffuse Scattering in Electron Diffraction (J.M. Cowley and J.K. Gjønnes).
4.3.1 Introduction.
4.3.2 Inelastic Scattering.
4.3.3 Kinematical and Pseudo-Kinematical Scattering.
4.3.4 Dynamcial Scattering: Bragg Scattering Effects.
4.3.5 Multislice Calculations for Diffraction and Imaging.
4.3.6 Qualitative Interpretation of Diffuse Scattering of Electrons.
4.4 Scattering from Mesomorphic Structures (P.S. Pershan).
4.4.1 Introduction.
4.4.2 The Nematic Phase.
4.4.3 Smectic-A and Smectic-C Phases.
4.4.4 Phases with In-Plane Order.
4.4.5 Discotic Phases.
4.4.6 Other Phases.
4.5 Polymer Crystallography (R.P. Millane and D.L. Dorset).
4.5.1 Overview.
4.5.2 X-Ray Fibre Diffraction Analysis.
4.5.3 Electron Crystallography of Polymers.
4.6 Reciprocal-Space Images of Aperiodic Crystals (W. Steurer and T. Haibach).
4.6.1 Introduction.
4.6.2 The n-Dimensional Description of Aperiodic Crystals.
4.6.3 Reciprocal-Space Images.
4.6.4 Experimental Aspects of the Reciprocal-Space Analysis of Aperiodic Crystals.
5.1 Dynamical Theory of X-Ray Diffraction (A. Authier).
5.1.1 Introduction.
5.1.2 Fundamentals of Plane-Wave Dynamical Theory.
5.1.3 Solutions of Plane-Wave Dynamical Theory.
5.1.4 Standing Waves.
5.1.5 Anomalous Absorption.
5.1.6 Intensities of Plane Waves in Transmission Geometry.
5.1.7 Intensity of Plane Waves in Reflection Geometry.
5.1.8 Real Waves.
5.2 Dynamical Theory of Electron Diffraction (A.F. Moodie, J.M. Cowley and P. Goodman).
5.2.1 Introduction.
5.2.2 The Defining Equations.
5.2.3 Forward Scattering.
5.2.4 Evolution Operator.
5.2.5 Projection Approximation – Real-Space Solution.
5.2.6 Semi-Reciprocal Space.
5.2.7 Two-Beam Approximation.
5.2.8 Eigenvalue Approach.
5.2.9 Translational Invariance.
5.2.10 Bloch-Wave Formulations.
5.2.11 Dispersion Surfaces.
5.2.12 Multislice.
5.2.13 Born Series.
5.2.14 Approximations.
5.3 Dynamical Theory of Neutron Diffraction (M. Schlenker and J.-P. Guigay).
5.3.1 Introduction.
5.3.2 Comparison Between X-Rays and Neutrons with Spin Neglected.
5.3.3 Neutron Spin and Diffraction by Perfect Magnetic Crystals.
5.3.4 Extinction in Neutron Diffraction (Nonmagnetic Case).
5.3.5 Effect of External Fields on Neutron Scattering by Perfect Crystals.
5.3.6 Experimental Tests of the Dynamical Theory of Neutron Scattering.
5.3.7 Applications of the Dynamical Theory of Neutron Scattering.
Author Index.
Subject Index.
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On the groups of H(Π,n
Results 1 - 10 of 20
- TOPOLOGY , 2003
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory
and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
Cited by 78 (16 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and
the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that
stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules
over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can
be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on
derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.
- Illinois J. Math , 1989
"... Perturbation theory is a particularly useful way to obtain relatively small differential com-plexes representing a given chain homotopy type. An important part of the theory is “the basic
perturbation lemma ” [RB], [G1], [LS] which is stated in terms of modules M and N of the same homotopy type. It ..."
Cited by 66 (10 self)
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Perturbation theory is a particularly useful way to obtain relatively small differential com-plexes representing a given chain homotopy type. An important part of the theory is “the basic
perturbation lemma ” [RB], [G1], [LS] which is stated in terms of modules M and N of the same homotopy type. It has been known for some time that it would be useful to have a perturbation
- Ann. Math
"... Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded
(dg) commutative algebra whose n-simplices Ωn are the dg algebra of differential forms on the geometric n- ..."
Cited by 22 (0 self)
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Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg)
commutative algebra whose n-simplices Ωn are the dg algebra of differential forms on the geometric n-simplex ∆ n. In [20], Sullivan reformulated Quillen’s
- J. Symbolic Comp , 1991
"... The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use
of computer algebra in an area not usually associated with that subject. Comparison of the complexes prod ..."
Cited by 14 (5 self)
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The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use of
computer algebra in an area not usually associated with that subject. Comparison of the complexes produced by the method discussed here with those produced by other methods shows
- Math. Proc. Cambridge Philos. Soc. 114 , 1993
"... Abstract. In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical
group A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our ..."
Cited by 12 (0 self)
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Abstract. In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group
A a sequence of simplicial sets {K(A,n)}n≥0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−,n) is right adjoint to the
functor ℘n, where℘n(X•) is defined as the fundamental groupoid of the n-loop complex � n (X•). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy
types of spaces Y with πi(Y) = 0foralli = n, n + 1andn ≥ 3; and also we obtain a classification theorem for those spaces: [−,Y] ∼ = H n (−,℘n(Y)).
- in Adams Memorial Symposium on Algebraic Topology , 1992
"... Dedicated to the memory of Frank Adams ..."
- In ‘Simposio di Topologia (Messina, 1964)’, Edizioni Oderisi, Gubbio , 1965
"... The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F)
such that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular ..."
Cited by 7 (0 self)
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The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F) such
that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular chains of E over a ring Λ). We work in the context of (semi-simplicial) twisted cartesian
products (thus we assume as do the proofs of the theorem given in [Gug60, Shi62, Szc61] the results of [BGM59] on the relation between fibre spaces and twisted cartesian products). In fact we prove a
general result on filtered chain complexes; this result applies to give proofs not only of Brown’s theorem but also of a theorem of G. Hirsch, [Hir53]. Our proof is suggested by the formulae (1) of
[Shi62, Ch. II, §1]. Let (X,d), (Y,d) be chain complexes over a ring Λ. Let (Y,d) ∇ f − → (X,d) − → (Y,d) be chain maps and let Φ: X → X be a chain homotopy such that Let X,Y have filtrations (1.1) f
∇ = 1; (1.2) ∇f = 1 + dΦ + Φd; (1.3) fΦ = 0; (1.4) Φ ∇ = 0; (1.5) Φ 2 = 0; (1.6) ΦdΦ = −Φ. and let ∇,f,Φ all preserve these filtrations. 0 = F −1 X ⊆ F 0 X ⊆ · · · ⊆ F p X ⊆ F p+1 X ⊆ · · · (1) 0 = F
−1 Y ⊆ F 0 Y ⊆ · · · ⊆ F p Y ⊆ F p+1 Y ⊆ · · · (2) Example 1 Let B,F be (semi-simplicial) complexes, let (X,d) = C(B × F), the normalised chains of B × F, let (Y,d) = C(B) ⊗Λ C(F), and let ∇,f,Φ be
the natural maps of the Eilenberg-Zilber theorem as constructed explicitly in [EML53]. The relations (1.1)-(1.4) are proved in [EML53] while (1.5), (1.6) are easily proved (cf. [Shi62, p.114]). The
filtrations on X,Y are induced by the filtration of B by its skeletons. The fact that ∇,f,Φ preserve filtrations is a consequence of naturality of these maps (cf. [Moo56, Ch. 5, p.13]). We now wish
to compare C(B ×F) with C(B ×τ F) where B ×τ F coincides with B ×F as a complex except that ∂0 in B ×τ F is given by ∂0(b,x) = (∂0b,τ(b,x)), b ∈ Bp,x ∈ Fp. Then the filtered groups of C(B × F) and C
(B ×τ F) coincide but the latter has a differential d τ. If τ satisfies the normalisation condition τ(s0b ′,x) = ∂0x,
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory
and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
Cited by 6 (5 self)
Add to MetaCart
A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and
the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent `the same homotopy theory'. We show that
stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules
over a `ring spectrum with several objects', i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can
be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on
derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. 1.
- J. Pure Appl. Algebra , 1993
"... ..." | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=1913280","timestamp":"2014-04-18T01:49:43Z","content_type":null,"content_length":"35767","record_id":"<urn:uuid:636bbe38-5a47-481f-aa37-b4b266355d52>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00049-ip-10-147-4-33.ec2.internal.warc.gz"} |
Null Termination
Been a while, but I finally have things to post!
In my recent endeavors to become more... intimately... familiar with the C++ programming language, I have taken to attempting Project Euler!
These are some fairly thought-provoking mathematical challenges that can be solved using the power of COMPUTERS! And they are especially handy in learning or practicing your programming language of
As I was moving from challenge to challenge, most of them are straight-forward after a little thought. However, the first one that provided me with a challenge (and inspiration) was a little
something they call Challenge #16.
Challenge: What is the sum of the digits of the number 2^1000?
At first you think, man this is pretty easy! C++ has a nice little function named pow in math.h! This is what I thought as well, but lo and behold 2 to the thousandth power is a fairly large number!
In fact it's approximately:
Which is exactly what the output of pow( 2.0, 1000 ) will give you, unfortunately we need all the digits and, as my hour or two messing with the precision of pow told me, you just can't get it from
So I am brought to the topic of this post, my own personal implementation of BigInteger. The goal of my BigInteger is meant to store an int only limited by the capacity of your imagination! (and
computer memory) As this is something semi crude that I have done in a few hours, it is neither that pretty nor complete, but it by golly it works!
The first issue that I ran into was choosing the appropriate structure to store this amazingly long integer, and I settled on a string. I may go back and use vectors but I have a bad feeling it would
negatively impact performance as inserting at the front of a vector is extremely inefficient and I would rather concatenate strings.
So my basic concept is I want each character to represent a digit in the number, and perform normal operations appropriately. (+, *, ^)
The main thing you have to understand is that raising a number to a power is multiplying it times itself that many times. For example, 2 ^ 4 would just be 2 multiplied four times!
2^4 = 2 * 2 * 2 * 2
The same goes for multiplication simplified to addition
2*4 = 2 + 2 + 2 + 2
You might be saying that you already understand basic math, however these concepts are important as we design our operator functions.
x ^ y calls (x * x )(y times) and i * j calls (i + i) (j times)
So my functions look a little like ** Please note this is extremely pseudo-pseudocode **
// Return the sum of this number and the other number
operator+(BigInt other){ return this + other: }
// Add this number to result, however many times other says to.
operator*(BigInt other) {
int result = 0;
{ result += this; }
return result;
//Multiply the result by this, however many times other says to.
operator^(BigInt other) {
int result = 1;
{ result *= this; }
return result;
You can see how these methods cascade into each other. I chose to do it iteratively because that is what came to mind first and is easily readable.
Well that's I all can I get down for right now, I'll detail my approach to actually adding the two numbers in the next post. And in either that one or the next I will go over overloading operator<<
and how the friend keyword works so we can get some custom output for our class.
Hey everyone, my name is Chad Campbell and I am currently a 3rd year Computer Science student at Rochester Institute of Technology. Nice to meet you! :)
I'm creating this blog as an effort to both record my experiences and expand my portfolio. I will be documenting my school projects, my programming learnings and a new personal project I have taken
on to create a game from scratch using SDL inside the C++ language.
I will try to keep this updated as possible and provide you with some nice (or bad) code examples along my journey for both feedback and teaching material for inexperienced programmers. Feel free to
comment on any of my posts with any feedback you'd like to leave.
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Writing Guidelines
Project Reports
Mathematics Department
School of Science and Engineering
Seattle University
TABLE OF CONTENTS
I. WHAT IS A PROJECT REPORT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
II. WHY DO WE WRITE MATHEMATICAL REPORTS? . . . . . . . . . . . . . . . . . . . . 2
III. HOW DO WE STRUCTURE A MATHEMATICAL REPORT? . . . . . . . . . . . . . 3
IV. SUGGESTIONS FOR WRITING A GOOD PROJECT REPORT . . . . . . . . . . . 4
A. Suggestions for Each Part of the Project Report . . . . . . . . . . . . . . . . . . . . . 4
1. The Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. The Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. The Main Body or Mathematical Argument . . . . . . . . . . . . . . . . . . 5
4. The Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
B. General Writing Suggestions for Project Reports . . . . . . . . . . . . . . . . . . . . 6
1. Connections and Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. Important Aspects of Any Mathematical Project Report:
Correctness, Completeness, Spelling, Grammar, Punctuation . . . . 6
3. What is an ÒElegantÓ Mathematical Paper? . . . . . . . . . . . . . . . . . . . 7
4. Acknowledgments for Assistance with Project Reports . . . . . . . . . . 7
V. WORKING EFFECTIVELY IN A GROUP OR TEAM . . . . . . . . . . . . . . . . . . . . 8
AND GRAPHING CALCULATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
VII. EVALUATING PROJECT REPORTS IN MATHEMATICS . . . . . . . . . . . . . . . . 10
A. Overview of Evaluating Project Reports in Mathematics . . . . . . . . . . . . . . 10
B. Sample Grading Scale for Written Projects in Mathematics . . . . . . . . . . . 10
I. WHAT IS A PROJECT REPORT?
The goal of most writing is to communicate something to someone else. In the case of a project report in mathematics, the goal is to communicate some ideas in mathematics, such as a new concept, a
new technique, or the solution to a complex problem, to someone who is not familiar with those ideas. The main body or mathematical argument part of a project report must contain appropriate material
to support all conclusions drawn about concepts or to support all solutions to problems.
The project or laboratory instructions you are given by your professor may consist of several questions to answer or several steps to follow with your project partners, often while you are in one of
the computer labs. However, a project report is not a list of answers or solutions to the problems given. Even though your professor may have listed various parts or numbered questions in the project
instructions, this usually does not mean that you should have these parts or questions numbered in your report. Your job is to tie all of these individual parts into a single, cohesive whole. Instead
of focusing on the different parts, try to find the overall purpose that your professor had in mind when writing the project questions or instructions. Your written report should communicate this
overall purpose to a reader who has about the same level of mathematical experience that you do. You can imagine your reader as a student in another section of the same course, but a student who is
not familiar with the problem addressed in your project.
In some of the projects, the questions that you are asked to answer will be leading you through the solution of a single complex problem. In that case, your project report will state and motivate the
main problem and describe the solution in detail. In other projects, the questions that you are asked to answer may involve different examples intended to lead you toward an understanding of some
general concept or method. In that case, your project report will be a mathematical exposition (somewhat like a textbook exposition) describing the mathematical concept in general terms and
illustrating with specific examples. In either case, your report will read as a single document and will be primarily text, with equations and graphics incorporated into the text when appropriate.
To Explain to Others
For most of your life so far, the writing you have done in mathematics classes has been primarily as homework or tests, and you have been explaining your work to people who know more mathematics than
you do, usually to your teachers. At this point in your education, you know far more mathematics than the average person has ever learned; in fact, you know more mathematics than most college
graduates remember. With each additional mathematics course you take, you further distance yourself from the average person on the street. You may feel that the mathematics you do is simple and
obvious. (DoesnÕt everyone know what a function is?) However, other people who need to understand these concepts may find this same mathematics complex and confusing. It becomes increasingly
important, therefore, that you learn to explain what you are doing to others who might be interested: your parents, your supervisors at work, the media.
To Communicate with Others
Regardless of perception, the mathematical sciences and writing are not at all far-removed from one another. Professional mathematicians, scientists, and engineers spend most of their time writing:
communicating with colleagues, submitting proposals, applying for grants, publishing papers, writing memos. Writing well is extremely important to mathematicians and scientists, since poor writers
have difficulty writing papers which will be accepted for publication, difficulty communicating effectively with employers, and difficulty obtaining funding for proposals. Writing well is extremely
important to engineers, since poor writers have a difficult time convincing a potential client to hire them and difficulty explaining a product or design to employers.
To Clarify to Yourself
One of the simplest reasons for writing in a mathematics class is that writing helps you to learn mathematics better. By explaining a difficult concept to other people, you clarify your own
understanding of the ideas and applications.
The Title
An appropriate title demonstrates the purpose and content of the project report.
The Introduction
The introduction gives a basic restatement of the problem, explains the significance or importance of the problem, and ends with a statement of the solution, when appropriate. The introduction of the
project report contains what is often called the thesis statement in writing in the humanities and social sciences.
The Main Body or Mathematical Argument
The project report continues with a set-up of the problem and a statement of any basic assumptions. The report outlines the approach to be taken and discusses any limiting conditions that were
required to solve the problem. The supporting argument also outlines the reasoning behind the formulas and theorems used and includes explanation of the calculations performed. This main body of the
report incorporates supporting graphs, tables, and diagrams in appropriate locations within the text.
The Conclusion
The conclusion of the project report critically and carefully analyzes the mathematical solution or solutions to determine appropriateness to the real world setting in which the problem was posed.
The conclusion should eliminate any inappropriate solutions with a brief justification of why the solutions were discarded. The conclusion then brings the appropriate solutions back into the real
world setting in which they were posed and interprets their significance in that setting.
This portion of the writing guidelines offers more specific suggestions for preparing your mathematics project report. Part A gives some ideas for individual sections of the report.
Part B gives general writing suggestions for mathematical writing and notes some crucial final steps to take before you or your group or team submits the final project report.
While you are preparing any project report, the textbook for your mathematics course will serve as a good example of appropriate mathematical writing. Study some of the examples in your text to see
good use of explanation, connections, transitions, centering of formulas, and other writing and formatting techniques.
1. The Title
The title of your project report should be appropriate in tone and structure to a formal mathematical presentation, such as a section of your textbook. It should have some descriptive information
about the problem to be solved and the general concepts you applied in solving the problem. The title should not be too wordy, but should give the reader enough information to anticipate the type of
problem and the approach you used in solving the problem.
2. The Introduction
In the introduction of your project report, you will first explain what the problem is, and you will often try to convince the reader that the question or problem is an interesting or worthwhile
problem to solve. The introduction will often include the result, answer, or solution to the problem, even though the solution process itself will not be explained until the main body or mathematical
argument section of your report. For some projects, you may choose to leave the reader in suspense and not give the solution until later in the report.
Clearly restate the problem to be solved. Do not assume that the reader knows the background of the problem or question. You do not need to restate every detail, but you should explain enough so that
someone who has never seen the assignment can read your paper and understand your work, without any further explanation from you. Outline the problem carefully.
Give some motivation to the problem. Try to answer the following questions: Why would a person outside your mathematics class be interested in knowing the solution to this question? Where does this
issue arise in the real world? In other words, give the reader some context for the project.
When possible, state the result in a complete sentence which stands on it own. In most project reports, the answer or solution will be in the introduction. If you can avoid variables in your
statement of the solution, do so; otherwise, remind the reader what the variables represent. If your solution does not appear until the end of the paper and you have made any significant assumptions
in the solution, restate them at that time. Do not assume that the reader has actually read every word and remembered all of the details. Also, if you do see the Òbigger pictureÓ or think that the
project has led to your understanding of a major concept (the ÒAh-Ha!Ó experience), the introduction would be a good place to discuss that Òbigger pictureÓ for the readerÕs benefit.
3. The Main Body or Mathematical Argument
In a mathematical argument, we want to persuade the reader that our solution to the problem or analysis of the mathematical property is correct. We often use calculations, statements of previous
results, and accompanying visuals (such as graphs, diagrams, and tables) to help us accomplish this.
Provide a paragraph which explains how the problem will be approached. Carefully outline the steps you are going to take, giving some explanation of why you are taking that approach. It is nice to
refer to this paragraph later when you are well into your calculations, to help the reader follow the sequence of steps you are taking.
Clearly state the assumptions which underlie the formulas that you are using. For example, what physical assumptions do you need to make? (No friction, no air resistance? That something is lying on
its side, or far away from everything else?) Sometimes things are so straightforward that there are no assumptions, but not often.
Clearly label diagrams, tables, graphs, or other visual representations of the mathematics, if these are used. In mathematics, even more than in literature, a picture can be worth a thousand words,
especially if it is well-labeled. (Please see section VI on appropriate use of software in creating visuals.) There are certain techniques you should use when adding visual components to mathematical
writing. For example, label all axes in graphs, usually with words. Give a diagram or graph a title describing what it represents. It should be clear from the picture what any variables in the
diagram represent. The idea is to make everything as clear and self-explanatory as possible. Incorporate your graphs, tables, or pictures into the body of your paper. Remember to introduce the visual
aid within the writing and to provide some context as to why it is important in the discussion.
Define all variables used. The more specific you are, the better. State the units of measurement. Clarify words such as ÒpositionÓ (Do you mean height above the ground?) and ÒtimeÓ (Do you mean time
since the experiment started?).
Explain how each formula is derived or give a reference in which the formula can be found. Some formulas, such as the formula for the area of a circular region, are very well-known and just need to
be identified. Others should be derived as part of your mathematical argument, or a source provided. Long formulas are often easier to read if they are centered on a separate line of your text. (See
your textbook for good examples of formatting text involving long formulas.)
4. The Conclusion
This is the part of your project report in which you will interpret and explain the mathematical solution to the problem or question presented in the original assignment.
Watch for solutions that do not make sense in the Òreal world.Ó When you eliminate solutions which would produce negative time or negative length, for example, explain your reasoning.
Watch for mathematical solutions that need interpretation to make sense. Negative current, for example, would need explanation based on the set-up of the original problem. In your interpretation,
always explain any changes you are making to the calculated solutions.
1. Connections and Transitions
Throughout the project report, it is important to develop a clear exposition that enables the reader to progress as smoothly as possible from the introduction through the mathematical argument to the
conclusion. In mathematical writing, it is particularly helpful to clarify connections and transitions with words and phrases such as the following:
á Therefore (also: so, hence, accordingly, thus, it follows that, we see that, from this we get, then)
á I am assuming that (also: assuming, where, M stands for; in more formal mathematics: let, given, M represents)
á Show (also: demonstrate, prove, explain why, find)
á This formula can be found on page ____ of ________.
á If (also: whenever, provided that, when)
á Notice that (also: note that, notice, recall)
á Since (also: because)
2. Important Aspects of Writing Any Mathematical Project Report
Throughout the work on the project and especially at the stage of the final project report to be submitted, each member of the group or team is always responsible for carefully checking the following
major aspects of the paper.
á Be certain that the mathematics is correct.
á Be certain that you solved the original problem given or completely answered the fundamental question in the assignment.
á Be certain that the spelling, grammar, and punctuation are correct.
It may surprise you that many students lose points on projects as a result of spelling, grammar, and punctuation errors. These detract considerably from the quality of the project. Be sure to
spell-check and proofread your work. In addition, ask a friend to read the project report for content as well as error detection.
Mathematical formulas are clauses or sentences and need proper punctuation. Put a period at the end of a computation if the computation ends the sentence. Use a comma if the formula or computation is
a clause or other part of a sentence. (You will find many examples of such uses within your textbook. Try to follow the format demonstrated in the text.) Do not confuse mathematical symbols with
English words. The symbols = and # are common examples of such misuse. The symbol Ò=Ó is used only in a mathematical formula or equation. Otherwise, the word equal is written out.
3. Finally, always keep in mind when writing any mathematical paper that ÒelegantÓ mathematics papers are the ones that are the easiest to read: clear explanations, uncluttered expositions on the
page, well-organized presentation.
4. Give Acknowledgment to Those Who Have Assisted With Your Project Report
It is extremely important to acknowledge the help provided by others, as well as citing appropriate references. In your project report, you should acknowledge any of the following resources: any book
you used, any computational or graphical software which helped you understand or solve the problem, any student with whom you spoke (whether in your class or not), any assistance provided by the
Writing Center, any proofreaders, and any professor with whom you spoke. Be as specific as possible.
For example, the Mathematics Department faculty thank the following people for their inspiration and assistance in our preparation of these writing guidelines for mathematics projects:
Dr. Annalisa Crannell of Franklin & Marshall College, who graciously allowed us the use of her ÒGuide to Writing in Mathematics ClassesÓ as the basis for our own writing guidelines development.
Dr. John Bean, Professor of English and Consulting Professor of Academic Writing at Seattle University, for his generous time commitment and thoughtful work with the Mathematics Department in
developing both these guidelines and the ÒGrading Scale for Written Projects in Mathematics.Ó
The Seattle University Science and Engineering Project Center and the Writing Center for use of their Writing Guidelines, particularly the suggestions for writing as a team.
Most of your mathematics projects will be carried out as a group or team. Everyone in the group should be actively engaged in work on the project, both during computer laboratory hours and outside
the lab. The ability to work successfully in a group is a crucial skill for people in almost every career. The projects in mathematics provide an opportunity to develop your teamwork skills, while
project report writing should increase your understanding of the concepts studied and your ability to apply these concepts to problem solving.
The Science and Engineering Project Center and the Writing Center, in their Writing Guidelines for senior design teams, offer the following suggestions for working as a project team. (These have been
adapted slightly to the writing of project reports in mathematics.)
1. Do as a team what teams do best. Do as individuals what individuals do best.
Teams are good at the following:
á Brainstorming for ideas
á Achieving creative solutions to problems
á Planning and organizing activities and setting goals
á Developing an organizational plan for a document and determining what ideas go into which parts
á Serving as readers who give feedback on drafts
á Seeing how individual pieces fit into the whole and making recommendations for improvements of individual pieces
Teams are not good at the following:
á Writing drafts of individual sections
á Editing for uniform voice and style
2. Recognize that the pieces of any document make sense only in relation to the whole. Never draft individual pieces until all team members understand the whole.
3. Talk about your document before you write it. As a team, talk your way through each
section of the document. Working together, take notes and plan each section of the document. Articulate its purpose and its content. Each member of the team should feel qualified to write any section
of the document. The Writing Center can facilitate this crucial organizational step for project reports.
4. If possible, all team members should use the same word-processing program and follow the same plan for font sizes, heading styles, and formatting. This will make it easier to combine sections of
the report.
5. Assign individual team members the responsibility of drafting sections of the document.
Copies of the drafts should be made for all group members. At a team meeting, members should review drafts for accuracy, completeness, clarity, development, and style. Individual writers should then
revise their sections. One team member who writes well should do final editing of the document so that it seems written in one voice.
6. Write your project report as you proceed with your work, rather than putting off writing
until the very end. The act of trying to write a section of the document can clarify your thinking and suggest additional paths to follow. The more you write as you go along, the easier it will be at
the end to produce an excellent project report.
Every mathematics course makes use of graphing calculators, mathematical software, or both. These can be particularly helpful in working on mathematics projects. In many courses you will have regular
computer laboratory days, using mathematical software such as Mathematica, Joy of Mathematica, MATLAB, or Minitab. You are expected to develop some level of proficiency with the provided software,
although you are not expected to be an expert in software use. However, you should develop the competency needed to use the software or your graphing calculator to help solve problems. In some
classes, you will be expected to learn to use software to present your papers in a neat, professional manner. In Mathematica, for example, you can create entire project reports, if you wish.
Alternatively, you can cut and paste into a word processing document, created with software such as Microsoft Word. Sometimes, you may want to sketch a picture to illustrate your writing in a way
that the laboratory software does not do, or does not do well. In that case, a program such as Mac Draw or Paint can be useful.
It is important to remember that projects are not simply an exercise in software use. A project report should never be just a sequence of graphs, tables, or other computer output, without appropriate
written discussion of the concepts involved.
Include visuals such as figures, graphs, charts, or tables only if you or your team can explain why a reader needs them. Visuals for their own sake confuse readers rather than help them. The document
should be able to stand on its own with all visuals removed, because essential information displayed on the visuals should also be discussed verbally in the document itself.
Please see earlier sections of these Writing Guidelines for suggestions on including graphs, tables, diagrams, and charts in your project reports. Your course textbook is also a good source of
examples of visuals included appropriately within text.
A. A Brief Overview of Evaluating Project Reports in Mathematics Courses
In the Mathematics Department at Seattle University, most courses from freshman Core classes to senior-level courses for mathematics majors have labs or projects, often carried out in groups or
teams, with well-written project reports required. These project reports are evaluated in a variety of ways, based on the level of the course and the nature of the particular project. Each course
instructor will give students information about the evaluation of projects in that class, as well as specific expectations for the project reports. However, grading of any project report will involve
the structure of the argument, the quality of the argument, and the clarity and professional appearance of the presentation.
Goals for students in all mathematics courses include development of the ability to communicate mathematical concepts, both orally and in writing. Group work on projects and the writing of project
reports are crucial parts of this development process. Therefore, the evaluation of projects is based on communication of ideas, as well as correctness of the mathematics.
B. Grading Scale for Written Projects in Mathematics
Dr. John Bean, Professor of English and Consulting Professor of Academic Writing at Seattle University, worked with the entire Mathematics Department faculty to develop the Grading Scale for Written
Projects in Mathematics which appears on the following page. This grading scale describes in the left-hand column the attributes mathematics faculty members consider appropriate for an excellent
project report. As you read from left to right in each row, reasons are given for higher to lower scores in each aspect of the report.
While not every project will be designed to fit this grading model, studying the grading scale will help you apply the writing guidelines in this booklet and will help you understand other grading
models provided by your instructor for specific projects in a particular mathematics class. | {"url":"http://www.mscs.mu.edu/~jones/math_project_writing_guidelines.htm","timestamp":"2014-04-19T15:05:56Z","content_type":null,"content_length":"77992","record_id":"<urn:uuid:d96a044b-51c8-46d0-bdc9-cbb6311577fe>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00527-ip-10-147-4-33.ec2.internal.warc.gz"} |
getting the proper normal [Archive] - OpenGL Discussion and Help Forums
Hi, I have the following problem:
For further calculations I need the normals of some planes to point in the correct direction, namely they are supposed to point to the viewer and not away from him.
So is there a way to get the viewers point as a vector that is also specified in the same coordinate system like the plane vectors, so I can compare the vectors? | {"url":"http://www.opengl.org/discussion_boards/archive/index.php/t-135052.html","timestamp":"2014-04-19T04:51:24Z","content_type":null,"content_length":"9322","record_id":"<urn:uuid:830097c0-a325-47b4-bb84-f77a78569235>","cc-path":"CC-MAIN-2014-15/segments/1398223203235.2/warc/CC-MAIN-20140423032003-00369-ip-10-147-4-33.ec2.internal.warc.gz"} |
Michael Artin’s Algebra is perhaps not yet considered a classic, but for many years it has been a serious contender for textbook of choice in the standard undergraduate abstract algebra course. So I
was very excited to have the opportunity to read it critically for the MAA Reviews. After a few sessions of careful reading I can now happily say that my enthusiasm was completely justified.
This was certainly a good read. The book is well-written and the author’s style adds an appealing personal touch, though I still wish the non-English epigraphs were accompanied by their translations
— at least the verbal ones. In terms of content, the book covers all the classical and standard material, but also introduces students to all the fun stuff that I have to pull out of my hat when I am
using a more traditional text. Case in point: Symmetries of plane figures and crystallographic groups. This is such a beautiful topic, and perfectly appropriate for this level, but Herstein and
Dummit & Foote do not touch it with a long pole. And of course Lang or Hungerford could never be expected to stoop so low…
Another case in point: Group representations. Artin justifies the inclusion of this topic in an undergraduate introductory text by saying “If chemists can do it, why can’t we?” I couldn’t agree more!
I believe that it is our duty to introduce our students to what group theory means to their physicist and chemist friends, and the road to that goes through a path into representation theory. And for
those who could not care less about representation theory, there is a chapter on quadratic number fields and one on Galois theory for alternative capstone experiences at the end of the term.
The book contains a lot more than what a typical one-semester algebra course could cover, and this is intentional: the author expects the instructor to pick and choose. This flexibility allows for
many different kinds of courses to be taught from the same text. For instance, if your students take abstract algebra immediately after calculus and a mainly computational introduction to linear
algebra, Artin’s book can be used to introduce abstract algebraic notions via the development of more advanced ideas in linear algebra, thus solidifying the linear algebra background of your students
as they learn basic algebra. If on the other hand your students took a proof-based linear algebra course already (like the typical student at my own institution who comes into the abstract algebra
classroom), then you can go very quickly through the linear algebra material that is covered in detail early on, and have the chance to touch upon more of the fun stuff later. It is highly likely
that some of the linear algebra material will be new to any student, as Artin touches upon the Jordan canonical form, the spectral theorem, bilinear forms, matrix exponentials, and other such
advanced topics. Even if these sound familiar to your students, a quick recap can never hurt.
Teaching a course following this book could be a joy. My minor annoyance for the term used for group actions (Artin calls them group operations) is quite easily overshadowed by my overwhelming
support for the early and often use of linear groups as examples. Unlike more traditional texts Artin does not solely depend on permutation groups as the major examples of groups; he develops a
substantial amount of theory for linear groups as well (including a brief introduction to Lie algebras). There is some amount of topology, geometry and analysis that is referred to in various
instances, which is not typical, but I think such references could help our students notice earlier than they usually do that in fact these branches of mathematics are not really unrelated, and that
they actually benefit from connections and interactions. Looking carefully, it is clear that the amount of topology, analysis and geometry required is not substantial and allows Artin to cover much
more significant mathematics. Therefore I would be more than willing to give him a break for writing a book which is technically not completely self-contained.
To many, Artin’s book may have seemed like an interesting experiment in the first edition. In this new edition the book has matured, and is, I believe, ready to compete against anybody’s personal
favorite. Take a look!
Gizem Karaali is assistant professor of mathematics at Pomona College and an editor of the Journal of Humanistic Mathematics. As a representation theorist by training, she is obviously quite partial
to introducing the subject in the undergraduate curriculum. | {"url":"http://www.maa.org/publications/maa-reviews/algebra-3","timestamp":"2014-04-21T15:55:06Z","content_type":null,"content_length":"99136","record_id":"<urn:uuid:dbf0ff70-72f0-4257-9f59-d85f674bf48e>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00558-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reply to comment
Submitted by Anonymous on September 14, 2010.
I don't believe it's coincidental that the Lie Group E8. image, with 30 points on each of its circular perimiters, is congruent with the Croft Spiral Sieve. Thirty (30)--the product of the first
three primes--is integral not only to the Lie Group, but is a fundamental building block of the prime number sequence: http://www.primesdemystified.com. It's like looking into the eye of creation | {"url":"http://plus.maths.org/content/comment/reply/2597/1459","timestamp":"2014-04-19T12:09:42Z","content_type":null,"content_length":"20390","record_id":"<urn:uuid:60aebfdf-de1a-49fd-98d4-fa69b4f2cd21>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00441-ip-10-147-4-33.ec2.internal.warc.gz"} |
PIMS/SFU/UBC Number Theory seminar
The UBC Number Theory seminar typically meets on Thursdays from 3:30–4:30 PM, in room MATH 126 (Mathematics building).
The seminar schedule is kept up to date. The default is to show all future number theory seminars, but you can see past seminars and so on by making different choices from the drop-down menus
(the ones that initially read "All Future" and "Number Theory Seminar" when you click on the link).
The UBC Number Theory seminar typically meets on Thursdays from 3:30–4:30 PM, in room MATH 126 (Mathematics building).
The seminar schedule is kept up to date. The default is to show all future number theory seminars, but you can see past seminars and so on by making different choices from the drop-down menus (the
ones that initially read "All Future" and "Number Theory Seminar" when you click on the link). | {"url":"http://www.math.ubc.ca/Research/NumberTheory/index.shtml?seminar","timestamp":"2014-04-19T09:26:55Z","content_type":null,"content_length":"11655","record_id":"<urn:uuid:6c3e9059-464c-45d4-b870-b976cd360d09>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00492-ip-10-147-4-33.ec2.internal.warc.gz"} |
petsc-3.4.4 2014-03-13
Type of generalized additive Schwarz method to use (differs from ASM in allowing multiple processors per subdomain).
typedef enum {PC_GASM_BASIC = 3,PC_GASM_RESTRICT = 1,PC_GASM_INTERPOLATE = 2,PC_GASM_NONE = 0} PCGASMType;
Each subdomain has nested inner and outer parts. The inner subdomains are assumed to form a non-overlapping covering of the computational domain, while the outer subdomains contain the inner
subdomains and overlap with each other. This preconditioner will compute a subdomain correction over each *outer* subdomain from a residual computed there, but its different variants will differ in
(a) how the outer subdomain residual is computed, and (b) how the outer subdomain correction is computed.
PC_GASM_BASIC - Symmetric version where the full from the outer subdomain is used, and the resulting correction is applied
over the outer subdomains. As a result, points in the overlap will receive the sum of the corrections
from neighboring subdomains.
Classical standard additive Schwarz.
PC_GASM_RESTRICT - Residual from the outer subdomain is used but the correction is restricted to the inner subdomain only
(i.e., zeroed out over the overlap portion of the outer subdomain before being applied). As a result,
each point will receive a correction only from the unique inner subdomain containing it (nonoverlapping covering
PC_GASM_INTERPOLATE - Residual is zeroed out over the overlap portion of the outer subdomain, but the resulting correction is
applied over the outer subdomain. As a result, points in the overlap will receive the sum of the corrections
from neighboring subdomains.
PC_GASM_NONE - Residuals and corrections are zeroed out outside the local subdomains.
Not very good.
See Also
Index of all PC routines
Table of Contents for all manual pages
Index of all manual pages | {"url":"http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/PC/PCGASMType.html","timestamp":"2014-04-17T11:12:56Z","content_type":null,"content_length":"3451","record_id":"<urn:uuid:2f6bf50e-6b86-4261-b7b3-9a0ba84f5f97>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00532-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kids.Net.Au - Encyclopedia > Strike price
In the context of
strike price
of an option is a key variable in a financial contract between two parties. Typically an option has positive monetary value if an underlying
financial instrument
(e.g. a
interest rate
inflation rate
) has a value above (or below depending on the particular type of contract, but not both) the strike price.
In the context of a call option, the payoff is <math>(S-K)^{+}</math> where S is the final of the underlying, K is the strike and where
<math>()^+(x)=\{^{x\ if\ x\geq 0}_{0\ otherwise}</math>
For a put option the corresponding payoff is <math>(K-S)^{+}</math>
For a digital option[?] <math>1_{S\geq K}</math> where <math>1_{\{\}}</math> is the indicator function.
All Wikipedia text is available under the terms of the GNU Free Documentation License | {"url":"http://encyclopedia.kids.net.au/page/st/Strike_price?title=Digital_option","timestamp":"2014-04-20T03:21:49Z","content_type":null,"content_length":"13058","record_id":"<urn:uuid:7cb8d7f0-f754-439b-80ab-2969d2591e7c>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00384-ip-10-147-4-33.ec2.internal.warc.gz"} |
linear differential equations??
April 19th 2008, 10:10 PM
linear differential equations??
I have a few questions about finding the general solution of first-order linear differential equations..
please help, I have no clue!! =/
April 19th 2008, 10:14 PM
I have a few questions about finding the general solution of first-order linear differential equations..
please help, I have no clue!! =/
the integrating factor method takes care of all these. see post #21 here
April 19th 2008, 10:20 PM
it doesn't make sense though..
is there a simpler way of explaining it?
I don't understand what "the integrating factor" is in the first place..
April 19th 2008, 10:33 PM
it is just something you multiply through by to make the left hand side the derivative given by the product and the integrating factor. for the first one:
$y' + 2y = 3e^t$
(with practice, you will be able to recognize the integrating factor immediately, but let's go through the method to see it)
the integrating factor is $e^{\int 2~dt} = e^{2t}$
multiply through by the integrating factor, we get:
$e^{2t}y' + 2e^{2t}y = 3e^{3t}$
now the left hand side is the result of differentiating $e^{2t}y$ by the product rule, thus
$(e^{2t}y)' = 3e^{3t}$
integrate both sides, we get:
$e^{2t}y = e^{3t} + C$
$\Rightarrow y = e^t + Ce^{-2t}$
the others are done similarly
April 20th 2008, 08:51 AM
the integrating factor method takes care of all these. see post #21 here
You always give that example, I think you may put a link in your signature with that. :D
April 20th 2008, 09:44 AM
Haha, yeah. It's kinda weird though. I do not want something in my signature as particular as a kind of differential equations problem. I want general things in my signature, like yours! | {"url":"http://mathhelpforum.com/calculus/35177-linear-differential-equations-print.html","timestamp":"2014-04-17T07:41:09Z","content_type":null,"content_length":"10205","record_id":"<urn:uuid:f0b5490f-d4d1-4643-9944-d906f0c5dd00>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00010-ip-10-147-4-33.ec2.internal.warc.gz"} |
MATH 530: Mathematical Models I (3)
An introduction to mathematical models useful in a large variety of scientific and technical endeavors. Topics include: model construction, Markov chain models, models for linear optimization, graphs
as models, and game theory. Prerequisite: MATH 223 and MATH 290, or MATH 143. LEC
View current sections... | {"url":"http://www2.ku.edu/~distinction/cgi-bin/index.php?id=2006&dt=courses_201011_under&classesSearchText=MATH+530","timestamp":"2014-04-20T10:56:31Z","content_type":null,"content_length":"987","record_id":"<urn:uuid:bceeeee6-7664-414f-a0fe-06806e81b86c>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00294-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Euler Zigzag Numbers
An alternating permutation is one in which the difference between each successive pair of adjacent elements changes sign—this is, each "rise" is followed by a "fall", and vice versa. For example,
the permutation {1324} is an alternating permutation.
The number of alternating permutations on elements is sometimes called the Euler zigzag number.
Flipping the image upside-down with the "flip" control toggles between the alternating permutations that begin with a rise and those that begin with a fall. | {"url":"http://demonstrations.wolfram.com/EulerZigzagNumbers/","timestamp":"2014-04-21T07:09:37Z","content_type":null,"content_length":"41276","record_id":"<urn:uuid:ae6f251c-e488-44cb-a2f7-472885c09466>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00293-ip-10-147-4-33.ec2.internal.warc.gz"} |
Buckeye, AZ Algebra 1 Tutor
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Automorphism Group of Paley Graph
up vote 2 down vote favorite
Hello all,
I would like an explanation as to the structure description of the automorphism group of a Paley graph.
Paley graphs are a specific case of Cayley graphs where the group is $Z_q$ (q is a prime power for some prime p = 1 mod 4) and the connection set is all the quadratic residues in GF(q).
I'll be satisfied even with the less general case where q is prime.
I'm pretty sure that the said group is a semi-direct product of CyclicGroup(q) and CyclicGroup(q-1/2) but I have trouble showing it in the general case...
Also posted on: http://math.stackexchange.com/questions/53668/automorphism-group-of-paley-graph
co.combinatorics graph-theory gr.group-theory
3 Crossposted to math.SE: math.stackexchange.com/questions/53668 Shaywei, you should know that it is not polite to post your question in multiple places simultaneously. If someone put a lot of work
into giving you a good answer here, only to hear from you that you already got the answer elsewhere, that would be quite frustrating. Please delete one of the two copies of your question, and
re-ask it only if you are unsuccessful in getting an answer for some time. – Zev Chonoles Jul 25 '11 at 15:36
OK. Thanks for comment. – Shaywei Jul 25 '11 at 15:43
1 I don't see how to delete one of the posts. Added relevant P.S instead. I will monitor both threads closely and as soon as I get an answer I will update both posts. – Shaywei Jul 25 '11 at 15:48
That sounds fair, I suppose there is no need to delete then. But for future reference, the "delete" button should be right below the tags of your question, next to an "edit" and "close" button. –
Zev Chonoles Jul 25 '11 at 16:04
As far as I can tell, the 'delete' button is not for the author of the thread. In fact, it is DISABLED for the original author. The purpose of the delete button is for other viewers to vote to
delete said post. That is, in this case, you are the one who should use the delete button to vote that this post will be deleted. Perhaps I am wrong though. – Shaywei Jul 25 '11 at 16:41
show 2 more comments
4 Answers
active oldest votes
A simple, "out-of-nothing", one-page proof for the case where $q$ is a prime by Peter Muller can be found at http://arxiv.org/PS_cache/math/pdf/0310/0310200v1.pdf. (It suffices to look
up vote 4 at Muller's Proposition 1, and indeed, it gives even more than one needs for isomorphisms of the Paley graph.)
down vote
add comment
It is a natural first guess to think that if $F$ is a finite field and $S$ is a subgroup of the multiplicative group $F^{\times}$ containing $-1$ which generates $F$ additively, then the
automorphism group of the Cayley graph (of the additive group $F^{+}$, using $S$ as the generating set) is the semidirect product of the additive group $F^{+}$ and the multiplicative group
$S$. But that group needs, in turn, to have its semidirect product taken with the group of automorphisms of $F$ (as an extension of its prime subfield). But even this modified conjecture
should have (verification, please?) at least two counterexamples:
up vote (i) $|F| = 2048$ and $|S| = 23$, in which case the automorphism group should be $2^{11}: M_{23}$
3 down (ii) $|F| = 243$ and $|S| = 22$, in which case the automorphism group should be $3^{5}: (M_{11}\times 2)$.
But since these apparent counterexamples involve sporadic groups, this version of the conjecture is probably largely on the right track. (This should not come into play in the Paley graph
question, but it should serve as a warning that sometimes the problem of determining the automorphism group for such a family of graphs can exhibit unexpected irregularities.)
add comment
It is easy to verify from the definition for each non-zero square $a$ in $GF(q)$ and each $b$ in $GF(q)$, the each map $$ t_{a,b}: x \mapsto ax+b $$ is an automorphism of the Paley graph.
Suppose $q=p^d$ where $p$ is prime. Then the Frobenius map $x\mapsto x^p$ is an automorphism of $GF(q)$ with order $d$, and this is also an automorphism of the Paley graph. Combining all
this we get a group of automorphisms of order $dq(q-1)$.
up vote 3 Proving that this is the entire automorphism group is difficult. One of the first proofs is in Carlitz, L. "A theorem on permutations in a finite field". Proc. Amer. Math. Soc. 11 (1960)
down vote 456–459. Even the case when $q$ itself is prime is non-trivial. The obvious approach is to use the theorem that a transitive group of prime degree which is not 2-transitive is solvable, and
then apply a theorem due to Galois that a solvable group of prime degree consists of translations, as above.
1 Shouldn't "where $q$ is prime" be "where $p$ is prime"? – DavidLHarden Jul 25 '11 at 17:10
... and shouldn't "$a$ in $GF(q)\setminus\{0\}$" be "$a$ is a quadratic residue in $GF(q)"? – Seva Jul 25 '11 at 17:59
Thanks. I've made both corrections. – Chris Godsil Jul 25 '11 at 21:33
add comment
This `answer' is a contribution to the question raised in DavidLHarden's answer: There are many more examples where the automorphism group is larger than the expected group, besides the two
nice Mathieu cases and the trivial case where $S=F^\times$.
For this let $p$ be an odd prime, $F$ be the field with $p^2$ elements, $\mathbb F_p$ be the subfield of order $p$, and $S$ be the subgroup of $F^\times$ of order $2(p-1)$. So distinct
elements $u,v\in F$ are joined if and only if $(u-v)^2\in\mathbb F_p$. Let $\omega\in F$ be of order $2(p-1)$. Identify the element $a+b\omega\in F$ with $(a,b)\in\mathbb F_p\times \mathbb
up vote F_p$. Then distinct pairs $(a,b)$, $(c,d)$ are joined if and only if $a=c$ or $b=d$. But any permutation on the first and another permutation on the second component preserves the graph
1 down structure, and so does the involution which switches the two components. Thus the automorphism group is at least as big as $(S_p\times S_p)\rtimes C_2$, where $S_p$ is the symmetric group on
vote $p$ letters. (One easily shows that this is the full automorphism group.)
Of course one can generalize this example to get even more examples. One also gets examples where the automorphism group has more interesting composition factors, like $\text{PSL}_2(q)$ for
certain prime powers $q$. Thus, I find the question interesting, and I'm not aware of results in the literature.
add comment
Not the answer you're looking for? Browse other questions tagged co.combinatorics graph-theory gr.group-theory or ask your own question. | {"url":"http://mathoverflow.net/questions/71236/automorphism-group-of-paley-graph/71255","timestamp":"2014-04-16T20:13:10Z","content_type":null,"content_length":"73453","record_id":"<urn:uuid:4929b12e-9e30-424b-8977-dc4034eccd9b>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00081-ip-10-147-4-33.ec2.internal.warc.gz"} |
Challenge Me: Math Workout
Challenge Me: Math Workout
Platform: Nintendo DS , DSI
Category: Puzzle
Developer: Oxygen Interactive
Publisher: Oxygen Games
Math and videogames, these are two things that I never thought I would really see melded into one. Sure, there have been math puzzles included in a game full of different types of puzzles, but never
have I seen math headline a game’s content. Well that day has come as Oxygen Games has recently released Challenge Me: Math Workout. Having already reviewed Challenge Me: Brain Puzzles I was somewhat
intrigued to see what a game full of math problems offered.
When I fired up Challenge Me: Math Workout I had a bit of déjà vu, as the visual presentation of the game was on par with Challenge Me: Brain Puzzles. The text and numbers are clean and clear looking
which helps make reading everything on screen somewhat easy. I am glad that everything is not too small as having to look at a game based on numbers could be quite a headache if things were too tiny.
The look and feel is very simple, and there is not a whole lot of pizzazz going on to make this game anymore exciting looking. I can say that it is slightly, and I do mean slightly, more colourful
looking then the Brain Puzzles game I just reviewed. As with my other Challenge Me review, everything in the visual department is serviceable for the content that is offered and it manages to get the
job done, it is just that it is not particularly exciting. That being said; how does one really jazz up math?
As with the visuals, again I get that feeling of déjà vu. There seems to have been minimal effort here too as the music is bland and can get grating while the sound effects are kept to a minimum. All
in all there is not a lot going on here. Sure, it sounds like I am being tough here, but I still can’t get figure out how one would ‘spice up’ the sound of a math based title to make it exciting.
Challenge Me: Math Workout is part of a series of games that are meant to test your brains ability in one form or another. So it is no surprise that the game follows the same line of content as its
cousin that was released before it, Brain Puzzles. Challenge Me: Math Workout’s game modes are separated into two types: Formulate and Hidden Logic. Both of these game types are card based games
that, of course, rely on one’s ability to use numbers and math to solve each problem. You can choose to play against up to three computer AI opponents or three real people using the Wi-Fi feature of
the DS.
Formulate is a game that pits you against the computer in a race to solve equations. You are given four face up cards with number values on them and you are required string together a working formula
from the cards that you are given. It sounds simple, but to add further challenge to this is the fact that each card also has a math symbol (e.g. division, multiplication, addition or subtraction) on
it that precedes the number. Your goal is to shuffle the cards around in order to get the correct equation. Although this may sound easy, once you experience it you will find that it is quite a
challenge indeed, especially since you are given a limited time period to do this.
Hidden Logic is a game where you play against the computer or other players. It is a game of deduction in such that you take turns guessing the numbers on the cards that your opponent(s) hold. There
are two sets of cards in front of each player, and they are color coded black or white. So in theory you have two sets of numbers that you must figure out. Once players have taken a turn they then
draw another card from the stack in front of them. There are some other rules to this game, such as the numbers only being revealed in ascending order, and no numbers repeating. In many ways this
game is like a card based version of battleship, as you race to guess the right numbers on your
opponent’s cards.
The biggest problem for me was that it was not that fun to play against the computer AI. There is no doubt that I am not the most proficient when it comes to math, but to continually get beaten by
the AI, no matter what I did, was frustrating to say the least. When playing Hidden Logic there was nary a time when I beat the AI, and it seemed like it had an uncanny ability to figure out other
players cards on a far too consistent basis. When taking a stab at Formulate, it was just as frustrating as I would rarely figure out the equation before the computer AI would. That being said
though, I think my issues with the latter could be also attributed to needing more time. Like I said, I am not a math whiz.
Another complaint I have about the game, which didn’t seem to affect my experience with Challenge Me: Brain Puzzles, is that sans the two main play modes I have mentioned there is not a whole lot
more to do in Challenge Me: Math Workout. There is the benefit of being able to play against another person or two via the DS Wi-Fi feature, but this too only lasts for so long. I think that for a
game that is based on math, there should have been more options and more modes to make the game a little more enticing.
On a more positive note is the fact that unlike Challenge Me: Brain Puzzles, Math Workout does manage to provide a little more explanation of how to play the game and the two available puzzle types.
So those looking for a bit of guidance will be a little luckier here then they were in Brain Puzzles.
At the end of the day Challenge Me: Math Workout does what it is supposed to do, it tests your cranial skills using a form of mathematics. It just isn’t that exciting. Those who pick this up looking
for a math trainer may find this game does not offer such, but what would you expect, the DS can’t make you a modern day Einstein ya know. | {"url":"http://www.game-boyz.com/content/node/11492","timestamp":"2014-04-19T20:05:57Z","content_type":null,"content_length":"72100","record_id":"<urn:uuid:6d44286f-652f-44af-8b87-7f4e2d1979ca>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00617-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: Quantile regression
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Re: st: Quantile regression
From David Hoaglin <dchoaglin@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: Quantile regression
Date Fri, 21 Sep 2012 22:43:12 -0400
Dear Vasan,
I'm puzzled. From the way in which you described your analysis in
your first message, I don't understand why you would use quantile
regression. As I recall, you wanted to compare the means of some
variables across quartiles of BMI for males and females. In that
description, it was not clear to me whether you wanted to compare the
mean of a variable in data from males among the quartiles of BMI and
similarly in data from females, or whether you wanted to compare the
female mean and the male mean within each quartile of BMI, or whether
you wanted to make both of these types of comparisons. I did not see
any mention of the numbers of observations or the source of the data
or, importantly, the scientific question that you are addressing.
As I read the command below, you are asking -qreg- the fit a
regression model to the median of BMI with predictors fast_glucose,
etc. (the median is the default quantile in -qreg-). This seems far
from what you set out to do.
Those of us who are following this thread would be better able to
advise you if you went back to the beginning and gave us more
information on the data and the context. I do not know, for example,
whether the data that you are analyzing are suitable for ANOVA. They
may be (perhaps after a transformation), and you may have given up on
ANOVA too quickly.
David Hoaglin
On Wed, Sep 19, 2012 at 5:33 PM, Vasan Kandaswamy
<vasan.kandaswamy@ki.se> wrote:
> Many thanks Nick.
> Now, I have given up on ANOVA since I cannot derive p values for gender seperately, but did a regression.
> A quantile regression this way comes up this way
> bysort bmi_q sex:sum g0mmol
> bysort sex: qreg bmi fast_glucose age pr ( adjusted for age)
> I tabulate the output this way
> BMI Q1 Q2 Q3 Q4 Beta (95%CI) P value
> Male 5.3 5.4 5.5 5.6 2.61 (1.46, 3.76) 8.91 x 10^-06
> Female 5.4 5.4 5.4 5.7 0.36 (-0.15, 0.86) 0.168
> IF you actually look at the mean glucose values in Q1-Q5, there is not much difference, but the regression shows a clear difference with p values of males significant, while females are not.
> Could you please explain of my approach is correct.
> The basic question I would like to ask is if the fold change from Q1 to Q5 is significant.
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Browse by Keyword: "sort"
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comparators Provides chainable comparator-function generators a la Java 8 Comparators
compare-numbers Compare two numbers, return -1/0/1. To be used to sort arrays
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dubh_sorter A program that will retrieve the latest information from a database and sort the dancer requests to the best outcome
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9. Visualizing
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 What Logics of Induction are There?
John D. Norton
9. Visualizing Deductive Structure
A great deal of the analysis here depends upon the deductive definability in preferred partitions of the inductive strengths. As a result, many properties of the deductive structure are reflected
within the inductive logic. One of the most important is this: whatever symmetries are present in the deductive structure must also be present in the inductive logic. So it is useful to develop a
picture of just what the logical structure and these deductive symmetries look like. That proves to be quite easy with the help of pictures provided by directed graphs.
Here's the simple case of a Boolean algebra with three atoms, a[1], a[2] and a[3]. The arrows represent deductive entailment. The graph depicts deductive inference emanating at the bottom from the
contradiction that, in classical logic, deductively entails everything. It proceeds up through the atoms a[1], a[2] and a[3], towards logically less specific propositions. These include the
disjunctions of atoms, such as (a[1] or a[2]), written as (a[1] v a[2]). All deduction terminates in the universally true Ω.
What matters for deductive structure is the way these arrows connect the nodes of the graph. The labels,a[1], a[2] and a[3], attached to the nodes have no intrinsic significance for the deductive
structure beyond their function of distinguishing nodes.
We could equally label them "Tom," "Dick" and "Harry" and still have the same three-atom deductive structure.
What this means, strictly speaking, is that labels for the nodes are inessential. A good representation of the deductive structure shows all the nodes as intrinsically the same.
We can get away with this in a picture in which the nodes look like identical spheres, for then the distinctness of the nodes is assured through the fact that they appear in a different part
spatially in the figure. Aside from that distinctness, all properties of the nodes arise through their presence in the network of deductive entailments, that is, in so far as they play a role in the
deductive structure. the contradiction is distinguished as the unique origin of all arrows of entailment. The universal Ω is distinguished as the unique endpoint of all inextensible chains of
deductive inference.
When we aren't using these graphical figures and our logic proceeds by writing formulae, however, we must use different symbols to distinguish the atoms.
The symmetries of the deductive structure follow immediately from this freedom to relabel the nodes while leaving the deductive structure unchanged. This relabeling is called a symmetry
transformation. It is a transformation on aspects of the structure whose effect is to leave the overall structure unchanged.
One of the simplest of these transformations cyclically permutes the atoms:
a[1] to a[2]
a[2] to a[3]
a[3] to a[1]
If we think of the graphs representing the three-atom Boolean algebra as a figure in three dimensional space,
then the symmetries become geometrical symmetries. A simpler permutation that just switches a[1] and a[3] but leaves a[2] unchanged is a
mirror reflection through a plane.
The cyclic permutation just shown becomes a rotation about the vertical axis through 120^o.
Once we start looking for geometrical ways to picture the symmetries, there is a simpler approach. We can also encode the three-atom algebra geometrically in a triangle. The corners are the three
atoms a[1], a[2] and a[3]. The sides are their disjunctions. The symmetries of the algebra are once again represented by the symmetries of the triangle. A cyclic permutation of the three atoms is a
rotation through 120^o.
This more compact representation gives us an easy figure for a four-atom algebra. It is just a tetrahedron, as shown at right.
In general, an N-atom algebra is represented by an N-1 simplex, which is a higher dimensional analog of the triangle and tetrahedron. Unfortunately drawing one is no longer easy.
We can display graphs that show disjunctive refinement. The graph as left shows the four-atom algebra that results when we apply a disjunctive refinement to a three-atom algebra.
In this case, we have refined only by altering a[1], which becomes b[1] or b[2]. The other two atoms, a[2] and a[3] remain undivided.
At first glance, this graph may appear bewildering. However, it does have a very orderly structure. It consists of three layers (aside from the contradiction and the universal proposition).
The first layer contains the four atoms b[1], b[2], a[2] and a[3], which sit at the corners of a gray square.
The second layer contains the six disjunctions of pairs of atoms,which sit at the corners of a gray hexagon.
The third layer contains the four disjunctions of three atoms, which sit a the corners of a gray square.
This four-atom algebra begins to illustrate the deductive fact that is expressed in inductive logic as the generic presence of independence. The largest layer is the middle layer whose proposition
are formed from 2 = 4/2 atoms. Since N=4, the effects that become pronounced for very large N are not yet visible. However as N grows much larger, this middle layer and those close to it come to
contain virtually all propositions. This is illustrated more accurately in the picture at right, which shows a Boolean algebra for very large N.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 What Logics of Induction are There?
John D. Norton | {"url":"http://www.pitt.edu/~jdnorton/Goodies/logics_induction/index9.html","timestamp":"2014-04-20T06:21:37Z","content_type":null,"content_length":"13495","record_id":"<urn:uuid:37edb739-63ed-47f4-a186-0246b6d2bf24>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00434-ip-10-147-4-33.ec2.internal.warc.gz"} |
The .Plan: A Quasi-Blog
My paper’s first main finding is a quantitative estimate of the objective component in the economics/finance refereeing process. Consider a scale defined by a single parameter that measures referee
accuracy, named lambda. Lambda can be from 0 to 1 and measures the fraction of the referee report that constitutes an objectively agreeable paper quality. ... λ = 1 means that every referee reports
the paper’s objective aspect. λ = 0 means that every referee reports noise. ...
The observed consensus estimates among referees were λ ≈ 0.30 for the Journal of Finance (JF) and Review of Financial Studies (RFS), λ ≈ 0.35 for Econometrica (ECMTA), the Quarterly Journal of
Economics (QJE) and the SFS Cavalcade; and λ ≈ 0.40 for the International Economic Review (IER), the Journal of Economic Theory (JET), the Journal of the European Economic Association (JEEA), and the
Rand Journal of Economics (Rand).
Roughly, referee reports were one part signal, two parts noise. ...
For economics journals, when two referees are consulted, the top-10p [percentile] paper receives two rejects with probability 14%, one reject and one non-reject with probability 47%, and two
non-rejects with probability 40%. With three referees, the top-10p papers receives a majority of reject recommendations with 30% probability, a majority of non-reject recommendations with 70%
For finance journals, with their lower lambdas and higher rejection probabilities, the higher than 50% reject probability for the top-10p paper results in a strange situation: The more referees are
consulted, the more likely it is that the referees will agree that the top-10p paper is bad. For this top-10p paper, with one referee, the probability that the majority of referees recommends
rejection is 38%; with three referees, it is almost 70%. (This also obviates the idea of using a tie-breaker referee when two referees disagree.) In fact, only the top-2p papers have a conditional
probability of rejection that is less than 50%, resulting in a majority rejection probability that does not increase with the number of referees. | {"url":"http://jamesjchoi.blogspot.com/2012/11/how-noisy-is-economicsfinance-peer.html?m=1","timestamp":"2014-04-18T18:11:08Z","content_type":null,"content_length":"57678","record_id":"<urn:uuid:8acd16f4-1e8f-46a9-81b7-a672e621dcdd>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00168-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Are integers real?
up vote 2 down vote favorite
Do you think that $\mathbb Z \subset \mathbb R$? On one hand this inclusion is quite handy. We like to write things like: $$ \sqrt{n} \quad \text{for $n\in \mathbb Z$} $$ which requires the number
$n$ to be a real number (where $\sqrt\cdot$ is defined). On the other hand it is difficult to obtain such an inclusion when comes to definition. One would like to be able to define whole numbers
without the need to define real numbers. This becomes more tricky when one notices that there are other inclusions which one would like to satisfy. For example I would like $1$ to be a polynomial
with whole coefficients, or maybe a polynomial with complex coefficients, or maybe a real function of one variable...
I would say that it is not possible to satisfy all these inclusions. So maybe we must not insist on saying that $\mathbb Z \subset \mathbb R$ in the first place. Are there alternatives?
One possibility I see is that of having many different sets isomorphic to $\mathbb Z$. We should use the name $\mathbb Z$ for integers as we use $V$ for vector spaces. We should say: let $\mathbb Z$
be any set of integers. Or: let $\mathbb R$ be a set of reals and let$\mathbb Z$ be the set of real integers $\mathbb Z \subset \mathbb R$. And so on...
Another possibility I see (but I'm not sure if it can be really founded) is that of redefining the meaning of equality $=$ and distinguish between equality and identity. We could try to take an
"object-oriented" approach where equality could be defined like any other operation. So one could define $1 = 1/1$ i.e. the integer $1$ is the same as the rational $1/1$ and one should define the sum
of integers and rational by converting the integer to a rational and then performing the sum between rationals. This also modifies the concept of 'set' since the set $\{1, 1/1 \}$ is equal to $\{1/1
\}$ and hence has a single element. This is, more or less, how types work in computer languages. Can this approach be made rigorous?
If you follow the approach of successors, you proceed from Natural numbers to integers to rational numbers and with a completion the reals. This inherently embeds integers in the reals.
Alternatively, if you introduce the real numbers as a model for a certain set with some axiomatic structure, you can identify the previously mentioned subsets via their structural properties
(which is more along the line of your comment about sets isomorphic to $\mathbb{Z}$. Is this in the direction you are thinking? – Daniel Spector Jan 28 '13 at 8:49
4 I don't understand why this is closed. Is the problem that it's too elementary? Or that it's been discussed elsewhere here? But then some comments along those lines would be nice (such as a
suggestion to use math.SX or a link to another question). Instead it's closed with no useful explanation. Fortunately there were already a couple of answers worth voting up. – Toby Bartels Jan 28
'13 at 10:30
@Daniel: you speak of an embedding of $\mathbb N$ into $\mathbb R$ not of a set inclusion. This means you couldn't write $\sqrt n$ for $n \in \mathbb N$ without an abuse of notation. My question
is whether it is possible to avoid the abuse of notation without having to much complications in writing formulas. Saying with other words: since we all understand what we are speaking about, why
is it so difficult to find a notation to formalize it in a correct way (i.e. without abusing the notation)? This problem has been addressed in computer languages, where we don't admit abuses of
notation. – Emanuele Paolini Jan 28 '13 at 11:34
3 Toby, I agree -- the question doesn't seem bad or MO-unworthy (although I'd get rid of the opening "Do you think..." which could suggest subjectivity). My hypothesis is that people in the math
blogosphere have gotten a little tired of foundational discussions, of which there has been a recent spate. However, some explanation of closure would be nice. Should a meta thread be opened? –
Todd Trimble♦ Jan 28 '13 at 12:57
1 I don't know if this is the sort of thing you're looking for, but you might be interested in Chapter 7 of Mohan Ganesalingam's thesis: people.ds.cam.ac.uk/mg262 . (Which, it turns out, has now
vanished from the web in anticipation of publication.) – HJRW Jan 28 '13 at 16:21
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closed as off topic by Stefan Geschke, Eric Wofsey, Michael Greinecker, Dmitri Pavlov, Chandan Singh Dalawat Jan 28 '13 at 9:47
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3 Answers
active oldest votes
We do in fact have many different sets isomophic to $\mathbb{Z}$. The correct definition of $\mathbb{Z}$ is a set with some operations on it that satisfy some axioms. This is easier to
see with $\mathbb{N}$, which is a set with a 0, a "plus one" operations, that satisfies mathematical induction. There is only one such set up to oeration-preserving isomorphism.
$\mathbb{R}$ can be axiomatized similarly, as a complete Archimedean ordered field (see the synthetic approach section at Wikipedia). Again there is only one such set up to
operation-preserving isomorphism. You can identify in that set a copy of $\mathbb{Z}$, so you can treat $\mathbb{Z}$ as a subset. But this is a matter of notation. If you had a
compelling reason to make it disjoint, you could require that, instead.
We say "the integers" or "the reals" because there is only one, up to isomorphism. Any theorem you prove about $\mathbb{Z}$ or $\mathbb{R}$ that doesn't use the internal representation
up vote 6 will transfer to any isomorphic copy, so we don't need to know which one.
down vote
accepted There is a whole mathematical notion of types that actually underpins languages with more-complicated type systems, such as ML or Haskell. You can think of $\mathbb{Z}$ and $\mathbb{R}$
as a type in a type system. Some typing systems have a notion of "subtype", but let's suppose that you don't. Then $\mathbb{Z}$ and $\mathbb{R}$ are types, and there's a designated
monomorphism $i$ from $\mathbb{Z}$ to $\mathbb{R}$. Then 2 + 3.5 is an overloaded operation that is syntactic sugar for $i(2) + 3.5$. I don't know if anyone has ever worked out a clear
account of what mathematicians do from this point of view, but what they do is not very complicated.
If you do want to allow subtypes, there's a notion of order-sorted algebra that allows the kind of overloading you probably have in mind. I don't know of a canonical link, but
introductions are easy to find.
+1 for 'I don't know if anyone has ever worked out a clear account of what mathematicians do from this point of view, but what they do is not very complicated.' – HJRW Jan 28 '13 at
Also, iirc, Chapter 7 of Ganesalingam's thesis (mentioned in a comment above) is more or less an attempt to do exactly that. – HJRW Jan 28 '13 at 17:20
I accepted this answer. I never heard of types used in mathematics, but this is in the direction of what I was thinking... Can you present some references? Is the notion of types
replacing set theory or using it? – Emanuele Paolini Jan 29 '13 at 13:19
There is a gigantic mathematical theory of types, that predates the invention of the computer, and has had considerable influence on the design of programming languages such as ML and
Haskell. Some people would like to replace set theory with type theory. I think the idea of one replacing the other is pointless, since for most mathematical applications I can express
anything I want to say equally well in either. – arsmath Jan 29 '13 at 14:23
The best introductions to type theory are more computational oriented than math oriented. I would recommend Girard's "Proofs and Types", which is available online. I don't know a good
reference explicitly for the definition of the reals inside a type theory. The main consumers of type theory tend to be more interested in constructive mathematics than in a
non-constructive topic such as classical analysis. Second-order arithmetic (en.wikipedia.org/wiki/Second-order_arithmetic) is like a type theory with two types, even though it is
couched in set-theoretic language. – arsmath Jan 29 '13 at 14:28
add comment
The answer to this question depends on the type of set theory that you are working in, and the way you decide to code the integers and real numbers inside set theory. For instance in
material set theory, we often (it is not mandatory) define the natural numbers to be finite Von Neumann ordinals,integers as certain classes of ordered pairs of naturals the rationals as
certain equivalence classes of certain ordered pairs of integers, then the reals as classes of Cauchy sequences. In this kind of set theory, in no way is the integers a subset of the reals.
But their is a cannonical inclusion of the integers into the reals.
In a structural set theory what is in the set does not matter so much as how different sets relate to one another. This is inherently a categorical discussion of what set theory is. Now in
category theory, a sub-object is an equivelence class of monomorphisms. In this kind of set theory the integers are a subset of the reals.
up vote
4 down For the differences between material and structural set theory see http://ncatlab.org/nlab/show/set+theory .
A subobject is defined here: http://ncatlab.org/nlab/show/subobject
Also a nice exposition of structural set theory is here: http://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html
Also relevant to material and structural set theory and relations to type theory are here: http://golem.ph.utexas.edu/category/2013/01/from_set_theory_to_type_theory.html
add comment
Expanding on the first part of Daniel Spector's comment, in all the cases you mentioned there is a canonical inclusion $\mathbb{Z}\hookrightarrow X$ for a certain set $X$ (reals,
polynomials with complex coefficients, etc. ). When we write $\sqrt{n}$ for $n\in\mathbb{Z}$ (or any analogous expression), you are doing a little abuse of notation, identifying $n$ with
up vote 2 its image under the canonical embedding of $\mathbb{Z}$ into the reals (or the appropriate set $X$). This usually doesn't cause any ambiguity and that's why we do it all the time.
down vote
Ok, so you are on the side where $\mathbb Z \subset \R$ is not assumed (or is considered an abuse of notation). – Emanuele Paolini Jan 28 '13 at 11:24
Yes, I don't feel the need of having a unique object in set theory that deserves to be called "the set of integers". The relevant thing is the canonical identification referred to above.
But I'm not an expert in logic and set theory, so I could be missing some subtlety... – Gian Maria Dall'Ara Jan 28 '13 at 22:20
add comment
Not the answer you're looking for? Browse other questions tagged foundations or ask your own question. | {"url":"http://mathoverflow.net/questions/120090/are-integers-real?sort=oldest","timestamp":"2014-04-18T18:15:43Z","content_type":null,"content_length":"70775","record_id":"<urn:uuid:106ece59-8455-4bda-939a-781ba162fd91>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00036-ip-10-147-4-33.ec2.internal.warc.gz"} |
Computational Failures and INFO
Next: Wrong Results Up: Failures Detected by LAPACK Previous: Invalid Arguments and XERBLA   Contents   Index
Computational Failures and INFO > 0
A positive value of INFO on return from an LAPACK routine indicates a failure in the course of the algorithm. Common causes are:
• a matrix is singular (to working precision);
• a symmetric matrix is not positive definite;
• an iterative algorithm for computing eigenvalues or eigenvectors fails to converge in the permitted number of iterations.
For example, if SGESVX is called to solve a system of equations with a coefficient matrix that is approximately singular, it may detect exact singularity at the i^th stage of the LU factorization, in
which case it returns INFO = i; or (more probably) it may compute an estimate of the reciprocal condition number that is less than machine precision, in which case it returns INFO = n+1. Again, the
documentation in Part 2 should be consulted for a description of the error.
When a failure with INFO > 0 occurs, control is always returned to the calling program; XERBLA is not called, and no error message is written. It is worth repeating that it is good practice always to
check for a non-zero value of INFO on return from an LAPACK routine.
A failure with INFO > 0 may indicate any of the following:
• an inappropriate routine was used: for example, if a routine fails because a symmetric matrix turns out not to be positive definite, consider using a routine for symmetric indefinite matrices.
• a single precision routine was used when double precision was needed: for example, if SGESVX reports approximate singularity (as illustrated above), the corresponding double precision routine
DGESVX may be able to solve the problem (but nevertheless the problem is ill-conditioned).
• a programming error occurred in generating the data supplied to a routine: for example, even though theoretically a matrix should be well-conditioned and positive-definite, a programming error in
generating the matrix could easily destroy either of those properties.
• a programming error occurred in calling the routine, of the kind listed in Section 7.2.
Next: Wrong Results Up: Failures Detected by LAPACK Previous: Invalid Arguments and XERBLA   Contents   Index Susan Blackford | {"url":"http://www.netlib.org/lapack/lug/node138.html","timestamp":"2014-04-17T15:27:56Z","content_type":null,"content_length":"6322","record_id":"<urn:uuid:544b843d-fa71-4173-8235-d3be2c9e32c2>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00150-ip-10-147-4-33.ec2.internal.warc.gz"} |
September 2
measurement order in cluster states
I should have included a link to
this paper by Michael Nielsen
in my previous post. It's a more introductory paper, perhaps easier to follow. It also includes a review of the quantum circuit model that's good.
I wanted to discuss measurement order more in the last post, but it had gotten too long. Somewhere in your cluster state are qubits that are your "output" qubits. If you've simply followed the most
straightforward method of mapping your circuit to the cluster state, they will be on the right-hand edge.
A fascinating fact: your output qubits are among the first to be measured! They go down in the first round, and you get back the answer to your computation. Viola! Done in one time step!
Well, not quite. The numbers you got back are
to the answer you want, but you don't yet know
they are related. Essentially, what you have is the answer, but encrypted. As you perform other measurements, you learn more information. You keep that data, and perform various bit flips on that
data based on measurement outcomes. Eventually, your answer is "decrypted".
I wonder, idly, if you could use this as a form of quantum network security. Start with a big cluster state, send half to your partner. Now neither of you can do anything without the other. Probably
theoretically possible, but not practically interesting.
Mayumi is also blogging these days. If you can't read Japanese, you won't follow the text, but she's posting pictures of the girls, which you might enjoy. Her blog is here. At the bottom of the page
is a calendar with links to dates she has blog entries. You'll just have to skip around to find the pictures.
I really should add pictures of the girls to the web somewhere myself...
David Deutsch, one of the preeminent quantum computing theorists, has suggested on his blog that recent advances in cluster state computing mean that it might be possible to actually build a quantum
computer within the next few years. See here and here. I have studied cluster state computing a little (n.b.: this is COMPLETELY DIFFERENT from classical cluster computing), and I know one thing for
I haven't the FOGGIEST notion how to build a machine to run such a computation.
Deutsch seems to think that this paper by Lim et al. is a serious, practical proposal. And, it happens to be, essentially, a form of quantum multicomputer, which is my thesis topic, so I'm very
interested in this. But, before we can decide how practical Lim's proposal is, we need to know at least a little bit about cluster state computing.
Be warned: cluster state computing is mathematically difficult stuff, and even more important, a real mind-bender of an idea. It is important, though, and I think the basic idea can be comprehended
without following the math, so I'm going to try to explain it. One particularly comprehensive paper on it is this one, but it's rough going. A better first step might be this one. Both of these are
by the originators of the concept, Raussendorf and Briegel, and their collaborator Browne (we'll call them RBB). They sometimes call cluster state computing the "one-way quantum computer" and
abbreviate it as QCc, a notation we'll adopt. Even if you follow little of the text and none of the mathematics, checking out their diagrams in those papers might be helpful, since I have none on
this blog. There are many papers on QCc, including how to do fault-tolerant QCc; I have read only a few, and won't go into details here.
Okay, my totally inept attempt to explain a little about it, in layman's terms:
If you know how to build and program a classical computer, how to design circuits, you can use that knowledge with the "standard" (circuit-based) model of quantum computation. Mathematical physicists
often deal more abstractly in the actual Hamiltonians involved, especially when they want to directly simulate another quantum system, rather than run what you and I think of as an "algorithm". But
you can stick with the circuit model pretty comfortably for most purposes, and that's how I've done my work on e.g. how to do integeer arithmetic on a quantum computer. But there's a third way:
cluster state computing.
If the circuit model is C or assembly language for a quantum computer, then QCc is Prolog. In QCc, you don't so much tell the computer what to do as you make statements about conditions the state of
the computer fulfills.
Cluster state computation is "build a huge entangled state, then selectively measure some of the connections, and magically the state transforms to the result you're looking for". That's an egregious
simplification, but it's the basic idea. It has been heralded, with good reason, as the biggest theoretical revolution in quantum computing in recent years.
I think the theorists are attracted by totally new paradigm (which is a wonderful new way to think - computation carried out strictly by measurement!). It is also supposed to open up entirely new
avenues of thought and new ways to "program" a quantum computer. But let's stick to one particular path - how to map the circuits we know to QCc.
For QCc, you start with a set of physical qubits that are in a 2-D grid (logically or physically) and can connect to their Manhattan neighbors. You initialize them, then perform some local operations
that entangle small groups into a specific state called the cluster state. You then probabilistically connect those small groups together to make larger groups, retrying until it works (it's
probabilistic, but we know when it works and when it doesn't, so we can just keep retrying until it does). Eventually, your entire machine is in one VERY large entangled state. The work up to this
point is independent of the computation (circuit or algorithm) you're going to execute - you don't need to know anything but how big a cluster state to build.
Now back up for a second. Those circuits we're talking about (for some examples, see my arithmetic page (which is out of date these days, but anyway...)) are drawn using horizontal lines and vertical
line segments connecting some of those lines in particular patterns. Each line represents a qubit, and the vertical line segments are gates being executed on the connected qubits. Time flows left to
right. The physical resources you need are the vertical axis (the number of lines) and the time you need is the horizontal axis. (It's also worth mentioning, if you're delving into the literature,
that a circuit is often called a "network". This is network in the sense that you create a netlist when doing circuit design (VLSI or PCB layout), not network in the Internet sense. There is research
(and even products already) in the quantum long-distance network sense, but that's not what we're talking about here.)
To turn a circuit into a cluster state computation, we are going to unroll the computation and lay it out flat on a large cluster state. The physically-laid out cluster state will look much like the
complete circuit diagram, instantiated all at once. To start with, our cluster state is a featureless 2-D grid. The derivation of the rules for mapping a circuit to it is difficult, but their
application is straightforward once you understand them. First, you cut out a number of the qubits, creating holes in the grid. This cutting is done by measuring those qubits along the z axis (recall
that a single qubit state can be thought of as a point anywhere on the unit sphere - a unit vector - and that measuring the qubit along a particular axis forces the qubit into a state aligned along
that axis in either the positive or negative direction).
At this point, we have a structure that is partially adapted to our planned circuit, but we haven't yet applied any of the gates. A "gate" in QCc is nothing but the measurement of a particular
pattern of nearby qubits along a specific set of axes (that's why this is also sometimes referred to as "measurement-based computing"). The choice of which qubits to measure and along which axes will
determine what gate is effectively executed.
So now we have a plan - we know, roughly, what measurements we are going to perform on which qubits and how that will get us where we want to go. Next question: in what order do we apply those
measurements? Ah, that's where the magic of QCc really shines. You might intuitively think that we more or less follow the circuit, starting from the left, applying the measurements to effect the
gates in more or less the order they appear in the circuit. Nope!
Some of the possible quantum gates fall into what is known as the Clifford group. This includes the control-NOT (CNOT) gate and a few single-qubit rotations such as NOT and Hadamard. It does not
include the CCNOT (control-control NOT, or Toffoli) gate. All of the gates in your circuit that are in the Clifford group can be executed at the same time! That is, if your circuit consists of only
Clifford group gates, the execution of your entire circuit takes one time step, regardless of the apparent dependencies among gates! This is one of the most exciting features of cluster state
computing. We have just traded space for time in a way that should make any parallel-computing guru green with envy.
Ah, but there's a catch: the non-Clifford-group gates. We need the Toffoli gate to run addition, and there are non-Clifford-group gates in the quantum Fourier transform, as well. How do those work?
Well, it turns out that the choice of which measurements to make (the choice of measurement axis, not which qubit to measure) depends on the results of previous measurements. It turns out that they
become dependent in a way that has some relationship to the original circuit, but is not identical. For both the QFT and addition, the number of timesteps winds up being N for an N-bit computation.
To quote RBB, "the temporal axis is converted into an additional spatial axis. The temporal axis in a QCc computation emerges anew."
For a Vedral-Barenco-Ekert (VBE) carry-ripple adder, QCc requires about 100x as many qubits as does the direct circuit implementation. However, it might be faster, depending on the relative
difficulty of preparing the cluster state and making single-qubit measurements for the QCc versus the cost of Toffoli gates for the circuit implementation. This is a technology-dependent question.
Moreover, the cluster states are HUGE, requiring an entangled state that is on the order of the total number of GATES, not qubits, you would have in the circuit version of the problem. This is not
practical, so you have to figure out how to buffer the state you need on a rolling basis with limited physical resources. This is understood, though I'm not familiar with the details, so that will
have to wait for another time.
The theorists believe that QCc opens up new paths to finding quantum algorithms. So far, though, what is best understood is how to map the known circuits to QCc, and there, as we have seen, QCc's
advantages are somewhat muted.
This brings us back to where we started - the Lim et al. proposal to implement a distributed cluster-state computer. But this posting has already gotten long, and I'm not through digesting the Lim
paper yet, so it will also have to wait for another time.
Important theoretical advance? Heck, yeah. Faster path to real implementations? I'm not sure, but personally I kind of doubt it.
Of course, I hope to be part of the team that proves myself either right or wrong in short order :-). | {"url":"http://rdvlivefromtokyo.blogspot.com/2005_09_01_archive.html","timestamp":"2014-04-19T01:54:14Z","content_type":null,"content_length":"39593","record_id":"<urn:uuid:3869dcae-35c2-42c2-bdfc-09fcfcc73ca8>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00268-ip-10-147-4-33.ec2.internal.warc.gz"} |
LSPI with Random Projections
Mohammad Ghavamzadeh, Alessandro Lazaric, Odalric-Ambrym Maillard and Rémi Munos
(2010) Technical Report. INRIA, France.
We consider the problem of reinforcement learning in high-dimensional spaces when the number of features is bigger than the number of samples. In particular, we study the least-squares temporal
difference (LSTD) learning algorithm when a space of low dimension is generated with a random projection from a high- dimensional space. We provide a thorough theoretical analysis of the LSTD with
random projections and derive performance bounds for the resulting algorithm. We also show how the error of LSTD with random projections is propagated through the iterations of a policy iteration
algorithm and provide a performance bound for the resulting least-squares policy iteration (LSPI) algorithm. | {"url":"http://eprints.pascal-network.org/archive/00007387/","timestamp":"2014-04-16T16:40:11Z","content_type":null,"content_length":"6513","record_id":"<urn:uuid:dd93d1ce-717b-4531-b919-4afdd254d6c1>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00103-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Wolfram Demonstrations Project
Distillation Lines for a Mixture of Chloroform, Acetone, and Methanol at 1 atm
Consider a ternary mixture composed of chloroform, acetone, and methanol at 1 atm. This Demonstration plots the (blue) distillation line that corresponds to the composition profiles in a staged
column at infinite reflux. You can drag the locator to change the position of this curve. This curve is found by solving the following equation:
where is the liquid mole fraction of component
at stage
and is the vapor mole fraction of component
at stage .
The azeotropes (labeled in the figure by for from
1 to 4) and pure components appear as the fixed points of . It turns out that it is equivalent to finding the fixed points of equation (1) and the singular points of
which gives the residue curves or the composition profile in a packed column at infinite reflux.
For more information, see
M. F. Doherty and M. F. Malone,
Conceptual Design of Distillation Systems
, New York: McGraw-Hill, 2001.
W. D. Seider, J. D. Seader, and D. R. Lewin,
Product and Process Design Principles
, New York: Wiley, 2004. | {"url":"http://demonstrations.wolfram.com/DistillationLinesForAMixtureOfChloroformAcetoneAndMethanolAt/","timestamp":"2014-04-18T00:13:48Z","content_type":null,"content_length":"44440","record_id":"<urn:uuid:32543fab-80c7-4dd9-bb13-cbdbcc25db8e>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00616-ip-10-147-4-33.ec2.internal.warc.gz"} |
the encyclopedic entry of cyclometric
Josip Plemelj
December 11
May 22
) was a
Slovene mathematician
Plemelj was born in the village of Grad on Bled (Grad na Bledu), Austria-Hungary (now Slovenia), he died in Ljubljana, Yugoslavia (now Slovenia). His father, Urban, a carpenter and crofter, died when
Josip was only a year old. His mother Marija, née Mrak, found bringing up the family alone very hard, but she was able to send her son to school in Ljubljana where Plemelj studied from 1886 to 1894.
After leaving and obtaining the necessary examination results he went to Vienna in 1894 where he had applied to Faculty of Arts to study mathematics, physics and astronomy. His professors in Vienna
were von Escherich for mathematical analysis, Gegenbauer and Mertens for arithmetic and algebra, Weiss for astronomy, Stefan's student Boltzmann for physics.
On May 1898 Plemelj presented his doctoral thesis under Escherich's tutelage entitled O linearnih homogenih diferencialnih enačbah z enolično periodičnimi koeficienti (Über lineare homogene
Differentialgleichungen mit eindeutigen periodischen Koeffizienten, About linear homogeneous differential equations with uniform periodical coefficients). He continued with his study in Berlin (1899/
1900) under the German mathematicians Frobenius and Fuchs and in Göttingen (1900/1901) under Klein and Hilbert.
In April 1902 he became a private senior lecturer at the University of Vienna. In 1906 he was appointed assistant at the Technical University of Vienna. In 1907 he became associate professor and in
1908 full professor of mathematics at the University of Chernivtsi (Russian Черновцы), Ukraine. From 1912 to 1913 he was dean of this faculty. In 1917 his political views led him to be forcibly
ejected by the Government and he fled to Bohemia (Moravska). After the First World War he became a member of the University Commission under the Slovene Provincial Government and helped establish the
first Slovene university at Ljubljana, and was elected its first rector. In the same year he was appointed professor of mathematics at the Faculty of Arts. After the Second World War he joined the
Faculty of Natural Science and Technology (FNT). He retired in 1957 after having lectured in mathematics for 40 years.
Plemelj had shown his great gift for mathematics early in elementary school. He mastered the whole of the high school syllabus by the beginning of the fourth year and began to tutor students for
their graduation examinations. At that time he discovered alone series for sin x and cos x. Actually he found a series for cyclometric function arccos x and after that he just inverted this series
and then guessed a principle for coefficients. Yet he did not have a proof for that.
Plemelj had great joy for a difficult constructional tasks from geometry. From his high school days originates an elementary problem - his later construction of regular sevenfold polygon inscribed in
a circle otherwise exactly and not approximately with simple solution as an angle trisection which was yet not known in those days and which necessarily leads to the old Indian or Babylonian
approximate construction. He started to occupy himself with mathematics in fourth and fifth class of high school. Beside in mathematics he was interested also in natural science and especially
astronomy. He studied celestial mechanics already at high school. He liked observing the stars. His eyesight was so sharp he could see the planet Venus even in the daytime.
Let us hear about his early days in school in his own words: "It was the April 1891 in fifth class. The class had two rows of desks with crossing in between and I was sitting on the most side inside
chair very rear. I think there were only two desks after me. Professor Borštner did not lecture. He had only given a lection from the book for the next lesson. He called to the blackboard two pupils
and there he was discussing the subject and furthermore with this he included the whole class for cooperation. He used to have such a habit gratefully to give geometrical constructional tasks which
he dictated from some collection he brought with. Once he gave amongst the other a task: Draught a triangle if one side, its altitude and a difference of two angles along it are given. Classmates had
appealed to me before the lesson if the task was a little bit hard. They could not solve this task after several lessons and he had asked them the other. Borštner used to come from before the
master's desk and he stopped ahead of me, sat toward me in the desk and hence he examined. After sometime he had said we should solve that task. Perhaps he was suspicious about that we had not yet
solved it so he turned to me and asked me if I had tried this construction. I had said to him I could not find any path for the solution. Then he said he would show it in the next lesson. This had
plucked up my courage to inspect it again. I had found a solution with subsidiary points, lines and so forth which seems to be inaccessible for human mind if the way which had inevitably led me to my
aim is hidden from. Next lesson professor Borštner had sat toward me in the desk again. After customary examination of my classmates he said: "Well, let us work out that construction task." I
whispered him I had succeeded till then and he said: "So, let me show how had you done this." He thought I would show him written on paper and he said: "Well, all-right." He had stepped aside and we
went before the blackboard. I drew a triangle ΔABC with an ordinary analysis: Given side AB = c, its altitude v[c] and the difference 0 < α − β < π.
Let us draw a perpendicular AA' from A to side AB and let AA' = 2 v[c]. Let us bind A' with C in the way to be A'C = AC = b and draw A'B = m. Along the side A'C let us gather out along A' an angle α
and on the left side of a triangle A'B' = c. On this way the risen triangle is ΔA'B'C ~ ΔABC. By B lies an angle <A'B'C = β and an angle <A'CB' = γ.
A triangle ΔBCB' is isosceles and <BCB' = 2 α, so we have < BB'C = π/2 − α. Now it is the angle <A'B'B = π/2 − α + β and we can construct over the side BA' a circumferential angle of a certain
circle. We get a point B' at once because it is A'B' = c. Because a triangle ΔBCB' is isosceles a point C lies on a symmetric of the side BB' where it intersects a parallel with AB in a given
altitude v[c]. With this the triangle ΔABC is drawn. Professor Borštner was gazing when he saw this curious solution and he held of his head: "Aber um Hergottswillen, das ist doch harsträubend, das
ist doch doch menschenunmöglich auf so einen Einfall zu kommen; sagen Sie mir doch, was hat Sie zu dieser Idee geführt!" I said to him I had not guessed this strange solution but I had asked myself
about a trigonometric determination of a triangle because I could not find the solution in the other way. Geometric interpretation of this solution had led me up to this pure geometrically
understandable construction. We did not spoke anymore about this with professor Borštner, but he had after that shown another easier solution, which I could imitate from my own construction and which
I had not perceived because I had precisely traced way.
Trigonometric solution is easy: with altitude v[c ]from point C to side AB it breaks in two parts v[c ]cot α and v[c ]cot β. Then we have:
$v_c \left(cotalpha + cotbeta\right) = cquad mbox\left\{or\right\}quad$
v_c sin(alpha + beta) = c sinalpha sinbeta.
We can then write:
$2v_c sin\left(alpha + beta\right) = c \left(cos\left(alpha - beta\right) - cos \left(alpha + beta\right)\right).,$
Because α + β = π − γ, this equation is:
$2 v_c singamma - c cosgamma = ccos \left(alpha - beta\right).,$
From this equation we have to obtain an angle γ. The easiest way is if we introduce a certain subsidiary angle μ. Namely we raise:
2 v[c] = m cos μ c = m sin μ.
Both equations give us for μ a uniform certain acute angle and for m a certain positive length. The equation for γ is then:
$m sin\left(gamma - mu\right) = c cos\left(alpha - beta\right).,$
We can consider this equation as a theorem of the sine for a certain triangle in which c and m are its sides and their opposite angles are γ - μ and π/2 ± (α - β) respectively (lightgreen triangle on
the picture below ). In this quoted construction this triangle is ΔBA'B', where BA' = m and A'B' = c, the angle <BB'A' = π/2 - α + α and the angle <A'BB' = γ - μ, as we can easily see. In my
construction this requested triangle is drafted twice. I saw later on we can interpret above equation with a triangle which has a side AB already. This leads us to very beautiful and short
construction. Requested triangle is ΔK'AB.
Straight line CK = a is symmetrically displaced in CK' = a and at the same time is AK = AK' = m. I had spoken as a professor in Černovice with two of my students about this elementary geometrical
problem and I said that my high school teacher had dictated this task from a certain collection. They brought me a collection indeed where there was the exact picture from Borštner's construction.
Unfortunately I had not written down a title of that book which was with no doubt professor Borštner's collection. Our teacher's library at classical gymnasium in Ljubljana does not have this book at
present. But I got from there a Wiegand's book entitled Geometrijske naloge za višje gimnazije (Geometrische Aufgaben für Obergymnasien, Geometric exercises for upper gymnasiums), which does not have
this exercise. I found in it a task: construct a triangle if we know one length of an angle symmetrical from one point, perpendicular on this symmetrical from the other point and an angle by the
third different solutions. The last has annotation: at the night of the January 1 1940 after the New Year's Eve 1939".
His main research fields were (linear) differential equations, integral equations, potential theory (of harmonic functions), theory of analytical functions and function theory. When he was studying
at Göttingen, Holmgren had reported about a theory which was developed by Fredholm for linear integral equations of the 1st and the 2nd degree. Mathematicians from Göttingen began to work on this new
research field under Hilbert's guidance. Plemelj was among the first who had done a beginning work and he had achieved fine results. He had used integral equations in a potential theory successfully.
His most important work in a potential theory is a book entitled Raziskave v teoriji potenciala (Potentialtheoretische Untersuchungen, Researches in a potential theory), "Preisschriften der fürstl.
Jablonowskischen Gesselschaft in Leipzig", (Leipzig 1911, pp XIX+100) which 1911 received award from Scientific society of prince Jablonowski in Leipzig in the amount of 1500 marks and Richard Lieben
award from University of Vienna in the amount of 2000 crowns. Argument for this was that this work was the most outstanding on the field of pure and applied mathematics which had been written by any
kind of 'Austrian' mathematician in the last three years. His most important work in general is with no doubt his original, marvellous and simple solution of the Riemann problem f[+]=g f[-] about the
existence of a differential equation with given monodromy group. He published his solution in 1908 in a treatise entitled Riemannovi razredi funkcij z dano monodromijsko grupo (Riemannsche
Funktionenscharen mit gegebener Monodromiegruppe, Riemannian classes of functions with given monodromy group), "Monatshefte für Mathematik und Physik" 19, W 1908, 211-246. In solving the Riemann -
problem Plemelj used equations about boundary values of holomorphic functions which he had discovered a short time before and which are now called after him Plemelj formulae, Sokhotsky-Plemelj or
sometimes (mainly in German literature) Plemelj-Sokhotsky formulae after Russian mathematician Sokhotsky (Юлиан Карл Васильевич Сохоцкий ) (1842-1927):
$f_+\left(z\right)=\left\{1over2ipi\right\} int_Gamma\left\{phi\left(t\right)-phi\left(z\right)over\left\{t-z\right\}\right\} dt$
+ phi(z)
$f\left(z\right)=\left\{1over2ipi\right\} int_Gamma\left\{phi\left(s\right)-phi\left(z\right)over\left\{t-z\right\}\right\} dt$
+ {1over2} phi(z)
$f_-\left(z\right)=\left\{1over2ipi\right\} int_Gamma \left\{phi\left(t\right)-phi\left(z\right)over\left\{t-z\right\}\right\} dt$
quad zinGamma
From his methods on solving the Riemann - problem had developed the theory of singular integral equations (MSC (2000) 45-Exx) which was entertained above all by the Russian school at the head of
Muskhelishvili (Николай Иванович Мусхелищвили) (1891-1976).
Also important are Plemelj's contributions to the theory of analytic functions in solving the problem of uniformization of algebraic functions, contributions on formulation of the theorem of analytic
extension of designs and treatises in algebra and in number theory.
1912 Plemelj published a very simple proof for the Fermat's last theorem for exponent n = 5, which was first given almost simultaneously by Dirichlet in 1828 and Legendre in 1830. Their proofs
difficult, while Plemelj showed how to use the ring we get if we extend the rational numbers by √ 5.
His arrival in Ljubljana 1919 was very important for development of mathematics in Slovenia. As a good teacher he had raised several generations of mathematicians and engineers. His most famous
student is Ivan Vidav. After the 2nd World War Slovenska akademija znanosti in umetnosti (Slovene Academy of Sciences and Arts) (SAZU) had published his three year's course of lectures for students
of mathematics: Teorija analitičnih funkcij (The theory of analytic functions), (SAZU, Ljubljana 1953, pp XVI+516), Diferencialne in integralske enačbe. Teorija in uporaba (Differential and integral
equations. The theory and the application).
Plemelj found a formula for a sum of normal derivatives of one layered potential in the internal or external region. He was pleased also with algebra and number theory, but he had published only few
contributions from these fields - for example a book entitled Algebra in teorija števil (Algebra and the number theory) (SAZU, Ljubljana 1962, pp XIV+278) which was published abroad as his last work
Problemi v smislu Riemanna in Kleina (Problems in the Sense of Riemann and Klein) (edition and translation by J. R. M. Radok, "Interscience Tract in Pure and Applied Mathematics", No. 16,
Interscience Publishers: John Wiley & Sons, New York, London, Sydney 1964, pp VII+175). This work deals with questions which were of his most interests and examinations. His bibliography includes 33
units, from which 30 are scientific treatises and had been published among the others in a magazines such as: "Monatshefte für Mathematik und Physik", "Sitzungsberichte der kaiserlichen Akademie der
Wissenschaften"; in Wien, "Jahresbericht der deutschen Mathematikervereinigung", "Gesellschaft deutscher Naturforscher und Ärzte" in Verhandlungen, "Bulletin des Sciences Mathematiques", "Obzornik za
matematiko in fiziko" and "Publications mathematiques de l'Universite de Belgrade". When French mathematician Charles Émile Picard denoted Plemelj's works as "deux excellents memoires", Plemelj
became known in mathematical world.
Plemelj was regular member of the SAZU since its foundation in 1938, corresponding member of the JAZU (Yugoslav Academy of Sciences and Arts) in Zagreb, Croatia since 1923, corresponding member of
the SANU (Serbian Academy of Sciences and Arts) in Belgrade since 1930 (1931). 1954 he received the highest award for research in Slovenia the Prešeren award. The same year he was elected for
corresponding member of Bavarian Academy of Sciences in Munich.
1963 for his 90th anniversary University of Ljubljana granted him title of the honorary doctor. Plemelj was first teacher of mathematics at Slovene university and 1949 became first honorary member of
ZDMFAJ, (Yugoslav Union of societies of mathematicians, physicists and astronomers). He left his villa in Bled to DMFA where today is his memorial room.
Plemelj did not do extra preparation for lectures; he didn't have any notes. He used to say that he thought over the lecture subject on the way from his home in Gradišče to the University. Students
are said to have got the impression that he was creating teaching material on the spot and that they were witnessing the formation of something new. He was writing formulae on the table beautifully
although they were composited from Greek, Latin or Gothic letters. He requested the same from students. They had to write distinct.
Plemelj is said to have had very refined ear for language and he had made a solid base for the development of the Slovene mathematical terminology. He had accustomed students for a fine language and
above all for a clear and logical phraseology. For example, he would become angry if they used the word 'rabiti' (to use) instead of the word 'potrebovati' (to need). For this reason he said: "The
engineer who does not know mathematics never needs it. But if he knows it, he uses it frequently".
Happy were they who had an opportunity to listen him during his lectures... His work lives on. And when all mathematical problems will be solved it shall still live...
See also
External links
• Plemelj's students: http://hcoonce.math.mankato.msus.edu/html/id.phtml?id=20359
1. Josip Plemelj, Iz mojega življenja in dela (From my life and work), Obzornik mat. fiz. 39 (1992) pp 188-192. | {"url":"http://www.reference.com/browse/cyclometric","timestamp":"2014-04-17T05:14:00Z","content_type":null,"content_length":"100185","record_id":"<urn:uuid:3a7d58e5-06bd-48ba-b1dc-fe8384c48351>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00076-ip-10-147-4-33.ec2.internal.warc.gz"} |
Teaching Textbooks
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I've heard that some of your products don't have automated grading. If so, which ones?
A. This is true. The Math 3, 4, 5, 6, and 7, along with Pre-Algebra (version 2.0), Algebra 1 (version 2.0), Algebra 2 (version 2.0), and Geometry (version 2.0) all have automated grading, but our
Pre-Calculus program does not. In case you are unfamiliar with our curriculum, the products with automated grading work like this: students watch a lesson on the computer and then start on the
assignment. As soon as they finish working a problem (by hand) they type their answer into the computer. The computer will then grade the problem and tell them if they got it right or wrong. The
grade for that problem is also stored in a digital gradebook. Once the problem has been graded, the program asks students if they want to view the solution. Students who click ‘yes,’ will then be
shown a step-by-step audiovisual solution to that problem.
Our one program that does not have automated grading, Pre-Calculus, works more like a traditional textbook with CD accompaniment. Students can either listen to the lecture on the CD or read a
printed lesson out of the book. Next, they will work out the problems in the book on a sheet of notebook paper, grade them with the print answer key, and then, if needed, go to the CDs for the
audiovisual solution to any problems that were missed. Our Pre-Calculus program does not record grades either.
Will your new 2.0 books work with the older CDs?
A: The 2.0 texts are entirely new editions, so they do not line up with the 1.0 CDs. The 2.0 CDs will also not work with the 1.0 texts. However, if you need replacement parts for the 1.0 version of
Pre-Algebra, Algebra 1, and Algebra 2, just call our toll-free number at 1-866-867-6284 (while supplies last).
How do I know if my child is ready for Pre-Algebra?
A. The best way to tell is to take our Pre-Algebra placement test. If your child is unable to take the placement test at this time, then you should know that if a student has had some exposure to
basic math and can handle the concept of using a letter (usually an x) to represent a missing number in an equation, then he or she is probably ready for Pre-Algebra.
The TT Pre-Algebra program includes a complete review of basic math (whole numbers, fractions, decimals, percents, units of measurement, etc.), but it explains things more conceptually than books
for younger students. This approach teaches students why the techniques they've been using for years really work. Knowing the whys is necessary preparation for Algebra 1 and other higher math
How do I know if my child is ready for Algebra 1?
A. The best way to tell is to take our Algebra 1 placement test. If you don't have time for this, then you should know that if a child is 13 or older and knows basic math fairly well, he is
probably ready for Algebra 1. To be more specific, if the student can easily add, subtract, multiply, and divide whole numbers and has made average grades on assignments involving fractions,
decimals, and percents, it’s time to move to our Algebra 1 program (especially since the book starts by assuming that the student has no knowledge of algebra). If the student is 12 or younger or
still needs work on basic math, then he or she should probably start with the Pre-Algebra Teaching Textbook™ (Pre-Algebra placement test).
How do I know if my child is ready for Algebra 2?
A. The best way to tell is to take our Algebra 2 placement test.
How do I know if my child is ready for Geometry?
A. The best way to tell is to take our Geometry placement test.
I'm no computer whiz, are the Teaching Textbook™ CDs easy to use?
A. Yes. The engineers who created our software made it as easy-to-use as a videotape. That means when you stick the disc in, it will just pop up and play. The product is so intuitive it doesn't
even come with instructions. There's no downloading or passwords or any of that stuff. Most important you won't have to worry about sitting on the line for hours with tech support.
[If you have cable or DSL, click here to see a product demonstration.]
The program does not automatically appear on the screen after I insert the CD. What should I do?
A. For Windows users, please go to "My Computer", double-click on the CD-ROM drive, and double-click on the file that has the ".exe" extension or icon that looks like this:
For Mac users, please double-click on the icon that looks like the above.
Are Teaching Textbooks™ MAC-compatible?
A. All currently sold products are compatible with MAC OS 10.4 through 10.7 (Lion). However, older software versions of Math 5, Math 6, & Math 7 (the serial number starts MATH5-1XXXX-) are NOT
MAC-compatible. Also, our 1st editions (version 1.0) of Pre-Algebra, Algebra 1, Algebra 2, Geometry, & Pre-Calculus are not 10.7 compatible, BUT we do have a 10.7 patch for each of those (see link
below). DO NOT download the patch if you currently own a 2.0 version of Algebra 1 or Pre-Algebra. The 2.0 versions of these products are the ones that have a digital gradebook. Questions? Call and
we'll be happy to walk you through any difficulties (toll free: 866-867-6284).
What are the system requirements for the CDs?
For Pre-Algebra v1.0, Algebra 1 v1.0, Algebra 2, Geometry, and Pre-Calculus:
The system requirements for PCs are
• A CPU of 133 MHz or faster
• A Windows 98 or later operating system
• A 4x CD-ROM drive
• Speakers
The system requirements for MAC are
• A CPU of 211 MHz or faster
• A MAC OS 9.0 or later (but with 10.7 (Lion), you must first download a patch from the page that has all the info. for that product)
• A 4x CD-ROM drive
• Speakers
For Math 3, Math 4, Math 5, Math 6, Math 7, Pre-Algebra v2.0, and Algebra 1 v2.0:
The system requirements for PCs are
• A CPU of 1.0 GHz or faster (2.0 Ghz recommended)
• A Windows XP or later operating system
• 1 GB RAM (2GB recommended)
• A 4x CD-ROM drive
• Speakers
The system requirements for Mac are
• A CPU Processor type of: G3 PowerPC / Intel Solo
• Processor Speed of: 500 MHz / 1.0 GHz or faster (2.0 Ghz recommended)
• Mac OS 10.4 or later
• 1 GB RAM (2 GB recommended)
• A 4x CD-ROM drive
How do I install my math program with parental controls (limited or standard account)?
A. You need to first set up the user account as a computer administrator, then you can install the program under that user name and activate it. Next, just change the account back to the standard
or limited account.
Can I use your program for more than one child?
A. Absolutely. These programs are reusable and can be used with more than one student on the same computer in the same year. For instructions on how to set up multiple student accounts if your
serial number starts with a 1, please call 1-866-867-6284 or email us at customerservice@teachingtextbooks.com. If your serial number starts with a 2, the program will ask you how many student
accounts you want to set up whenever you're going through the standard installation process. You can have up to 3 students on one computer and it will track their grades separately.
I am using a Mac and the buddies have disappeared and it just gives me a white screen when I try to login to the parent area.
A. This is normally caused by pulling the icon out of the “Applications\Teaching Textbook\Math x” folder and moving it to the desktop. You will need to put the icon back in order for the program to
work correctly. If you would like more information on how to put a shortcut, for the program, in your dock, please contact Customer Service at 1-866-867-6284 or email
Does your program work with anti-virus programs?
A. Yes, but you may need to add the program into the trusted application and exclusions, in order to keep the anti-virus program from possibly deleting the data file that stores the grades. In
order to do that you may have to contact the anti-virus company for help in adding our program(s) into theirs. | {"url":"http://www.teachingtextbooks.com/v/vspfiles/tt/FAQs.htm","timestamp":"2014-04-17T21:22:51Z","content_type":null,"content_length":"42098","record_id":"<urn:uuid:95517d57-9ac0-4cbc-aac5-417d61634144>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00321-ip-10-147-4-33.ec2.internal.warc.gz"} |
Buckeye, AZ Algebra 1 Tutor
Find a Buckeye, AZ Algebra 1 Tutor
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Buckeye, AZ Trigonometry Tutors | {"url":"http://www.purplemath.com/buckeye_az_algebra_1_tutors.php","timestamp":"2014-04-18T23:45:05Z","content_type":null,"content_length":"24087","record_id":"<urn:uuid:386ef9bf-ffe0-468c-bb27-c74ddf4dd1b2>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00369-ip-10-147-4-33.ec2.internal.warc.gz"} |
At the start of the German Year of Mathematics, the Oberwolfach research institute has released an exhibition and the software they used to produce it. The software, surfer, is a really nice GUI that
sits on top of surf and lets you rotate and zoom your algebraic surfaces as well as pick colours very comfortably.
They have a whole bunch of Really Pretty Images at the exhibition website, and I warmly recommend a visit. If you can get hold of the exhibition, they also have produced real models – with a
3d-printer – of some of the snazzier surfaces, so that one could have a REALLY close encounter with them.
But also, I’d really like to show you some of my own minor experiments with the program.
This is the interior of a Klein Bottle, using the “standard” realization as an algebraic surface given by Mathworld. In other words, I’m using
for the defining equation. It kinda looks a bit like a Sousaphone in my opinion. | {"url":"http://blog.mikael.johanssons.org/archive/category/jahr-der-mathematik/","timestamp":"2014-04-16T04:11:12Z","content_type":null,"content_length":"31952","record_id":"<urn:uuid:02e2dbbb-e53c-4482-ae00-3258d9404253>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00611-ip-10-147-4-33.ec2.internal.warc.gz"} |
Six Sigma Levels
In Statistics, Sigma represents the Standard Deviation which measures how much variation a process shows from the perfection. The calculation is based on the number of the defects occurring per
million of opportunities.
DPMO is Defects per million opportunities
DPMO = (Number of defects)*1000000/(Number of units)/(Number of opportunities for defects per unit
The relationship between the sigma level and DPMO clearly suggests the efficiency of the process. Higher sigma level means the lower value for the DPMO and increased levels of process efficiency.
Vice Versa lower sigma level means higher value for the DPMO and decreased levels of process efficiency. Therefore a process should aims to achieve the higher Sigma Levels.
Relationship between Sigma Levels and DPMO is as follows
• One sigma = 690,000 DPMO = 31% efficiency
• Two sigma = 308,000 DPMO = 69.2% efficiency
• Three sigma = 66,800 DPMO = 93.32% efficiency
• Four sigma = 6,210 DPMO = 99.379% efficiency
• Five sigma = 230 DPMO = 99.977% efficiency
• Six sigma = 3.4 DPMO = 99.9997% efficiency
The performance of the processes can be compared throughout an entire organization by determining the sigma levels of the processes. Determining the sigma levels is independent of the process, as it
calculates only opportunities and defects. Different processes may differ in number of opportunities for making mistakes and different number of units produced, however by comparing the DPMO of
these processes we can make out which process is performing better. | {"url":"http://www.sixsigma.in/six-sigma-levels.html","timestamp":"2014-04-17T00:48:07Z","content_type":null,"content_length":"13741","record_id":"<urn:uuid:6ac24241-2923-4f93-8e95-af24cc85d9fe>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00456-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability, that a graph G does not contain a cycle
up vote 4 down vote favorite
Hello, given graph $G=(V,E)$ with $n=|V|$ and $k=|E|$, what is the probability that it does not contain any cycle $C_l$ for $l\geq3?$
The requested clarification: My intention was to form the question in such a way, that there is no information about any distribution of the edges, and n and k are parameters. This lack of
information should be in fact the information. You construct graphs in any possible ways, and you have to decide which constructs are more possible to occur and which are less expected. This probably
implies the uniform distribution.
graph-theory random-graphs
5 You need to specify how you are choosing your random graph, otherwise this isn't a well-posed question. Do you mean over the uniform distribution over all graphs with n vertices and k edges? –
Steve Flammia Mar 2 '11 at 0:44
4 If $k\ge n$ then the graph is guaranteed to contain a cycle. Steve is right - this question needs work. – Gerry Myerson Mar 2 '11 at 1:02
Usually you specify that given n vertices , you have a protocol to choose edges. For example on each doubleton of vertices I pick an edge with a uniform probability p (it may depend on n or not).
If p =1/2 , the probability is rather low anyway. – Jérôme JEAN-CHARLES Mar 2 '11 at 1:45
add comment
2 Answers
active oldest votes
I will assume the uniform distribution on all (labelled) graphs with $n$ vertices and $k$ edges. An acyclic graph on $n$ vertices has at most $n-1$ edges, so let me further assume $k<
n$. More precisely, $k=n-c$, where $c$ is the number of connected components, i.e., the graph is acyclic iff it is union of $c$ disjoint trees. Now, the number of trees with $m$
vertices is $m^{m-2}$ by Cayley’s formula, hence the requested probability is
$$p_{n,k}=\frac1{(n-k)!\binom{\binom n2}k}\sum_{\substack{n_1+\cdots+n_{n-k}=n\\n_1,\dots,n_{n-k}>0}}\binom n{n_1\,\dots\,n_{n-k}}\prod_{i=1}^{n-k}n_i^{n_i-2}.$$
up vote 20 down
vote accepted We have $\binom{\binom n2}k\approx\left(\frac{en^2}{2k}\right)^kk^{-1/2}$ by Stirling’s approximation (where $f\approx g$ means $c_1f\le g\le c_2f$ for some positive constants
$c_1,c_2$), hence in the simplest case $k=n-1$,
$$p_{n,n-1}\approx\frac1{\sqrt n}\left(\frac2e\right)^n.$$
add comment
I will assume what seems the more natural question: uniform probability over all labeled graphs with exactly $k$ edges and $n$ vertices (rather than on isomorphism classes of graphs). The
number of labeled forests with $k$ edges on $n$ vertices is the integer sequence http://oeis.org/A138464 which doesn't seem to indicate that a closed form expression is known. To obtain
up vote 5 the probability divide this number by (($n$ choose 2) choose $k$).
down vote
Good - but it's not our job to write the question for the OP. Let's hope we get some clarification on what OP wants. – Gerry Myerson Mar 2 '11 at 12:01
add comment
Not the answer you're looking for? Browse other questions tagged graph-theory random-graphs or ask your own question. | {"url":"http://mathoverflow.net/questions/57062/probability-that-a-graph-g-does-not-contain-a-cycle","timestamp":"2014-04-20T13:26:29Z","content_type":null,"content_length":"59824","record_id":"<urn:uuid:ae059d57-056c-4164-acda-4def54e5bb9a>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00582-ip-10-147-4-33.ec2.internal.warc.gz"} |
More C++ Worries.
More C++ Worries.
Can anyone help me with this extra credit problem?
e^x= 1 +(x/1) + (x^2/2) + (x^3/6) = (x^4/24)...X^n/n!
I have to read in x from the keyboard and use the exponential function on x and write the result.
Then do the approximation for the first four terms, starting at 1 and write the results.
Then write the difference between the two.
Well... What's n?
You could just use a loop - i think that'd be the easiest way... Have you made any headway on this since yesterday?
Still Baffeled
n is just the number of what power we raised it to. For example X^2/2!.
I know I need the following libraries
I am so glade you are there because this is due in a couple of hourse and I am TOTALLY stressed out!!!;)
Well, to do this...
To multiply a number and an exponent you just do this:
int number1 = 2;
int number2 = 3;
int result = 0;
result = number1 * exp(number2); // exp(...) is for exponent
cout << result; | {"url":"http://cboard.cprogramming.com/cplusplus-programming/11857-more-cplusplus-worries-printable-thread.html","timestamp":"2014-04-21T03:07:48Z","content_type":null,"content_length":"7655","record_id":"<urn:uuid:1f577474-cb45-4fe2-bc49-bf347e766b53>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00168-ip-10-147-4-33.ec2.internal.warc.gz"} |
Pennsylvania $275 Million Cash Extravaganza Instant Lottery Game
[35 Pa.B. 3473]
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(TWYEGT), 29 (TWYNIN), 30 (THIRTY), 31 (THYONE), 32 (THYTWO), 33 (THYTHR), 34 (THRFOR), 35 (THYFIV), 36 (THYSIX), 37 (THYSVN), 38 (THYEGT), 39 (THYNIN) and 40 (FORTY). The play symbols and their
captions located in the ''YOUR NUMBERS'' area are: 1 (ONE), 2 (TWO), 3 (THREE), 4 (FOUR), 6 (SIX), 7 (SEVEN), 8 (EIGHT), 9 (NINE), 10 (TEN), 11 (ELEVN), 12 (TWLV), 13 (THRTN), 14 (FORTN), 15
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(g) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a Gold Bar Symbol (GBAR), and a prize symbol of $500 (FIV HUN) appears under the Gold Bar Symbol (GBAR) on a
single ticket, shall be entitled to a prize of $1,000.
(h) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a 5X Symbol (5TIMES), and a prize symbol of $200 (TWO HUN) appears under the 5X Symbol (5TIMES) on a single
ticket, shall be entitled to a prize of $1,000.
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''YOUR NUMBERS'' play symbol, on a single ticket, shall be entitled to a prize of $500.
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(p) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a 5X Symbol (5TIMES), and a prize symbol of $40$ (FORTY) appears under the 5X Symbol (5TIMES) on a single ticket,
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''YOUR NUMBERS'' play symbol, on a single ticket, shall be entitled to a prize of $100.
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(s) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a 5X Symbol (5TIMES), and a prize symbol of $20$ (TWENTY) appears under the 5X Symbol (5TIMES) on a single
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single ticket, shall be entitled to a prize of $50.
(x) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols matches any of the ''WINNING NUMBERS'' play symbols and a prize symbol of $40$ (FORTY) appears under the matching
''YOUR NUMBERS'' play symbol, on a single ticket, shall be entitled to a prize of $40.
(y) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a Safe Symbol (SAFE), and a prize symbol of $40$ (FORTY) appears under the Safe Symbol (SAFE) on a single ticket,
shall be entitled to a prize of $40.
(z) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a Gold Bar Symbol (GBAR), and a prize symbol of $20$ (TWENTY) appears under the Gold Bar Symbol (GBAR) on a
single ticket, shall be entitled to a prize of $40.
(aa) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols matches any of the ''WINNING NUMBERS'' play symbols and a prize symbol of $25$ (TWY FIV) appears under the matching
''YOUR NUMBERS'' play symbol, on a single ticket, shall be entitled to a prize of $25.
(bb) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a Safe Symbol (SAFE), and a prize symbol of $25$ (TWY FIV) appears under the Safe Symbol (SAFE) on a single
ticket, shall be entitled to a prize of $25.
(cc) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols matches any of the ''WINNING NUMBERS'' play symbols and a prize symbol of $20$ (TWENTY) appears under the matching
''YOUR NUMBERS'' play symbol, on a single ticket, shall be entitled to a prize of $20.
(dd) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a Safe Symbol (SAFE), and a prize symbol of $20$ (TWENTY) appears under the Safe Symbol (SAFE) on a single
ticket, shall be entitled to a prize of $20.
(ee) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols is a Gold Bar Symbol (GBAR), and a prize symbol of $10^.00 (TEN DOL) appears under the Gold Bar Symbol (GBAR) on a
single ticket, shall be entitled to a prize of $20.
(ff) Holders of tickets upon which any one of the ''YOUR NUMBERS'' play symbols matches any of the ''WINNING NUMBERS'' play symbols and a prize symbol of $10^.00 (TEN DOL) appears under the
matching ''YOUR NUMBERS'' play symbol, on a single ticket, shall be entitled to a prize of $10.
8. Number and Description of Prizes and Approximate Odds: The following table sets forth the approximate number of winners, amounts of prizes and approximate odds of winning:
When Any of Your Numbers Approximate
Match Any of the Winning Approximate No. Winners Per
Numbers, Win With Prize (s) of: Win Odds of 1 In: 19,200,000 Tickets
$10 × 2 $20 30 640,000
$10 w/Gold Bar $20 30 640,000
$20 w/Safe $20 30 640,000
$20 $20 30 640,000
$25 w/Safe $25 30 640,000
$25 $25 30 640,000
$10 × 4 $40 75 256,000
$20 × 2 $40 75 256,000
$20 w/Gold Bar $40 75 256,000
$40 w/Safe $40 75 256,000
$40 $40 75 256,000
$10 × 5 $50 150 128,000
$25 w/Gold Bar $50 150 128,000
$10 w/5X $50 150 128,000
$50 w/Safe $50 150 128,000
$50 $50 150 128,000
$10 × 10 $100 250 76,800
$20 × 5 $100 250 76,800
$20 w/5X $100 250 76,800
$50 w/Gold Bar $100 250 76,800
$100 $100 300 64,000
$10 × 20 $200 1,600 12,000
$20 × 10 $200 1,600 12,000
$25 × 8 $200 1,600 12,000
$40 w/5X $200 1,600 12,000
$100 w/Gold Bar $200 1,600 12,000
$100 × 2 $200 1,600 12,000
$200 w/Safe $200 1,600 12,000
$200 $200 1,600 12,000
$25 × 20 $500 4,800 4,000
$25 × 10 + $50 × 5 $500 4,800 4,000
$50 × 10 $500 4,800 4,000
$100 × 5 $500 4,800 4,000
$200 w/Gold Bar + $100 $500 4,800 4,000
$100 w/5X $500 4,800 4,000
$500 w/Safe $500 4,800 4,000
$500 $500 4,800 4,000
$50 × 20 $1,000 12,000 1,600
$50 × 10 + $100 × 5 $1,000 15,000 1,280
$100 × 10 $1,000 15,000 1,280
$200 × 5 $1,000 15,000 1,280
$200 w/5X $1,000 15,000 1,280
$500 × 2 $1,000 15,000 1,280
$500 w/Gold Bar $1,000 15,000 1,280
1,000 w/Safe $1,000 20,000 960
$1,000 $1,000 20,000 960
$200 × 10 + $500 × 5 + $100 × 5 $5,000 60,000 320
$500 × 10 $5,000 60,000 320
$1,000 × 5 $5,000 60,000 320
$1,000 w/5X $5,000 60,000 320
$5,000 $5,000 120,000 160
$20,000 $20,000 120,000 160
$1,000,000 $1,000,000 960,000 20
Safe = Win prize automatically.
Gold Bar = Win double the prize shown.
5X = Win 5 times the prize shown.
Prizes, including top prizes, are subject to availability at the time of purchase.
9. Retailer Incentive Awards: The Lottery may conduct a separate Retailer Incentive Game for retailers who sell Pennsylvania $275 Million Cash Extravaganza instant lottery game tickets. The
conduct of the game will be governed by 61 Pa. Code § 819.222 (relating to retailer bonuses and incentives).
10. Unclaimed Prize Money: For a period of 1 year from the announced close of Pennsylvania $275 Million Cash Extravaganza, prize money from winning Pennsylvania $275 Million Cash Extravaganza
instant lottery game tickets will be retained by the Secretary for payment to the persons entitled thereto. If no claim is made within 1 year of the announced close of the Pennsylvania $275
Million Cash Extravaganza instant lottery game, the right of a ticket holder to claim the prize represented by the ticket, if any, will expire and the prize money will be paid into the State
Lottery Fund and used for purposes provided for by statute.
11. Governing Law: In purchasing a ticket, the customer agrees to comply with and abide by the State Lottery Law, 61 Pa. Code Part V (relating to State Lotteries) and the provisions contained in
this notice.
12. Termination of the Game: The Secretary may announce a termination date, after which no further tickets from this game may be sold. The announcement will be disseminated through media used to
advertise or promote Pennsylvania $275 Million Cash Extravaganza or through normal communications methods.
GREGORY C. FAJT,
[Pa.B. Doc. No. 05-1188. Filed for public inspection June 17, 2005, 9:00 a.m.]
No part of the information on this site may be reproduced for profit or sold for profit.
This material has been drawn directly from the official Pennsylvania Bulletin full text database. Due to the limitations of HTML or differences in display capabilities of different browsers, this
version may differ slightly from the official printed version. | {"url":"http://www.pabulletin.com/secure/data/vol35/35-25/1188.html","timestamp":"2014-04-19T06:51:36Z","content_type":null,"content_length":"25561","record_id":"<urn:uuid:66f534ad-94f9-4da2-9e13-47cdc5dc071a>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00445-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Author Message
gad49 Posted: Saturday 22nd of Sep 09:28
I think God would have been in a really bad mood that he came up with something called algebra to trouble us! I’ve spent hours trying to figure out a solution to this algebra problem
which relates to what is easy in algebra? and I still can’t solve it. I’m particularly stuck on trinomials, reducing fractions and inequalities. Can anyone throw some light on how to go
about solving such problems? I’ve tried various means that I could think of, but none helped. I need some urgent help now. Anybody?
espinxh Posted: Sunday 23rd of Sep 18:36
Hey pal! Learning what is easy in algebra? online can be a disaster if you are not a pro at it. I wasn’t an expert either and really regretted my decision until I found Algebrator .
This little program has been my partner since then. I’m easily able to solve the questions now.
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I myself have been using this software since a year now, and it has never let me down. It won’t just solve a question for you, but it’ll also explain every step that was taken to arrive
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alhatec16 Posted: Monday 24th of Sep 14:55
Algebrator is the program that I have used through several algebra classes - College Algebra, College Algebra and Algebra 2. It is a really a great piece of math software. I remember of
going through problems with subtracting exponents, simplifying fractions and perpendicular lines. I would simply type in a problem from the workbook, click on Solve – and step by step
solution to my math homework. I highly recommend the program.
From: Notts, | {"url":"http://www.softmath.com/algebra-software-4/what-is-easy-in-algebra.html","timestamp":"2014-04-21T04:38:42Z","content_type":null,"content_length":"34144","record_id":"<urn:uuid:09856807-555c-488b-928d-6525ca0c6530>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00394-ip-10-147-4-33.ec2.internal.warc.gz"} |
Automatic sampling apparatus
An automatic sampling apparatus for individually obtaining samples from a plurality of receptacles in a rectangular array, includes a sample table for supporting a rack containing the array of
receptacles, and a sampling device movable in two dimensions above said sample table and receptacles. Further, a control device is provided for receiving positional data from the sampling device in a
teaching mode representing the two-dimensional position of the sampling device above the sample table, and for calculating the pitch spacing between receptacles in a single row, and row spacing
between adjacent rows of receptacles, wherein the pitch spacing is calculated by arithmetic division of the length of one row by the number of receptacles in a row minus one, and the row spacing is
calculated by arithmetic division of the distance between farthest rows by the number of rows minus one. Additionally, the control device is provided for controlling the sample device to individually
obtain samples in a stepping manner from the receptacles based on the calculated pitch and row spacings. An operation panel, operatively coupled with the control device, has input keys to position
the sampling device for the teaching mode by an operator.
Inventors: Nishimura; Takashi (Kyoto, JP)
Assignee: Shimadzu Corporation (Kyoto, JP)
Appl. No.: 06/883,807
Filed: July 9, 1986
Current U.S. Class: 700/64 ; 422/505; 422/67; 73/863.01; 73/864.25
Current International Class: G01N 1/00 (20060101); G01N 35/00 (20060101); G06F 015/46 (); G01N 035/06 ()
Field of Search: 364/167,496-500,478,479 422/50,62-67,75,99,100 436/47,48,50,55 73/864.24,864.25,863.91,863.92,863.01 | {"url":"http://patents.com/us-4757437.html","timestamp":"2014-04-21T02:01:01Z","content_type":null,"content_length":"33283","record_id":"<urn:uuid:6b495250-acc3-473c-90ce-4187a2aa2e01>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00531-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Perrin: Theory of Codes
Results 1 - 10 of 156
- HANDBOOK OF FORMAL LANGUAGES , 1997
"... ..."
, 1998
"... We prove that a sequence is primitive substitutive if and only if the set of its derived sequences is finite; we defined these sequences here. ..."
Cited by 59 (7 self)
Add to MetaCart
We prove that a sequence is primitive substitutive if and only if the set of its derived sequences is finite; we defined these sequences here.
, 1990
"... We define the nondeterministic complexity of a finite automaton and show that there exist, for any integer p>=1, automata which need \Theta(k^{1/p}) nondeterministic transitions to spell words
of length k. This leads to a subdivision of the family of recognizable M-subsets of a free monoid into a hi ..."
Cited by 28 (2 self)
Add to MetaCart
We define the nondeterministic complexity of a finite automaton and show that there exist, for any integer p>=1, automata which need \Theta(k^{1/p}) nondeterministic transitions to spell words of
length k. This leads to a subdivision of the family of recognizable M-subsets of a free monoid into a hierarchy whose members are indexed by polynomials, where M denotes the Min--Plus semiring.
, 1999
"... this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund. ..."
Cited by 26 (9 self)
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this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund.
- Computers and Mathematics with Applications 47 , 2004
"... Codes play an important role in the study of combinatorics on words. Recently, we introduced pcodes that play a role in the study of combinatorics on partial words. Partial words are strings
over a finite alphabet that may contain a number of “do not know ” symbols. In this paper, the theory of code ..."
Cited by 22 (8 self)
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Codes play an important role in the study of combinatorics on words. Recently, we introduced pcodes that play a role in the study of combinatorics on partial words. Partial words are strings over a
finite alphabet that may contain a number of “do not know ” symbols. In this paper, the theory of codes of words is revisited starting from pcodes of partial words. We present some important
properties of pcodes. We give several equivalent definitions of pcodes and the monoids they generate. We investigate in particular the Defect Theorem for partial words. We describe an algorithm to
test whether or not a finite set of partial words is a pcode. We also discuss two-element pcodes, complete pcodes, maximal pcodes, and the class of circular pcodes. A World Wide Web server interface
has been established at
- In: Theoretical Computer Science , 2002
"... The computation language of a DNA-based system consists of all the words (DNA strands) that can appear in any computation step of the system. In this work we define properties of languages which
ensure that the words of such languages will not form undesirable bonds when used in DNA computations ..."
Cited by 18 (8 self)
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The computation language of a DNA-based system consists of all the words (DNA strands) that can appear in any computation step of the system. In this work we define properties of languages which
ensure that the words of such languages will not form undesirable bonds when used in DNA computations. We give several characterizations of the desired properties and provide methods for obtaining
languages with such properties. The decidability of these properties is addressed as well. As an application we consider splicing systems whose computation language is free of certain undesirable
bonds and is generated by nearly optimal comma-free codes. 1 Introduction DNA (deoxyribonucleic acid) is found in every cellular organism as the storage medium for genetic information. It is composed
of units called nucleotides, distinguished by the chemical group, or base, attached to them. The four bases, are adenine, guanine, cytosine and thymine, abbreviated as A, G, C, and T . (The names of
- International Conference on Semigroups and Groups in honor of the 65th birthday of Prof , 2004
"... We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We define the groups as partial transformation groups and we give a faithful
representation in the Cuntz C ⋆-algebra. For the finitely presented simple group Tfin we show that the word-lengt ..."
Cited by 14 (7 self)
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We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We define the groups as partial transformation groups and we give a faithful
representation in the Cuntz C ⋆-algebra. For the finitely presented simple group Tfin we show that the word-length and the table size satisfy an n log n relation, just like the symmetric groups. We
show that the word problem of Tfin belongs to the parallel complexity class AC 1 (a subclass of P). We show that the generalized word problem of Tfin is undecidable. We study the distortion functions
of Tfin and we show that Tfin contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions,
the following three sets are the same: the set of distortions of Tfin, the set of all Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing
machines. 1
- In Preliminary Proceedings of 10th International Workshop on DNA-Based Computers, DNA 2004 (University of Milano-Bicocca , 2004
"... A very basic problem in all DNA computations is nding a good encoding. Apart from the fact that they must provide a solution, the strands involved should not exhibit any undesired behaviour like
forming secondary structures. ..."
Cited by 14 (1 self)
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A very basic problem in all DNA computations is nding a good encoding. Apart from the fact that they must provide a solution, the strands involved should not exhibit any undesired behaviour like
forming secondary structures.
, 1997
"... Classically, several properties and relations of words, such as "being a power of a same word", can be expressed by using word equations. This paper is devoted to study in general the expressive
power of word equations. As main results we prove theorems which allow us to show that certain properties ..."
Cited by 13 (5 self)
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Classically, several properties and relations of words, such as "being a power of a same word", can be expressed by using word equations. This paper is devoted to study in general the expressive
power of word equations. As main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations. In particular,
"the primitiveness" and "the equal length" are such properties, as well as being "any word over a proper subalphabet".
- IEEE Trans. Inform. Theory , 2001
"... Abstract—a variable-length code is a fix-free code if no codeword is a prefix or a suffix of any other codeword. This class of codes is applied to speed up the decoding process, for the decoder
can decode from both sides of the compressed file simultaneously. In this paper, we study some basic prope ..."
Cited by 11 (1 self)
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Abstract—a variable-length code is a fix-free code if no codeword is a prefix or a suffix of any other codeword. This class of codes is applied to speed up the decoding process, for the decoder can
decode from both sides of the compressed file simultaneously. In this paper, we study some basic properties of fix-free codes. We prove a sufficient and a necessary condition for the existence of
fix-free codes, and we obtain some new upper bounds on the redundancy of optimal fix-free codes. Index Terms—Fix-free code, prefix code, redundancy. I. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=257607","timestamp":"2014-04-17T07:43:10Z","content_type":null,"content_length":"34104","record_id":"<urn:uuid:8975fe1f-322e-4b84-b67d-35ed2a6ee596>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00608-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Algebra Solver Demos
Author Message
nadyc Posted: Sunday 19th of Nov 07:08
Hi, This morning I began working on my math assignment on the topic Intermediate algebra. I am currently not able to complete the same since I am unfamiliar with the fundamentals of
binomial formula, decimals and quadratic formula. Would it be possible for anyone to assist me with this?
From: Cyprus -
espinxh Posted: Monday 20th of Nov 17:12
Algebrator is one of the best resources that can render help to people like you. When I was a novice, I took support from Algebrator . Algebrator covers all the basics of Remedial
Algebra. Rather than using the Algebrator as a step-by-step guide to solve all your homework assignments, you can use it as a tutor that can offer the basics of dividing fractions,
multiplying matrices and binomials. Once you understand the basics, you can go ahead and solve any tough assignments on Algebra 2 in no time.
From: Norway
LifiIcPoin Posted: Tuesday 21st of Nov 08:35
I used Algebrator also, especially in Remedial Algebra. It helped me so much, and you won't believe how uncomplicated it is to use! It solves the exercise and it also explains
everything step by step. Better than a teacher!
From: Way Way
Vild Posted: Wednesday 22nd of Nov 07:55
Algebrator is a user friendly product and is surely worth a try. You will find lot of interesting stuff there. I use it as reference software for my math problems and can swear that it
has made learning math much more enjoyable.
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Gocon G Posted: Thursday 23rd of Nov 07:44
Recommended by teachers! I must say this tool sounds really interesting. Can I use it once?
From: San
Diego, CA
Techei-Mechial Posted: Friday 24th of Nov 20:33
I'm sorry. I should have included the connection our first time around: http://www.softmath.com/algebra-policy.html. I don't have any knowledge about a test copy, but the recognized
sellers of Algebrator , as opposed to some suppliers of imitation software, put up an entire satisfaction guarantee. Hence, you can order the official copy, test the package and send
it back if one is not gratified by the performance and functionality. Even though I think you are gonna love this program, I am very interested in learning from anyone should there be
something for which the software doesn't excel. I don't desire to recommend Algebrator for something it cannot do. Only the next one discovered will likely be the first one! | {"url":"http://softmath.com/algebra-software/math-soft/how-to-cube-root-on-ti-83.html","timestamp":"2014-04-18T03:10:46Z","content_type":null,"content_length":"99583","record_id":"<urn:uuid:c277762b-7565-49c1-88ba-7dfe3d439d18>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00430-ip-10-147-4-33.ec2.internal.warc.gz"} |
Geographic tracks from starting and ending points
[lat,lon] = track2(lat1,lon1,lat2,lon2)
[lat,lon] = track2(lat1,lon1,lat2,lon2,ellipsoid)
[lat,lon] = track2(lat1,lon1,lat2,lon2,units)
[lat,lon] = track2(lat1,lon1,lat2,lon2,ellipsoid,units)
[lat,lon] = track2(lat1,lon1,lat2,lon2,ellipsoid,units,npts)
[lat,lon] = track2(track,...)
mat = track2(...)
[lat,lon] = track2(lat1,lon1,lat2,lon2) computes great circle tracks on a sphere starting at the point lat1,lon1 and ending at lat2,lon2. The inputs can be scalar or column vectors.
[lat,lon] = track2(lat1,lon1,lat2,lon2,ellipsoid) computes the great circle track on the ellipsoid defined by the input ellipsoid. ellipsoid is a referenceSphere, referenceEllipsoid, or
oblateSpheroid object, or a vector of the form [semimajor_axis eccentricity]. If ellipsoid = [], a sphere is assumed.
[lat,lon] = track2(lat1,lon1,lat2,lon2,units) and
[lat,lon] = track2(lat1,lon1,lat2,lon2,ellipsoid,units) are both valid calling forms, which use the input string units to define the angle units of the inputs and outputs. If the input string units
is omitted, 'degrees' is assumed.
[lat,lon] = track2(lat1,lon1,lat2,lon2,ellipsoid,units,npts) uses the scalar input npts to determine the number of points per track computed. The default value of npts is 100.
[lat,lon] = track2(track,...) uses the track string to define either a great circle or a rhumb line track. If track = 'gc', then great circle tracks are computed. If track = 'rh', then rhumb line
tracks are computed. If the track string is omitted, 'gc' is assumed.
mat = track2(...) returns a single output argument where mat = [lat lon]. This is useful if a single track is computed. Multiple tracks can be defined from a single starting point by providing scalar
inputs for lat1,lon1 and column vectors for lat2,lon2.
A path along the surface of the Earth connecting two points is a track. Two types of track lines are of interest geographically, great circles and rhumb lines. Great circles represent the shortest
possible path between two points. Rhumb lines are paths with constant angular headings. They are not, in general, the shortest path between two points.
% Set up the axes.
axesm('mercator','MapLatLimit',[30 50],'MapLonLimit',[-40 40])
% Calculate the great circle track.
[lattrkgc,lontrkgc] = track2(40,-35,40,35);
% Calculate the rhumb line track.
[lattrkrh,lontrkrh] = track2('rh',40,-35,40,35);
% Plot both tracks.
See Also
azimuth | distance | reckon | scircle1 | scircle2 | track | track1 | trackg | {"url":"http://www.mathworks.com/help/map/ref/track2.html?nocookie=true","timestamp":"2014-04-19T15:09:08Z","content_type":null,"content_length":"42322","record_id":"<urn:uuid:2f331129-969f-4fa2-ad88-1d4879bcf5b3>","cc-path":"CC-MAIN-2014-15/segments/1397609537271.8/warc/CC-MAIN-20140416005217-00182-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Astronomical Tests of the Cold Dark Matter Scenario - J.P. Ostriker
Annu. Rev. Astron. Astrophys. 1993. 31: 689-716
Copyright © 1993 by . All rights reserved
3.3. The Velocity Distribution of Galaxies
The prime fact is that our Local Group has a velocity with respect to the CBR of about v[0] ^-1, as directly measured by the dipole anisotropy of the CBR, with modest corrections for Local Group
kinematics (Lubin & Villela 1986). If we interpret this in the conventional manner, it shows the effect of gravitational forces due to large-scale density irregularities, which provide the
acceleration that, integrated over time, has caused our peculiar velocity. CDM models for structure with relatively low amplitude in the perturbation spectrum ([0] < 1) had difficulty producing such
a large velocity but could always appeal to statistical fluctuations: We live, unluckily, in a galaxy which is on the tail of the peculiar velocity distribution. Utilizing the COBE normalization,
this problem is diminished.
The second disturbing fact is that, in the frame of reference at rest with respect to the galaxies on average (the "Local Standard of Rest" frame in the notation of galactic stellar kinematics), the
Local Group has a peculiar velocity v[p] < v[0] and that, furthermore, so do most other galaxies. That is, the flow is quite cold with bulk motion exceeding random motion on all scales where
measurements have been made.
This large coherence of the velocity field has two consequences. First, our velocity is not anomalous; most galaxies have a similar velocity with respect to the CBR. Second, this result is not likely
to be in error, despite the well acknowledged difficulty of measuring proper velocities. The measurements indicate small proper velocities, and it is difficult to imagine systematic errors (or random
ones) which would conspire to indicate small velocities with respect to us over a significant part of the sky if the truth were that there is a large directed velocity that cancels out our large
directed velocity with respect to the CBR. The numerical basis for these remarks was presented in a paper by Groth et al (1989) in the framework of an analysis of the velocity correlation function
(see also Gorski et al 1989).
Now let's address some of the quantitative details. In an exercise of precognition, Bertschinger et al (1990) compared the COBE normalized (i.e. unbiased) CDM model with observed large-scale
velocities. The observed average peculiar velocities (with respect to the CBR) within spheres of 4000 and 6000 km/s centered on the Local Group were found to be 388 ± 67 km s^-1 and 327 ± 82 km s^-1,
only modestly in excess of the predictions (287 km s^-1 and 224 km s^-1 on the same scales). However, this same model predicts a velocity dispersion on small scales which is now much too large.
Simple theory allows one to estimate the one-dimensional pairwise velocity dispersion of galaxies separated by 1 h^-1 Mpc should be v[||, rms] = 970 ± 160 km s^-1 in a standard CDM model (with COBE
normalization [8] = 1.08 ± 0.18), a result far in excess of the observed value 340 ± 40 by Davis & Peebles 1983). Numerical simulations by Cen & Ostriker (1993) and Ueda et al (1993) confirm that the
CDM prediction normalized to COBE is far too high.
It is extremely unlikely that the small-scale velocity dispersion of galaxies is larger than the quoted value by a factor of 2.8 as required to bridge the differences (since unrecognized errors in
observational procedures would have caused too large, not too small, a velocity dispersion to be "measured"). But, can the theoretical normalizations be in error? Is it possible that galaxies suffer
a large enough "velocity bias" with respect to dark matter particles to lower the predicted dispersion greatly from that given by linear theory? Couchman & Carlberg (1992) argue on the basis of
numerical hydro simulations that, on small scales, galaxies could have a considerably smaller velocity dispersion than dark matter particles. But, on the several megaparsec scale, Katz et al (1992)
find little evidence for velocity bias. Cen & Ostriker (1992b), in a detailed hydrodynamic study, which allowed for galaxy formation, found a small effect. On the (2-20) h^-1 Mpc scales, the ratio of
galaxy-to-dark matter proper velocities is expected to be close to 0.80; "velocity bias" is a real but quite small effect on appropriate scales. Thus, the standard CDM model, when normalized to the
COBE-determined amplitude is in serious conflict with the observed pairwise velocity dispersion of galaxies.
Finally, let us examine the velocity field in a way that does not depend on the COBE result. Both the bulk flow <v> and the velocity dispersion ratio <v> / R. Then <v>[R] will measure the
gravitational forces due to regions bigger than R, and [R] measures the fluctuations on scales smaller than R. The Mach number, M(R) v> / [R], thus measures the ratio of large-scale to small-scale
Ostriker & Suto (1990) showed how the Mach number could be calculated from standard linear theory and examined several of the popular cosmological scenarios. As expected, the standard CDM model
predicts a "hot" flow M < 1 on scales hR > 10 Mpc, where the observations, as noted previously, strongly indicate a very cold flow. In a far more detailed study, Strauss et al (1993, references
therein for earlier work) compared detailed numerical reconstructions of a standard CDM universe with the best currently available proper velocity data. The most conclusive results were from the AHM
spiral galaxy data, using primarily Tully-Fisher distances. On the scale H[0]R = 1500 km s^-1, they found a bulk flow of 460 km s^-1 with a 3-D velocity dispersion of 456 km s^-1, or a Mach number
close to unity. Examining the simulated data in a fashion as close as possible to the real data, they found that fewer than 5% of the observers in a hypothetical CDM universe would find such a high
Mach number. Thus, this test rejects the standard CDM scenario at the 95% confidence level; it is also independent of the COBE normalization.
To summarize this subsection, standard CDM models appear to predict significantly larger velocity dispersions for galaxies than are observed, whether one normalizes the theory to COBE or to the
observed large-scale flows. | {"url":"http://ned.ipac.caltech.edu/level5/Sept01/Ostriker/Ostriker3_3.html","timestamp":"2014-04-16T08:19:21Z","content_type":null,"content_length":"9758","record_id":"<urn:uuid:d85d36b6-76b4-40d0-b7c1-79b6ce865110>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00353-ip-10-147-4-33.ec2.internal.warc.gz"} |
2.1 Integers
Next: 2.2 Rational Numbers Up: 2 Numbers Previous: 2 Numbers
The integer number system is a basic system of numbers. The set of all integers is denoted by
This number system is particularly simple and forms the basis for all of the other number systems presented here. Although the reader is assumed to be familiar with the integers, some semi-formal
discussion follows, which serves to refresh the reader's memory and to illustrate common features of all number systems. The integers can be constructed from the natural numbers; purists construct
the naturals using set theory [41, 17, 59].
Integers can be combined through addition and multiplication. Operators abstract the notion of combining numbers, by allowing for unary and 0-ary operators. The terms function and operator are
interchangable. n-ary operator n-tuple of numbers to a single number. Formally stated,
Addition and multiplication are binary operators. An n-ary function F, of n+1-tuples of numbers: Boldface is used to indicate vectors.
A set of numbers n-ary operation
Integers are closed under addition and multiplication: the sum or product of any two integers is another integer.
Since binary operators are so prevalent several properties of binary operators will be relevant. A binary operator
it is associative if it has identity and it has where i is the identity for g has an inverse
An n-ary function
where g is undefined for agument g is injective (invertible) if An inverse operator of a function g is written formally as An n-ary function
Negation is the total inverse of addition. Subtraction is defined as the sum of a number with another number's additive inverse:
A serious limitation of the integers is the lack of a total inverse of multiplication. Division is defined as the product of a number with another number's multiplicative inverse: It follows that the
integers are not closed under division.
Addition and multiplication over the integers jointly satisfy the following distributive law:
multiplication is said to distibute over addition. Addition and multiplication over the integers do not satisfy the following, alternative, distributive law: The first distributive law will be
hereafter referred to as ``the'' distributive law.
Another nice property of the integers is that comparing any pair of integers will always result in exactly one of three orderings. Equivalently, every pair of distinct integers contains a larger
The comparison operator
Common Practice
Almost all computers have hardware dedicated to performing very quick operations on integers. Many systems strictly limit the magnitude of the integers to guarantee certain limits on computational
resource requirements, while some do not. Although the manipulations of integers by computers is a fascinating and vitally important research area we will envision integers as a basic data type with
rudimentary operations.
Next: 2.2 Rational Numbers Up: 2 Numbers Previous: 2 Numbers | {"url":"http://www.dgp.utoronto.ca/people/mooncake/thesis/node16.html","timestamp":"2014-04-16T07:14:38Z","content_type":null,"content_length":"11379","record_id":"<urn:uuid:4c9b71c8-79ae-4d45-a593-b1b5e2a6c6ca>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00085-ip-10-147-4-33.ec2.internal.warc.gz"} |
Untyped algorithmic equality for Martin-Löf’s logical framework with surjective pairs
- IN 23RD CONFERENCE ON THE MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, MFPS XXIII, ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE , 2007
"... ..."
"... Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance
of not only the compiled program but also the type checker such static terms need to be erased as early as ..."
Cited by 7 (1 self)
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Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not
only the compiled program but also the type checker such static terms need to be erased as early as possible, preferably immediately after type checking. To this end, Pfenning’s type theory with
irrelevant quantification, that models a distinction between static and dynamic code, is extended to universes and large eliminations. Novel is a heterogeneously typed implementation of equality
which allows the smooth construction of a universal Kripke model that proves normalization, consistency and decidability.
- IN FROCOS’05: PROCEEDINGS OF THE 5TH INTERNATIONAL WORKSHOP ON FRONTIERS OF COMBINING SYSTEMS , 2005
"... We present one way of combining a logical framework and first-order logic. The logical framework is used as an interface to a first-order theorem prover. Its main purpose is to keep track of the
structure of the proof and to deal with the high level steps, for instance, induction. The steps that i ..."
Cited by 4 (0 self)
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We present one way of combining a logical framework and first-order logic. The logical framework is used as an interface to a first-order theorem prover. Its main purpose is to keep track of the
structure of the proof and to deal with the high level steps, for instance, induction. The steps that involve purely propositional or simple first-order reasoning are left to a first-order resolution
prover (the system Gandalf in our prototype). The correctness of this interaction is based on a general meta-theoretic result. One feature is the simplicity of our translation between the logical
framework and first-order logic, which uses implicit typing. Implementation and case studies are described.
"... In this paper, we present an explicit substitution calculus which distinguishes between ordinary bound variables and meta-variables. Its typing discipline is derived from contextual modal type
theory. We first present a dependently typed lambda calculus with explicit substitutions for ordinary varia ..."
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In this paper, we present an explicit substitution calculus which distinguishes between ordinary bound variables and meta-variables. Its typing discipline is derived from contextual modal type
theory. We first present a dependently typed lambda calculus with explicit substitutions for ordinary variables and explicit meta-substitutions for meta-variables. We then present a weak head
normalization procedure which performs both substitutions lazily and in a single pass thereby combining substitution walks for the two different classes of variables. Finally, we describe a
bidirectional type checking algorithm which uses weak head normalization and prove soundness.
"... Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance
of not only the compiled program but also the type checker such static terms need to be erased as early as ..."
Add to MetaCart
Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not
only the compiled program but also the type checker such static terms need to be erased as early as possible, preferably immediately after type checking. To this end, Pfenning’s type theory with
irrelevant quantification, that models a distinction between static and dynamic code, is extended to universes and large eliminations. Normalization, consistency, and decidability are obtained via a
universal Kripke model based on algorithmic equality. 1. Introduction and Related | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=270986","timestamp":"2014-04-16T11:13:43Z","content_type":null,"content_length":"22341","record_id":"<urn:uuid:04a40df5-42f8-4af4-bf45-2996245028b8>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00171-ip-10-147-4-33.ec2.internal.warc.gz"} |
Work in progress
1. R. Sknepnek
Faceting of shells with non-linear elastic response.
2. C. M. Funkhouser, R. Sknepnek, M. Olvera de la Cruz
Elastic model for blebbing in nuclear lamina.
3. M. F. Demers, R. Sknepnek, M. Olvera de la Cruz
Curvature-driven morphologies in multicomponent vesicles.
1. C. Leung, L.C. Palmer, B. Qiao, R. Sknepnek, et al.
The Faceting of Nanocontainers with Crystallized Ionic Membrane Walls
submitted (2012).
Published and Accepted
1. C. M. Funkhouser, R. Sknepnek, M. Olvera de la Cruz
Topological defects in the buckling of elastic membranes
accepted to Soft Matter (2012), doi:10.1039/C2SM26607E.
2. Z. Yao, R. Sknepnek, C. K. Thomas, M. Olvera de la Cruz
Shapes of pored membranes
accepted to Soft Matter (2012), doi:10.1039/b803770c.
3. R. Sknepnek, G. Vernizzi, M. Olvera de la Cruz
Charge renormalization of bilayer elastic properties
J. Chem. Phys. 137, 104905 (2012), doi:10.1063/1.4751481.
4. M. F. Demers, R. Sknepnek, M. Olvera de la Cruz
Curvature-driven effective attraction in multicomponent membranes
Phys. Rev. E 86, 021504 (2012), doi:10.1103/PhysRevE.86.021504.
5. T. Li, R. Sknepnek, R. J. Macfarlane, C. A. Mirkin, and M. Olvera de la Cruz
Modeling the Crystallization of Spherical Nucleic Acid Nanoparticle Conjugates with Molecular Dynamics Simulations
Nano Lett., 2012, 12 (5), pp 2509, doi: 10.1021/nl300679e.
6. R. Sknepnek and M. Olvera de la Cruz
Nonlinear elastic model for faceting of vesicles with soft grain boundaries
Phys. Rev. E 85, 050501(R) (2012), doi: 10.1103/PhysRevE.85.050501.
7. R. Sknepnek, G. Vernizzi, M. Olvera de la Cruz
Buckling of multicomponent elastic shells with line tension
Soft Matter, 8, 636 (2012), doi:10.1039/C1SM06325A.
cover page article
8. R. Sknepnek, G. Vernizzi, M. Olvera de la Cruz
Shape change of nano-containers via a reversible ionic buckling
Phys. Rev. Lett. 106, 215504 (2011).
9. P. Guo, R. Sknepnek, M. Olvera de la Cruz
Electrostatic Driven Ridge Formation on Nanoparticles Coated with Charged End Group Ligands
J. Phys. Chem. C 115, 6484 (2011).
10. G. Vernizzi, R. Sknepnek, M. Olvera de la Cruz
Platonic and Archimedean geometries in multi-component elastic membranes
Proc. Natl. Acad. Sci. USA, 108, 4292 (2011).
11. M. D. Donakowski, J. M. Godbe, R. Sknepnek, K. E. Knowles, M. Olvera de la Cruz, E. A. Weiss
A Quantitative Description of the Binding Equilibria of para-Substituted Aniline Ligands and CdSe Quantum Dots
J. Phys. Chem. C 114, 22526 (2010).
12. J. Zhang, R. Sknepnek, J. Schmalian
Spectral analysis for the Fe-based superconductors: On anisotropic spin fluctuations and fully gapped s^+/--wave superconductivity
Phys. Rev. B 82, 134527 (2010)
13. P. K. Jha, R. Sknepnek, G. I. Guerrero-Garcia, M. Olvera de la Cruz
A Graphics Processing Unit Implementation of Coulomb Interaction in Molecular Dynamics
J. Chem. Theory Comput. 6, 3058 (2010)
14. J. A. Anderson, R. Sknepnek, A. Travesset
Design of polymer nanocomposites in solution by polymer functionalization
Phys. Rev. E 82, 021803 (2010)
15. J. Zhang, R. Sknepnek, R. M. Fernandes, J. Schmalian
Orbital coupling and superconductivity in the iron pnictides
Phys. Rev. B 79, 220502(R) (2009)
16. R. Sknepnek, G. Samolyuk, Y.B. Lee, J. Schmalian
Anisotropic pairing in the iron pnictides
Phys. Rev. B 79, 054511 (2009)
17. R. Sknepnek, J. A. Anderson, M. H. Lamm, J. Schmalian, and A. Travesset
Nanoparticle Ordering via Functionalized Block Copolymers in Solution
ACS Nano 2, 1259 (2008)
18. S. Papanikolaou, R. M. Fernandes, E. Fradkin, P. W. Phillips, J. Schmalian, R. Sknepnek
Universality of liquid-gas Mott transitions at finite temperatures
Phys. Rev. Lett. 100, 026408 (2008)
19. T. Vojta and R. Sknepnek
Quantum phase transitions of the diluted O(3) rotor model
Phys. Rev. B 74, 094415 (2006)
20. D. Dalidovich, R. Sknepnek, A. J. Berlinsky, J. Zhang, and C. Kallin
Spin structure factor of the frustrated quantum magnet Cs[2]CuCl[4]
Phys. Rev. B 73, 184403 (2006)
21. B. Fendler, R. Sknepnek, Thomas Vojta
Dynamics at a smeared phase transition
J. Phys. A: Math. Gen. 38, 2349 (2005)
22. T.Vojta and R. Sknepnek
Critical points and quenched disorder: From Harris criterion to rare regions and smearing
Phys. Stat. Sol. (b) 241, 2118 (2004)
23. R. Sknepnek, T. Vojta, M. Vojta
Exotic vs. conventional scaling and universality in a disordered bilayer quantum Heisenberg antiferromagnet
Phys. Rev. Lett. 93, 097201 (2004)
24. R. Sknepnek and T. Vojta
Smeared phase transition in a three-dimensional Ising model with planar defects: Monte-Carlo simulations
Phys. Rev. B 69, 174410 (2004)
25. R. Sknepnek, T. Vojta, R. Narayanan
Order parameter symmetry and mode coupling effects at dirty superconducting quantum phase transitions
Phys. Rev. B 70, 104514 (2004)
26. T. Vojta and R. Sknepnek
The quantum phase transition of itinerant helimagnets
Phys. Rev. B 64, 052404 (2001)
27. V. Miljkovic, S. Milosevic, R. Sknepnek, I. Zivic
Pattern recognition in damaged neural networks
Physica A 295, 526 (2001)
1. Magnetic and superconducting quantum critical behavior of itineranr electronic systems, PhD thesis, University of Missouri- Rolla (2004) download PDF.
2. Numerical Analysis of the Hopfield model of neural networks with diluted synapses, Diploma thesis, Belgrade University, Serbia (2000) (in Serbian).
1. New Perspectives in Strongly Correlated Electrostatics in Soft Matter
Aspen Center for Physics, Aspen, Colorado, Aug.-Sept., 2010.
2. Physical Principles of Multiscale Modeling. Analysis and Simulation in Soft Condensed Matter
Kavli Institute for Theoretical Physics, University of California Santa Barbara, April, 2012.
talk: Thin-shell Model for Faceting of Multicomponent Elastic Vesicles
Invited Talks
1. Faceting of multicomponent elastic vesicles
Physics Colloquium, Department of Physics and Astronomy, University of Maryland Baltimore County (2012).
download PowerPoint slides
2. Faceting of multicomponent elastic shells
Seminar, Institute for Physical Science and Technology, University of Maryland (2011).
3. Faceting of multicomponent elastic shells
Physics Colloquium, Department of Physics and Astronomy, Syracuse University (2011).
download PowerPoint slides
4. Block copolymer guided self-assembly of nanoparticles
Condensed Matter & Biological Physics Seminar, Department of Physics and Astronomy, Syracuse University (2011).download PowerPoint slides
5. Faceting of multicomponent elastic shells Seminar of Faculty of Physics, University of Belgrade, Serbia (2010). download PowerPoint slides
6. Self-assembly of nanoparticles via end-functionalized triblock copolymers
Theory group meeting, Department of Chemistry, Northwestern University (2008) download PowerPoint slides
7. Spin Structure Factor of the Frustrated Quantum Magnet Cs[2]CuCl[4]
Iowa State University (2006) download PowerPoint slides
8. Spin Structure Factor of the Frustrated Quantum Magnet Cs[2]CuCl[4]
Duke University (2006) download PowerPoint slides
9. Quantum Phase Transitions of Itinerant Electrons
Belgrade University, Serbia (2002).
Contributed Talks and Posters
1. R. Sknepnek, G. Vernizzi, M. Olvera de la Cruz
Buckling of Multicomponent Elastic Shells
Gordon Research Conference in Soft Condensed Matter Physics (2011) (poster)
2. R. Sknepnek, C. Leung, L.C. Palmer, G. Vernizzi, S. I. Stupp, M. J. Bedzyk, M. Olvera de la Cruz
Faceting of multicomponent charged elastic shells
APS March Meeting (2011) download PowerPoint slides
3. C. Leung, R. Sknepnek, L.C. Palmer, G. Vernizzi, M. Greenfield, S. I. Stupp, M. J. Bedzyk, M. Olvera de la Cruz
Crystallization Induced by Electrostatic Correlations in Vesicles of Mixed-Valence Ionic Amphiphiles APS March Meeting (2011)
4. P. Guo, R. Sknepnek, M. Olvera de la Cruz
Ridge formation of charged end group ligands grafted on faceted nanoparticle APS March Meeting (2011)
5. R. Sknepnek, G. Vernizzi, M. Olvera de la Cruz
Symmetry Selection via a Reversible Ionic Buckling of Elastic Membranes
MRS Fall Meeting, Boston (2010)
6. R. Sknepnek, G. Vernizzi, M. Olvera de la Cruz
The buckling transition of ionic shells and electrostatics
APS March Meeting, Portland (2010)
7. J. Zwanikken, R. Sknepnek, M. Olvera de la Cruz
Effective interactions between pH-responsive particles
APS March Meeting, Portland (2010)
8. G. Vernizzi, R. Sknepnek, M. Olvera de la Cruz
The shapes of two-component crystalline shell
APS March Meeting, Portland (2010)
9. J. Zhang, R. Sknepnek, J. Schmalian
Spectral information in the fluctuation induced superconducting state for iron based superconductors
APS March Meeting, Portland (2010)
10. R. Sknepnek, J. A. Anderson, M. H. Lamm, J. Schmalian, and A. Travesset
End-Functionalized Triblock Copolymers as a Guide for Nanoparticle Ordering
APS March Meeting, Pittsburgh (2009) (talk) download PowerPoint slides
11. J. Anderson, R. Sknepnek, A. Travesset
Phases of functionalized polymer-inorganic composites in solution studied via molecular dynamics
APS March Meeting, Pittsburgh (2009) (talk)
12. J. Zhang, R. Sknepnek, J. Schmalian
On the magnetic fluctuations and unconventional superconducting pairing in iron pnictides
APS March Meeting, Pittsburgh (2009) (talk)
13. M. H. Lamm, R. Sknepnek, L. Wang, M. Nilsen-Hamilton
Molecular dynamics simulation study of multimerization of the Mms6 protein from Magnetospirillum magneticum strain AMB-1
APS March Meeting, Pittsburgh (2009) (talk) download PowerPoint slides
14. Rastko Sknepnek, Joshua Anderson, Monica Lamm, Joerg Schmalian, Alex Travesset
Ordering of nanoparticles mediated by end-functionalized triblock copolymers
AIChE Meeting, Philadelphia (2008) (talk) download PowerPoint slides
15. R. Sknepnek, J. A. Anderson, M. H. Lamm, J. Schmalian, and A. Travesset
End-functionalized triblock copolymers as a robust template for assembly of nanoparticles
APS March Meeting, New Orleans (2008) (talk) download PowerPoint slides
16. S. Papanikolaou, R. M. Fernandes, E. Fradkin, P. W. Phillips, J. Schmalian, R. Sknepnek
Mott transition and Universality at finite temperatures
APS March Meeting, New Orleans (2008) (talk).
17. R. Sknepnek, J. Liu, J. Schmalian
On the role of inhomogeneities for correlated d-wave superconductors
APS March Meeting, Colorado (2007) (talk) download PowerPoint slides
18. R. Sknepnek, D. Dalidovic, A. J. Berlinsky, J. Zhang, and C. Kallin
Dynamical properties of the anisotropic triangular quantum antiferromagnet with Dzyaloshinskii-Moriya interaction
APS March Meeting, Baltimore (2006) (talk) download PowerPoint slides
19. R. Sknepnek and T. Vojta
Smeared phase transition in a three-dimensional Ising model with planar defects: Monte-Carlo simulations
CIAR Summer School 2005, Vancouver (2005) (poster)
20. R. Sknepnek, T. Vojta, R. Narayanan
Order parameter symmetry and mode coupling effects at dirty superconducting quantum phase transitions
APS March Meeting, Austin (2003) (talk)
21. R. Sknepnek and T. Vojta
The quantum phase transition of itinerant helimagnets
65^th Spring Conference of German Physics Society, Hamburg, Germany (2001) (poster)
22. R. Sknepnek
Numerical simulation of Ehrenfest s dog-flea model
XIII International Conference for Physics Students, Coimbra, Portugal (1998) (talk) | {"url":"http://aztec.ms.northwestern.edu/~sknepnek/publications.html","timestamp":"2014-04-18T13:06:52Z","content_type":null,"content_length":"22760","record_id":"<urn:uuid:6c77b27b-892c-4bc7-8546-b2adad167b79>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00012-ip-10-147-4-33.ec2.internal.warc.gz"} |
Area of an oblique triangle question
February 5th 2013, 01:37 PM
Area of an oblique triangle question
I need help with this question:
"There is a triangle OAB. The length of AO is 10m and AOB has an angle of 0.8^c. This triangle also has an arc between points A and B."
Find the area of the triangle OAB
Find the area of the sector OAB
I have drawn the triangle to its correct description, see the attached image.
I have tried using the formula for the area of the sector 0.5r^2θ, but I got the answer to be 2193. This might be right, just seemed too big. But I do not know the formula for the area of the
triangle! (I suppose this triangle is different to the formula 0.5xbxh)
Could you please help me work out how to find the area of this triangle?
Thanks in advance
February 5th 2013, 02:53 PM
Re: Area of an oblique triangle question
I need help with this question:
"There is a triangle OAB. The length of AO is 10m and AOB has an angle of 0.8^c. This triangle also has an arc between points A and B."
Find the area of the triangle OAB
Find the area of the sector OAB
I have drawn the triangle to its correct description, see the attached image.
This webpage will help you.
I cannot help you because I have no idea what $0.8^c$ could mean.
February 5th 2013, 04:02 PM
Prove It
Re: Area of an oblique triangle question
This webpage will help you.
I cannot help you because I have no idea what $0.8^c$ could mean.
It means 0.8 radians... C is the symbol for radians, it stands for "number of lengths of the radius on the Circumference".
I suppose we're assuming that it's a circular arc, which means OB = OA, correct?
February 5th 2013, 10:44 PM
Re: Area of an oblique triangle question
Yes OB=OA. And 0.8 radians = 57.3 degrees
February 5th 2013, 10:47 PM
Prove It
Re: Area of an oblique triangle question
If the angle is measured in radians, the area of the triangle can be found using \displaystyle \begin{align*} A = \frac{1}{2}ab\sin{(C)} \end{align*} and the area of a sector can be found using \
displaystyle \begin{align*} A = \frac{1}{2}r^2\theta \end{align*}. | {"url":"http://mathhelpforum.com/trigonometry/212625-area-oblique-triangle-question-print.html","timestamp":"2014-04-17T11:56:08Z","content_type":null,"content_length":"8111","record_id":"<urn:uuid:1674b7ca-59a9-469c-8666-30618a38efc9>","cc-path":"CC-MAIN-2014-15/segments/1397609527423.39/warc/CC-MAIN-20140416005207-00190-ip-10-147-4-33.ec2.internal.warc.gz"} |
Theory and realization of novel algorithms for random sampling in digital signal processing
Lo, King Chuen (1996) Theory and realization of novel algorithms for random sampling in digital signal processing. Doctoral thesis, Durham University.
Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier
transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N(^2). The objectives of
this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling is maintained : Two methods of
inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% of
the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the
auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods can be confirmed by using
convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A technique to evaluate the
auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly
spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for
the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully exploit its advantages,
random sampling has been adopted in measurement Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the
transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational
complexity is N"^. The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling
is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of
which can save at least 75% , of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the
evaluation. (2) To study the auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods
can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A
technique to evaluate the auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled
sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which
trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully
exploit its advantages, random sampling has been adopted in measurement instruments where computing a spectrum is either minimal or not required. Such applications in instrumentation are easily found
in the literature. In this thesis, two applications in digital signal processing are introduced. (5) To suggest an inverse transformation for random sampling so as to complete a two-way process and
to broaden its scope of application. Apart from the above, a case study of realizing in a transputer network the prime factor algorithm with regular sampling is given in Chapter 2 and a rough
estimation of the signal-to-noise ratio for a spectrum obtained from random sampling is found in Chapter 3. Although random sampling is alias-free, problems in computational complexity and noise
prevent it from being adopted widely in engineering applications. In the conclusions, the criteria for adopting random sampling are put forward and the directions for its development are discussed.
Item Type: Thesis (Doctoral)
Award: Doctor of Philosophy
Thesis Date: 1996
Copyright: Copyright of this thesis is held by the author
Deposited On: 09 Oct 2012 11:52 | {"url":"http://etheses.dur.ac.uk/5239/","timestamp":"2014-04-18T08:15:01Z","content_type":null,"content_length":"35060","record_id":"<urn:uuid:5f269ec8-99ce-4306-ac72-c277f41e8a57>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00552-ip-10-147-4-33.ec2.internal.warc.gz"} |
umber definition
Prime Number definition, examples
A prime number is a number whose only factors are 1 and itself. That means there is no whole number that evenly
the prime number.
Some often-confused facts about prime numbers
• Zero is not a prime a number
• The number one, 1, is also not a prime number. Although the definition of a prime number seems to apply to 1, you have to count 1 twice --sorry no 'double dipping' for prime numbers. 1 is not
• The only even prime number is 2, because all of the other even prime numbers are multiples of 2 and therefore violate the definition of a prime number (only divisible by 1 and itself)
• Read more about how to determine if a number is prime | {"url":"http://www.mathwarehouse.com/arithmetic/numbers/prime-number/","timestamp":"2014-04-18T20:42:16Z","content_type":null,"content_length":"15988","record_id":"<urn:uuid:84a0ea59-5106-4555-b9b1-4bd2c388ccc4>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00151-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Seeing New Connections
September 18, 2011
Ingrid Daubechies, honored for “her multifaceted and enduring contributions to mathematics, science, and engineering, especially her fundamental work in the development of the foundations and
applications of wavelets,” gave SIAM’s 2011 John von Neumann lecture in Vancouver, at ICIAM. Described in the citation as “a brilliant mathematical scientist who has opened new directions of research
and new approaches to signal and image processing and data analysis,” Daubechies was also recognized (as those who attended the lecture would agree) as “a gifted communicator who has greatly
facilitated the spread and appreciation of mathematical ideas, and a tireless scientific leader who both serves and inspires the mathematical community.” Photos by VisionPhoto.ca.
From the SIAM President
Nick Trefethen
A happy moment for me at ICIAM in July was the opportunity to introduce Ingrid Daubechies as SIAM's John von Neumann lecturer. People seemed to enjoy the story of how I first heard of Daubechies. It
was back around 1987, at MIT, and I was waiting to use the Xerox machine in the copy room in Building 2. Gil Strang was ahead of me in line and he told me, You know, Nick, something important is
happening and it's called wavelets. You should read the new paper on compactly supported orthogonal wavelets by a young physicist from Belgium.
That paper, "Orthonormal Bases of Compactly Supported Wavelets," now lists more than 6000 citations at Google Scholar. Since then, Daubechies has been a worldwide leader in the development of this
field, and her
Ten Lectures on Wavelets
is perhaps SIAM's most influential book ever.
Her von Neumann lecture, called "Sparsity in Data Analysis and Computation," began with a discussion of the application of wavelets that many of us know best, namely image compression. Traditionally
(and in the original JPEG standard), one compressed images by decomposing them into different wave numbers by a Fourier transform and then discarding certain low-order information. The trouble is,
sines and cosines are global, unlike the features in many images, and often one can't throw away so much. Wavelets combine a separation of wavelengths with a separation of spatial positions and often
do a better job. Daubechies outlined the fascinating phenomenon that although Fourier approaches are optimal among linear approximations, where you have to choose the basis in advance, they are not
optimal among nonlinear approximations, where the basis is allowed to depend on the data.
The audience of close to 1000 loved her story of watching a soccer game on television with her family one day some months ago, finding herself bored till she noticed that the grass in the TV image
had a scale-invariant structure she recognized. She exclaimed to her family, "They're using wavelet compression!"
Daubechies' lecture then turned from image compression to compressed sensing, which introduces the crucial new element of randomness. In compressed sensing, one samples a signal by a collection of
random measurements, and the number of measurements is (relatively) not very large. It might seem that nothing could come from such measurements, and so it would be, except for the crucial assumption
that the signal sought is sparse. Once sparseness is assumed, the random measurements turn out to be enough to reconstruct the signal with very high probability. The importance of these ideas was
underlined by the fact that earlier the same day in Vancouver, the ICIAM Collatz Prize had been awarded to compressed sensing pioneer Emmanuel Candès.
Overcoming her reservations about discussing compressed sensing with Emmanuel Candès in the audience, Ingrid Daubechies presented the important ideas underlying the area in her John von Neumann
lecture. Candès, she pointed out, had received the ICIAM Collatz Prize that morning for "his outstanding contributions to numerical solution of wave propagation problems and compressive sensing, as
well as anisotropic extensions of wavelets." Shown here at the ICIAM opening ceremonies are Taketomo Mitsui (left), who presented the prize, and Candès.
What made Daubechies' lecture most remarkable for me was where she took the discussion next, turning to the last two words of her lecture title. Random sampling combined with sparsity, she pointed
out, is closely related to a whole new set of algorithms that have transformed computer science in recent decades. Ideas like fast primality testing, zero-knowledge proof, and public-key encryption
have at their heart a targeted use of randomness. Suppose that we do a certain random test on a 1000-digit number
and each time it passes the test, the chance that
is not prime is cut in half. After 20 successful experiments, we know that n is prime with a risk of error of just one in a million. This kind of cumulative random process is now used in
probabilistic algorithms all across computing, and Daubechies, referring in particular to the Johnson–Lindenstrauss lemma, discussed how all of these methods rely on the nonlinear exploitation of
I found myself thinking of a curious symmetry. To achieve randomness in science or technology, our best bet is exponentials. You can toss a coin, but the outcome isn't so random because it is
sensitive only algebraically to the details of the throw. For truer randomness you need a chaotic system with exponential sensitivities, like a pinball machine or the Lorenz equations. Run such a
system for a moment and your randomness might be 99%. If that's not enough, run it a little longer to get 99.99%. The point is that with each new step, your knowledge about the system shrinks by a
constant factor, soon reaching zero for practical purposes.
And to achieve certainty, our best bet is exponentials again. At the level of fundamental physics, anything can happen because of quantum tunnelling. But some things "never" happen in practice, such
as the radioactive decay of an iron-56 atom. Why? Because the frequency of quantum events shrinks exponentially with the width of a potential barrier. Thickening up that barrier in a physics
experiment is like adding another level of error correction in an electronic circuit or taking another step of a random algorithm or making another compressed sensing measurement. With each new step,
your uncertainty about the system shrinks by a constant factor, soon reaching zero for practical purposes.
Daubechies' stimulating talk left us all seeing new connections. The next issue of
SIAM News
will have a collection of articles touching other aspects of ICIAM 2011. | {"url":"http://www.siam.org/news/news.php?id=1905","timestamp":"2014-04-20T14:47:56Z","content_type":null,"content_length":"13639","record_id":"<urn:uuid:9afd655e-e705-42b8-bc4e-702ad9cafbad>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00386-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Post a reply
This problem appeared in the Bafflers thread as exercise #2 and has some algebraic solutions.
A ship is sailing on a course from the origin. It follows the path y= 3.14 x. It has a speed of 1 / 3 units per second. A submarine is located at (5,0). The sub knows the ship is oblivious to its
existence. The sub would like to torpedo the warship. Torpedoes travel at 1 / 2 units per second and travel in straight lines. The sub commander's hobby is mathematics. He fires and sinks the ship!
What is the equation of the torpedoes flight?
One solution is to realize that what is required is triangle with one side being the line y = 3.14 x the base being the x axis from 0 to 5 and some line passing through (5,0) and intersecting y =
3.14 x. See fig 1. With the condition that the red line is 1.5 times longer than the other side of the triangle. Let's use Geogebra to solve the problem.
1)Draw the point (5,0) and call it Sub.
2)Enter in the input bar f(x) = 3.14 x
3)Place a slider on the drawing. Set it at Min=0 and Max = 10 with an increment of .1
It should be called a.
4)Enter (a,3.14*a) and call it Ship. Move the slider and you will see the point is constrained along y = 3.14 x.
5)Make a point at (0,0) and call it Start and hide it.
6)Enter Distance[Ship,Sub]/Distance[Start,Ship]
7) Set in options, rounding = 15 decimals, immediately you will see b = some value. That is the ratio of the torpedoes distance to the ships distance.
8)Move the slider using shift arrows until you get b = 1.497666014479171
9)Right click the top part of the slider to get the properties and set the increment to .01
Make sure you click the top part of the slider to select it after pressing close.
10)Press Shift left arrow until you get b = 1.499835795083569 and a = 1.038
11)Repeat 9 but set the increment to .001
12)Repeat the above loop always getting an answer as close to 1.5 as possible and slightly smaller.
I get b = 1.49999993088931 with an increment of 0.000001, how did you do?
13)Draw a line between Ship and Sub using the line tool and read off the equation of that line. I got
See the second drawing to check your work. | {"url":"http://www.mathisfunforum.com/post.php?tid=17496&qid=203534","timestamp":"2014-04-18T13:30:35Z","content_type":null,"content_length":"19081","record_id":"<urn:uuid:911978f6-e2b1-41ab-b12d-8c90702d58dc>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00160-ip-10-147-4-33.ec2.internal.warc.gz"} |
Are physical constants irrational? - JREF Forum
Originally Posted by
Magic Pancake
Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers?
One can pick units for which some constants are rational numbers. But even if one were to do this, some of them still
must be
irrational. For example, with the right set of units one could make hbar a rational number. But then h would necessarily be irrational. Conversely, if one makes h rational, then hbar must be
irrational. Same thing for k and epsilon (two different forms of the constant for Coulomb's law): they're related by irrational numbers, so they cannot both be rational (but they can both be
I sure can't see a ready fraction in any of them, but then again these numbers are determined experimentally, so how can you tell for sure.
You can't. But as mentioned above, since units are arbitrary and generally we don't pick them based upon fundamental constants, the probability is rather overwhelmingly in favor of any fundamental
constant with units being irrational.
How about the universe itself? Is there a countable number of states for it?As a layman I understand that units of dimensions in the universe are quantified in various Plank units, so that would seem
to say that these must be countable, but is this the case?
No, that is not the case.
It's a common misconception that the Planck scales represent some sort of fundamental quantization of the universe. That is not the case. Or to be more precise, we have no evidence that this is the
case. The significance of these scales is something rather different, namely the intersection of quantum mechanics and general relativity.
Let's think about length and mass/energy. In general relativity, a given mass will have an associated length scale, the Schwarzchild radius, which describes how small you need to get that mass in
order to turn it into a black hole. If you're much further away from that mass than this length scale, then even if it IS a black hole, the gravitational field at that distance will still behave like
Newtonian gravity. Which means you can essentially ignore general relativity. Conversely, if you're at distances from your mass of similar magnitude to this length scale, then even if it hasn't
actually collapsed into a black hole, general relativistic effects will still be strong. So for a given mass, this distance characterizes the crossover from where you can ignore GR to where you must
include GR. Now, the larger you make your mass, the longer this length scale will be.
We can do something very similar with quantum mechanics. Pick an energy, and you can associate a length scale, for example, the wavelength of a photon with that energy or the de Broglie wavelength of
a particle. If we're at distances much larger than this wavelength, we can often ignore quantum effects, but below this length we definitely need to consider quantum mechanics. So that idea of a
crossover is the same as with GR, but unlike GR, this length gets
as we increase out mass.
Now, let's start out by considering a very small mass/energy, much smaller than an electron. The quantum length scale is large, and the GR length scale is tiny. Now we start increasing the mass we're
considering. The quantum length scale decreases, and the GR length scale increases. At some point, if we increase our mass enough, the two length scales will cross. This happens at the Planck mass,
and the length is the Planck length. That's really the only thing either of those things definitely indicate.
But that's not
the end of the story. If it were, they might be nothing more than a curiosity. The question is, what
at these scales? And the answer is that we just don't know. The problem is that quantum mechanics and general relativity don't agree with each other. That's usually not a problem: almost everything
in the universe interacts under conditions where we're in the quantum OR general relativity regime (or neither). So we can almost always ignore either quantum mechanics or general relativity, if not
both. That means that we can get away without resolving that conflict. But if we got down to the Planck scales (which require fantastically large energy densities), then we wouldn't be able to ignore
either GR or quantum mechanics, and we can't make predictions without knowing how to resolve that conflict. Now, some people
that some additional quantization of space itself happens at these scales, and that this additional quantization may resolve the conflict, but that's just speculation. We really don't know. And since
we can't get anywhere near that density for practical reasons, we can't do direct experimentation to study such conditions either. But as it stands now, the Planck scales should be thought of as a
limit to our understanding, not as a limit to space itself. | {"url":"http://forums.randi.org/showthread.php?p=8268374","timestamp":"2014-04-19T19:49:28Z","content_type":null,"content_length":"368162","record_id":"<urn:uuid:6a90dd25-7557-4458-8ff2-2c944a24b212>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00105-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math 262A, Fall 1999
6/27/01: When the class was taught, everyone had accounts on a computer where all the software was installed. The instructions given here were for that environment. The pathnames given here no longer
exist. These instructions are illustrative only.
To use the software on your own computer, you will have to follow the links to the sites where the software came from, and obtain and install it yourself.
Go here for instructions on setting up.
• Koepf's worksheets for Hypergeometric Summation
The current software and worksheets accompanying Koepf's book Hypergeometric Summation can be downloaded from Koepf's web site; scroll down to "Hypergeometric Summation" under "Textbooks and
Monographs." Also look here.
To use them on euclid, type
cd ~m262f99/KOEPF/worksheetsV.4
xmaple &
and then open a worksheet from "Open" in the "File" menu. You can also set up netscape so that it will automatically open the file in xmaple when you select it below.
Koepf's worksheets for the text, by chapter. Maple V.4: chap1.mws, ..., chap13.mws.
Koepf's worksheets for exercises, by chapter. Maple V.4: exer2.mws, ..., exer13.mws.
Additional software. Maple V.4: hsum.mpl qsum.mpl
Programs for Petkovsek, Wilf, and Zeilberger's A=B
These are the web sites for Zeilberger's and Petkovsek's programs. To use these programs on euclid, type
cd ~m262f99/A=B/Maple
xmaple &
cd ~m262f99/A=B/Mathematica
mathematica &
as appropriate, and then follow the directions in the book A=B.
Zeilberger's programs for A=B. Maple programs EKHAD qEKHAD.
Petkovsek's programs for A=B. Mathematica programs gosper.m Hyper q-Hyper WZ.
Christian Krattenthaler. Mathematica. See his directions for hyp for hypergeometric series, and for rate for guessing formulas for sequences.
cd ~m262f99/KRATT
mathematica &
Guessing sequences:
xmaple &
Selected programs from INRIA
Salvy, Zimmermann, & Murray: Maple. Generating functions package Gfun.
Sample files are in ~m262f99/GFUN
xmaple &
Chyzak: Maple. Multivariable generating function package Mgfun. See his web site and papers.
Demos/instructions are in ~m262f99/CHYZAK/Doc and ~m262f99/CHYZAK/Tests | {"url":"http://www.math.ucsd.edu/~gptesler/262/software.html","timestamp":"2014-04-21T09:36:42Z","content_type":null,"content_length":"5011","record_id":"<urn:uuid:204a9cd8-fedd-40c1-8b94-2d52f90ab2c6>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00286-ip-10-147-4-33.ec2.internal.warc.gz"} |
Tutorial 6 : Keyboard and Mouse
Welcome for our 6th tutorial !
We will now learn how to use the mouse and the keyboard to move the camera just like in a FPS.
The interface
Since this code will be re-used throughout the tutorials, we will put the code in a separate file : common/controls.cpp, and declare the functions in common/controls.hpp so that tutorial06.cpp knows
about them.
The code of tutorial06.cpp doesn’t change much from the previous tutorial. The major modification is that instead of computing the MVP matrix once, we now have to do it every frame. So let’s move
this code inside the main loop :
// ...
// Compute the MVP matrix from keyboard and mouse input
glm::mat4 ProjectionMatrix = getProjectionMatrix();
glm::mat4 ViewMatrix = getViewMatrix();
glm::mat4 ModelMatrix = glm::mat4(1.0);
glm::mat4 MVP = ProjectionMatrix * ViewMatrix * ModelMatrix;
// ...
This code needs 3 new functions :
• computeMatricesFromInputs() reads the keyboard and mouse and computes the Projection and View matrices. This is where all the magic happens.
• getProjectionMatrix() just returns the computed Projection matrix.
• getViewMatrix() just returns the computed View matrix.
This is just one way to do it, of course. If you don’t like these functions, go ahead and change them.
Let’s see what’s inside controls.cpp.
The actual code
We’ll need a few variables.
// position
glm::vec3 position = glm::vec3( 0, 0, 5 );
// horizontal angle : toward -Z
float horizontalAngle = 3.14f;
// vertical angle : 0, look at the horizon
float verticalAngle = 0.0f;
// Initial Field of View
float initialFoV = 45.0f;
float speed = 3.0f; // 3 units / second
float mouseSpeed = 0.005f;
FoV is the level of zoom. 80° = very wide angle, huge deformations. 60° – 45° : standard. 20° : big zoom.
We will first recompute position, horizontalAngle, verticalAngle and FoV according to the inputs, and then compute the View and Projection matrices from position, horizontalAngle, verticalAngle and
Reading the mouse position is easy :
// Get mouse position
int xpos, ypos;
glfwGetMousePos(&xpos, &ypos);
but we have to take care to put the cursor back to the center of the screen, or it will soon go outside the window and you won’t be able to move anymore.
// Reset mouse position for next frame
glfwSetMousePos(1024/2, 768/2);
Notice that this code assumes that the window is 1024*768, which of course is not necessarily the case. You can use glfwGetWindowSize if you want, too.
We can now compute our viewing angles :
// Compute new orientation
horizontalAngle += mouseSpeed * deltaTime * float(1024/2 - xpos );
verticalAngle += mouseSpeed * deltaTime * float( 768/2 - ypos );
Let’s read this from right to left :
• 1024/2 – xpos means : how far is the mouse from the center of the window ? The bigger this value, the more we want to turn.
• float(…) converts it to a floating-point number so that the multiplication goes well.
• mouseSpeed is just there to speed up or slow down the rotations. Finetune this at will, or let the user choose it.
• += : If you didn’t move the mouse, 1024/2-xpos will be 0, and horizontalAngle+=0 doesn’t change horizontalAngle. If you had a “=” instead, you would be forced back to your original orientation
each frame, which isn’t good.
We can now compute a vector that represents, in World Space, the direction in which we’re looking
// Direction : Spherical coordinates to Cartesian coordinates conversion
glm::vec3 direction(
cos(verticalAngle) * sin(horizontalAngle),
cos(verticalAngle) * cos(horizontalAngle)
This is a standard computation, but if you don’t know about cosine and sinus, here’s a short explanation :
The formula above is just the generalisation to 3D.
Now we want to compute the “up” vector reliably. Notice that “up” isn’t always towards +Y : if you look down, for instance, the “up” vector will be in fact horizontal. Here is an example of to
cameras with the same position, the same target, but a different up.
In our case, the only constant is that the vector goes to the right of the camera is always horizontal. You can check this by putting your arm horizontal, and looking up, down, in any direction. So
let’s define the “right” vector : its Y coordinate is 0 since it’s horizontal, and its X and Z coordinates are just like in the figure above, but with the angles rotated by 90°, or Pi/2 radians.
// Right vector
glm::vec3 right = glm::vec3(
sin(horizontalAngle - 3.14f/2.0f),
cos(horizontalAngle - 3.14f/2.0f)
We have a “right” vector and a “direction”, or “front” vector. The “up” vector is a vector that is perpendicular to these two. A useful mathematical tool makes this very easy : the cross product.
// Up vector : perpendicular to both direction and right
glm::vec3 up = glm::cross( right, direction );
To remember what the cross product does, it’s very simple. Just recall the Right Hand Rule from Tutorial 3. The first vector is the thumb; the second is the index; and the result is the middle
finger. It’s very handy.
The code is pretty straightforward. By the way, I used the up/down/right/left keys instead of the awsd because on my azerty keyboard, awsd is actually zqsd. And it’s also different with qwerZ
keyboards, let alone korean keyboards. I don’t even know what layout korean people have, but I guess it’s also different.
// Move forward
if (glfwGetKey( GLFW_KEY_UP ) == GLFW_PRESS){
position += direction * deltaTime * speed;
// Move backward
if (glfwGetKey( GLFW_KEY_DOWN ) == GLFW_PRESS){
position -= direction * deltaTime * speed;
// Strafe right
if (glfwGetKey( GLFW_KEY_RIGHT ) == GLFW_PRESS){
position += right * deltaTime * speed;
// Strafe left
if (glfwGetKey( GLFW_KEY_LEFT ) == GLFW_PRESS){
position -= right * deltaTime * speed;
The only special thing here is the deltaTime. You don’t want to move from 1 unit each frame for a simple reason :
• If you have a fast computer, and you run at 60 fps, you’d move of 60*speed units in 1 second
• If you have a slow computer, and you run at 20 fps, you’d move of 20*speed units in 1 second
Since having a better computer is not an excuse for going faster, you have to scale the distance by the “time since the last frame”, or “deltaTime”.
• If you have a fast computer, and you run at 60 fps, you’d move of 1/60 * speed units in 1 frame, so 1*speed in 1 second.
• If you have a slow computer, and you run at 20 fps, you’d move of 1/20 * speed units in 1 second, so 1*speed in 1 second.
which is much better. deltaTime is very simple to compute :
double currentTime = glfwGetTime();
float deltaTime = float(currentTime - lastTime);
Field Of View
For fun, we can also bind the wheel of the mouse to the Field Of View, so that we can have a cheap zoom :
float FoV = initialFoV - 5 * glfwGetMouseWheel();
Computing the matrices
Computing the matrices is now straightforward. We use the exact same functions than before, but with our new parameters.
// Projection matrix : 45° Field of View, 4:3 ratio, display range : 0.1 unit <-> 100 units
ProjectionMatrix = glm::perspective(FoV, 4.0f / 3.0f, 0.1f, 100.0f);
// Camera matrix
ViewMatrix = glm::lookAt(
position, // Camera is here
position+direction, // and looks here : at the same position, plus "direction"
up // Head is up (set to 0,-1,0 to look upside-down)
Backface Culling
Now that you can freely move around, you’ll notice that if you go inside the cube, polygons are still displayed. This can seem obvious, but this remark actually opens an opportunity for optimisation.
As a matter of fact, in a usual application, you are never _inside_ a cube.
The idea is to let the GPU check if the camera is behind, or in front of, the triangle. If it’s in front, display the triangle; if it’s behind, *and* the mesh is closed, *and* we’re not inside the
mesh, *then* there will be another triangle in front of it, and nobody will notice anything, except that everything will be faster : 2 times less triangles on average !
The best thing is that it’s very easy to check this. The GPU computes the normal of the triangle (using the cross product, remember ?) and checks whether this normal is oriented towards the camera or
This comes at a cost, unfortunately : the orientation of the triangle is implicit. This means that is you invert two vertices in your buffer, you’ll probably end up with a hole. But it’s generally
worth the little additional work. Often, you just have to click “invert normals” in your 3D modeler (which will, in fact, invert vertices, and thus normals) and everything is just fine.
Enabling backface culling is a breeze :
// Cull triangles which normal is not towards the camera
• Restrict verticalAngle so that you can’t go upside-down
• Create a camera that rotates around the object ( position = ObjectCenter + ( radius * cos(time), height, radius * sin(time) ) ); bind the radius/height/time to the keyboard/mouse, or whatever
• Have fun ! | {"url":"http://www.opengl-tutorial.org/beginners-tutorials/tutorial-6-keyboard-and-mouse/","timestamp":"2014-04-16T10:24:47Z","content_type":null,"content_length":"31835","record_id":"<urn:uuid:cbe19da1-e99e-4dd0-98b8-afec1539aa11>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00251-ip-10-147-4-33.ec2.internal.warc.gz"} |
EPA On-line Tools for Site Assessment Calculation
Retardation Coefficient
The retardation coefficient expresses how much slower a contaminant moves than does the water itself.
where R is the retardation coefficient, r[b] is the bulk density, k[d] is the sorption coefficient, and q is the porosity.
A simplified method of accounting for sorption is in common usage. This method is based on the assumption that sorption of organics occurs to the naturally occurring organic carbon in the aquifer.
Obviously, the properties of the chemical are important also. Thus a sorption coefficient, k[d] (also called the soil/water distribution coefficient) is defined from the fraction organic carbon, f
[oc] in the aquifer and the organic carbon partition coefficient, K[oc] of the chemical:
The K[oc] values used in OnSite are tabulated on the chemical properties page. Use of K[oc] --- f[oc] as input parameters refects hydrophobic sorption theory and the observation that sorption of
organic chemicals occurs as a partitioning to aquifer organic carbon. These two parameters are used internally to calculate the solid/water distribution coefficient, k[d]
Bulk Density
The bulk density appears in the definition of the retardation coefficient. Bulk density appears because concentrations in the water phase are measured relative to volume of water and sorbed
concentrations are measured relative to the solid mass.
where the factor of 2.65 is the density of quartz sand, giving the bulk density units of g/cm^3. | {"url":"http://www.epa.gov/athens/learn2model/part-two/onsite/ard_onsite.html","timestamp":"2014-04-16T19:29:08Z","content_type":null,"content_length":"15974","record_id":"<urn:uuid:b349f447-6169-48db-9810-128c3c6a9dec>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00342-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions - Re: An Interesting Point
Date: Nov 6, 2012 7:51 PM
Author: Robert Hansen
Subject: Re: An Interesting Point
To get this thread back to the point, a student cannot know what data is needed until they have arrived at a solution to the problem. Thus, it is incorrect to say "Students have difficulty knowing what data is important and what data is not." What the students lack is sufficient analytical skill to take the whole of the problem in and through experience and instinct, fashion a solution. It is like playing chess. Before you make your next move you must analyze the whole board and through that analysis you arrive at a strategy and that is when determine which piece you are going to move. Up until that point, when you have settled on a strategy, any piece could be the piece you move.
Dan countered my point with another question, "How much would it cost to wash all of the Windows in Seattle?". I suppose he was thinking that the students would have to go find out how many windows are in Seattle and what size they are. Thus, this would be a "data" driven problem. But this problem, is no different than any other problem. The student would first have to have a model that provides total cost (the goal) and then they would have to do the work to get the data their model requires.
The ability to put mathematics to use relies on the ability to analyze the situation and model it in a manner that fulfills your goal. The ability to do this rests on experience and instincts that are developed over time. Like I said before, teachers may be saying one thing but meaning another, but given that some curriculums focus on data gathering rather than algorithm, I would say that many teachers actually do not understand the cognitive task of solving a problem. The do not understand that the student must actually be able to analyze the problem and arrive at a solution all at once. It is this inability to pull it all together and understand the math in context is what sinks them.
The time that these teachers spend having the students "guess" a solution would be better spent teaching then the art of solving itself. When a student can analyze and solve then they have graduated to the stage where math is useful.
Bob Hansen | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=7919007","timestamp":"2014-04-19T21:07:58Z","content_type":null,"content_length":"3104","record_id":"<urn:uuid:38bd60ac-ad62-4439-b44c-e6e09fa60826>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00547-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability problem.
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
Last edited by gAr (2014-01-15 05:20:24)
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
New problem:
i. What is the expected value of the variable c in the following snippet of code?
ii. If 'a' is a n bit integer, what is the expected value of c till all n bits are set?
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
Hi gAr;
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: Probability problem.
Hi bobbym,
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
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: Probability problem.
Hi bobbym,
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."
Re: Probability problem.
A modified version of the previous problem:
iii. What is the expected value of the variable c in the following snippet of code?
iv. If 'a' is a n bit integer, what is the expected value of c till all n bits are set?
"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?
"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay." | {"url":"http://www.mathisfunforum.com/viewtopic.php?id=19518&p=14","timestamp":"2014-04-21T12:26:00Z","content_type":null,"content_length":"45073","record_id":"<urn:uuid:ba0e71d8-3783-45d7-95fc-80658e28bc6d>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00625-ip-10-147-4-33.ec2.internal.warc.gz"} |
On an N x N board, suppose that prior to the preliminary end of the game (the first two consecutive passes in territory rules I, equivalent to the first pass in area rules III) Black has played M1
moves and White has played M2 moves. We will continue to use 1 to designate Black and 2 to designate White. Let Li (i = 1, 2) be the number of stones the board. Then Black has lost M1 - L1 prisoners
and White has lost M2 - L2. Let Ti be the number of stones played after the preliminary end, and Pi be the number passed as prisoners. Let Qi be the number of stones on the board at the end of the
game, and let Si be the amount of territory surrounded.
The number of prisoners captured after the preliminary end is Li + Ti - Qi. The total number of prisoners is therefore Mi + Ti - Qi + Pi.
Under area rules III, the scores are:
Black Q1 + S1 - (M1 - M2)/2
White Q2 + S2 + (M1 - M2)/2
The final term (M1 - M2)/2 is the half point added and subtracted when Black makes the last competitive move, which occurs when M1 - M2 = 1. The difference D(area III) between Black's score and
White's is:
D(area III) = black - white = (S1 - S2) + (Q1 - Q2) - (M1 - M2)
Under territory rules I, the scores (territory minus prisoners) are:
Black S1 - M1 - T1 + Q1 - P1
White S2 - M2 - T2 + Q2 - P2
D(ter. I) = (S1 - S2) + (Q1 - Q2) - (M1 - M2) - (T1 + P1 - T2 - P2)
By the rule of equal number of moves after the preliminary end of the game,
T1 + P1 = T2 + P2
T1 + P1 - T2 - P2 = 0
D(ter. I) = (S1 - S2) + (Q1 - Q2) - (M1 - M2)
D(ter. I) = D(area III)
So we have proved that area rules III and territory rules I are in complete agreement, even though one of them counts stones plus territory while the other counts territory minus prisoners. This
comes from the addition of the last competitive move condition to area rules III.
Another consequence of this proof is that since area rules II do not have the half-point adjustment, when Black makes the last competitive move, area rules II are one point different from area rules
III and territory rules I. Note that this does not depend on the size N of the N x N go board. There is a misconception that the one-point difference arises only when N is an odd number, but we have
seen that this is not true.
Table 2 gives the numerical values for the Go-Miyamoto game. | {"url":"http://gobase.org/studying/rules/ikeda/?sec=e4030000","timestamp":"2014-04-16T19:26:37Z","content_type":null,"content_length":"18263","record_id":"<urn:uuid:30b7d556-4514-44e9-8ec0-d70139d64e17>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00439-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problem Set - II 9/26/11
(Due at the beginning of class on 10/12/11)
CS 7800 Advanced Algorithms
This problem set will be graded out of 50 points. It will count for 20% of your final grade.
1) Random questions
a) Find the minimum and maximum of n numbers using no more than (3n/2) - 2 comparisons. [2]
b) Find the minimum and second minimum of n numbers using no more than n + log[2]n – 2 comparisons. [2]
c) Give an example, along with proof, of a monotone increasing function from the natural numbers to the natural numbers that grows faster than any computable function. [2]
d) A Platonic solid is the 3D analogue of a regular polygon; all its faces are congruent regular polygons. Use Euler’s formula to argue that there are exactly 5 Platonic Solids. [4]
2) Greedy
a) Given a list of n natural numbers d[1], d[2]… d[n], show how to decide in polynomial time whether there exists an undirected simple graph whose node degrees are precisely the numbers d[1], d[2]… d
[n]. A simple graph is one that does not contain self loops or multi-edges. [4]
b) Given a network (undirected graph) with an available bandwidth for each link (edge), the bottleneck bandwidth of a path is defined to be the minimum bandwidth of any edge on the path and the
bottleneck bandwidth of a pair of nodes, u, v, is defined to be the maximum, over all paths connecting u to v, of the bottleneck bandwidth of that path. Prove that there exists a tree such that for
each pair of nodes their bottleneck bandwidth in the tree is the same as in the graph. Give a polynomial time algorithm to find such a tree. [4]
c) Given two sets of n numbers a[1], a[2]…, a[n] and b[1], b[2]…b[n], find, in polynomial time a permutation Π such that ∑[i] |a[i] - b [Π(][i)]| is minimized.? [2]
3) Divide and Conquer
a) You are given n FPGA chips and an FPGA tester. The tester takes two FPGA chips and tells you whether the chips are equivalent (from a logic standpoint) or not. Give an algorithm that runs O(nlogn)
tests to decide whether more than half the chips are equivalent. [2]
b) The setup is the same as in 2a). Give an algorithm that runs O(n) tests to decide whether more than half the chips are equivalent. [4]
c) You are given a complete binary tree with n nodes (n = 2^d -1 for some d). Each node is labeled with a unique real number. A node is a local minimum if its label is less than the labels of its
neighbors. You can find out the label of a node only by probing it. Show how to find a local minimum using O(log n) probes. [4]
4) Matroids
a) Prove that the exchange property does not imply hereditariness, i.e. give an example of a set system which contains the empty set and has the exchange property but is not hereditary. [3]
b) A matching in a graph is defined to be a collection of edges such that no two edges share an endpoint. An edge in a matching is said to cover its endpoints. A vertex is said to be covered by a
matching if it is covered by some edge in the matching. A subset of vertices is said to be coverable if there exists a matching that covers each vertex in the subset. Let F be the family of coverable
i) Prove that (V,F) has the hereditary property. [1]
ii) Given two matchings M[1] and M[2] prove that M[1] ∆ M[2] (∆ denotes the symmetric difference, i.e. M[1 ]–[ ]M[2] U M[2] – M[1]) consists of alternating cycles and paths i.e. paths and cycles
where the edges alternate between M[1] edges and M[2] edges. [2]
iii) Prove that (V, F) has the augmentation property. (Hint use 4a ii)) [4]
5) More problems
a) Given an array of n numbers find in linear time that contiguous sub-array which has the largest sum. [4]
b) Suppose I give you a graph with n nodes and tell you that it is 3-colorable (i.e. the vertices can b colored with one of 3 colors such that no edge has both its endpoints of the same color) but I
don’t tell you the coloring. The following subparts will show how you can color this graph with no more than 4*n^1/2 colors.
i) Consider the following step – if the graph has a vertex of degree >= n^1/2 then remove the vertex and its neighbors from the graph. Prove that you can run this step only at most n^1/2 times before
you have no vertex with degree >= n^1/2 left. [2]
ii) Prove that after running 5b i) as many times as you can, the residual graph can be colored with n^1/2 colors in polynomial time. [1]
iii) Prove that the entire graph can be colored with 4*n^1/2 colors. [2]
iv) Can you give a polynomial-time algorithm to color any 3-colorable graph with 3 colors? [1] | {"url":"http://www.ccs.neu.edu/course/cs7800f11/ProblemSet-II.htm","timestamp":"2014-04-19T05:21:09Z","content_type":null,"content_length":"48948","record_id":"<urn:uuid:1deede4a-8158-4bcf-9c1e-7fb1e8bffe67>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00291-ip-10-147-4-33.ec2.internal.warc.gz"} |
umber definition
Prime Number definition, examples
A prime number is a number whose only factors are 1 and itself. That means there is no whole number that evenly
the prime number.
Some often-confused facts about prime numbers
• Zero is not a prime a number
• The number one, 1, is also not a prime number. Although the definition of a prime number seems to apply to 1, you have to count 1 twice --sorry no 'double dipping' for prime numbers. 1 is not
• The only even prime number is 2, because all of the other even prime numbers are multiples of 2 and therefore violate the definition of a prime number (only divisible by 1 and itself)
• Read more about how to determine if a number is prime | {"url":"http://www.mathwarehouse.com/arithmetic/numbers/prime-number/","timestamp":"2014-04-18T20:42:16Z","content_type":null,"content_length":"15988","record_id":"<urn:uuid:84a0ea59-5106-4555-b9b1-4bd2c388ccc4>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00151-ip-10-147-4-33.ec2.internal.warc.gz"} |
Counting by fives - Skip Counting - Printables, Worksheets, and Lessons
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Math Counting by Fives
Skip Counting Printables, Worksheets, and Lessons
Counting by 5s Chart (only fives in chart - numbers 5 to 100) - Fill in Missing Numbers
One Page Printables - Rows Have 10 Columns
Counting by 5s chart: Fill in 2 numbers (only fives in chart - numbers 5 to 100)
Counting by 5s chart: Fill in 4 numbers (only fives in chart - numbers 5 to 100)
Counting by 5s chart: Fill in 6 numbers (only fives in chart - numbers 5 to 100)
One Page Printables - Rows Have 5 Columns
Counting by 5s chart: Fill in 2 numbers (only fives in chart - numbers 5 to 100)
Counting by 5s chart: Fill in 4 numbers (only fives in chart - numbers 5 to 100)
Counting by 5s chart: Fill in 6 numbers (only fives in chart - numbers 5 to 100)
Counting by 5s on a Hundreds Chart - Fill in Missing Numbers
One Page Printables - Rows Have 10 Columns
Hundreds Chart - Blanks are only fives: Fill in 5 numbers
Hundreds Chart - Blanks are only fives: Fill in 10 numbers
Hundreds Chart - Blanks are only fives: Fill in 15 numbers
Hundreds Chart - Blanks are only fives: Fill in 20 numbers
Two Page Printables (landscape) - Rows Have 10 Columns
Hundreds Chart - Blanks are only fives: Fill in 5 numbers
Hundreds Chart - Blanks are only fives: Fill in 10 numbers
Hundreds Chart - Blanks are only fives: Fill in 15 numbers
Hundreds Chart - Blanks are only fives: Fill in 20 numbers
Two Page Printables (landscape) - Rows Have 5 Columns
Hundreds Chart - Blanks are only fives: Fill in 5 numbers
Hundreds Chart - Blanks are only fives: Fill in 10 numbers
Hundreds Chart - Blanks are only fives: Fill in 15 numbers
Hundreds Chart - Blanks are only fives: Fill in 20 numbers
Fives are Blank Sequences
10 Number Sequences
Numbers One to One Hundred: Fill in 1 to 3 numbers (only fives can be blank)
Numbers One to One Hundred: Fill in 3 to 5 numbers (only fives can be blank)
5 Number Sequences
Numbers One to One Hundred: Fill in 1 number (only fives can be blank)
Numbers One to One Hundred: Fill in 1 to 2 numbers (only fives can be blank)
Countby by Fives Sequences
10 Number Sequences
Fill in 1 to 3 numbers (only fives in sequences - numbers 5 to 100)
Fill in 3 to 5 numbers (only fives in sequences - numbers 5 to 100)
5 Number Sequences
Fill in 1 number (only fives in sequences - numbers 5 to 100)
Fill in 1 to 2 numbers (only fives in sequences - numbers 5 to 100)
Counting by 5's Dot to Dots
Ancient Egypt count by fives dot to dot
Birds count by fives dot to dot
Deserts count by fives dot to dot
Dinosaurs count by fives dot to dot
Explorers count by fives dot to dot
Fresh Water Animals count by fives dot to dot
Grasslands count by fives dot to dot
Insects count by fives dot to dot
Inventions count by fives dot to dot
Mammals count by fives dot to dot
Oceans count by fives dot to dot
Polar Region Animals count by fives dot to dot
RainForest count by fives dot to dot
Reptiles count by fives dot to dot
More Skip Counting
Counting by twos
Counting by threes
Counting by fours
Counting by sixes
Counting by sevens
Counting by eights
Counting by nines
Counting by tens
Hundreds Chart
Five by Five with Gum (Grades 1-2)
Have a suggestion or would like to leave feedback?
Leave your suggestions or comments about edHelper! | {"url":"http://www.edhelper.com/counting_by_fives.htm","timestamp":"2014-04-16T04:22:36Z","content_type":null,"content_length":"17757","record_id":"<urn:uuid:3bf392c2-09a2-49e2-af58-c05ac450d3b3>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00193-ip-10-147-4-33.ec2.internal.warc.gz"} |
Paging is one of the most prominent problems in the field of online algorithms. We have to serve a sequence of page requests using a cache that can hold up to k pages. If the currently requested
page is in cache we have a cache hit, otherwise we say that a cache miss occurs, and the requested page needs to be loaded into the cache. The goal is to minimize the number of cache misses by
providing a good page-replacement strategy. This problem is part of memory-management when data is stored in a two-level memory hierarchy, more precisely a small and fast memory (cache) and a
slow but large memory (disk). The most important application area is the virtual memory management of operating systems. Accessed pages are either already in the RAM or need to be loaded from the
hard disk into the RAM using expensive I/O. The time needed to access the RAM is insignificant compared to an I/O operation which takes several milliseconds. The traditional evaluation framework
for online algorithms is competitive analysis where the online algorithm is compared to the optimal offline solution. A shortcoming of competitive analysis consists of its too pessimistic
worst-case guarantees. For example LRU has a theoretical competitive ratio of k but in practice this ratio rarely exceeds the value 4. Reducing the gap between theory and practice has been a hot
research issue during the last years. More recent evaluation models have been used to prove that LRU is an optimal online algorithm or part of a class of optimal algorithms respectively, which
was motivated by the assumption that LRU is one of the best algorithms in practice. Most of the newer models make LRU-friendly assumptions regarding the input, thus not leaving much room for new
algorithms. Only few works in the field of online paging have introduced new algorithms which can compete with LRU as regards the small number of cache misses. In the first part of this thesis we
study strongly competitive randomized paging algorithms, i.e. algorithms with optimal competitive guarantees. Although the tight bound for the competitive ratio has been known for decades,
current algorithms matching this bound are complex and have high running times and memory requirements. We propose the algorithm OnlineMin which processes a page request in O(log k/log log k)
time in the worst case. The best previously known solution requires O(k^2) time. Usually the memory requirement of a paging algorithm is measured by the maximum number of pages that the algorithm
keeps track of. Any algorithm stores information about the k pages in the cache. In addition it can also store information about pages not in cache, denoted bookmarks. We answer the open question
of Bein et al. '07 whether strongly competitive randomized paging algorithms using only o(k) bookmarks exist or not. To do so we modify the Partition algorithm of McGeoch and Sleator '85 which
has an unbounded bookmark complexity, and obtain Partition2 which uses O(k/log k) bookmarks. In the second part we extract ideas from theoretical analysis of randomized paging algorithms in order
to design deterministic algorithms that perform well in practice. We refine competitive analysis by introducing the attack rate parameter r, which ranges between 1 and k. We show that r is a
tight bound on the competitive ratio of deterministic algorithms. We give empirical evidence that r is usually much smaller than k and thus r-competitive algorithms have a reasonable performance
on real-world traces. By introducing the r-competitive priority-based algorithm class OnOPT we obtain a collection of promising algorithms to beat the LRU-standard. We single out the new
algorithm RDM and show that it outperforms LRU and some of its variants on a wide range of real-world traces. Since RDM is more complex than LRU one may think at first sight that the gain in
terms of lowering the number of cache misses is ruined by high runtime for processing pages. We engineer a fast implementation of RDM, and compare it to LRU and the very fast FIFO algorithm in an
overall evaluation scheme, where we measure the runtime of the algorithms and add penalties for each cache miss. Experimental results show that for realistic penalties RDM still outperforms these
two algorithms even if we grant the competitors an idealistic runtime of 0.
We propose a variation of online paging in two-level memory systems where pages in the fast cache get modified and therefore have to be explicitly written back to the slow memory upon evictions.
For increased performance, up to alpha arbitrary pages can be moved from the cache to the slow memory within a single joint eviction, whereas fetching pages from the slow memory is still
performed on a one-by-one basis. The main objective in this new alpha-paging scenario is to bound the number of evictions. After providing experimental evidence that alpha-paging can adequately
model flash-memory devices in the context of translation layers we turn to the theoretical connections between alpha-paging and standard paging. We give lower bounds for deterministic and
randomized alpha-paging algorithms. For deterministic algorithms, we show that an adaptation of LRU is strongly competitive, while for the randomized case we show that by adapting the classical
Mark algorithm we get an algorithm with a competitive ratio larger than the lower bound by a multiplicative factor of approximately 1.7. | {"url":"http://publikationen.ub.uni-frankfurt.de/solrsearch/index/search/searchtype/authorsearch/author/%22Andrei+Negoescu%22/start/0/rows/10/author_facetfq/Andrei+Negoescu","timestamp":"2014-04-18T05:36:56Z","content_type":null,"content_length":"28415","record_id":"<urn:uuid:3926b143-e3c7-420e-b773-7c93587389a0>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00417-ip-10-147-4-33.ec2.internal.warc.gz"} |
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MathGroup Archive: July 1998 [00344]
[Date Index] [Thread Index] [Author Index]
Re: Q: Combining NDSolve with FindRoot
• To: mathgroup at smc.vnet.net
• Subject: [mg13382] Re: Q: Combining NDSolve with FindRoot
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Thu, 23 Jul 1998 03:32:30 -0400
• Organization: University of Western Australia
• References: <6okkoi$1lf@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com
[Contact the author to obtain the notebook or go to http://smc.vnet.net/paul.nb -
Anil Trivedi wrote:
> Trying to learn mathematica, I thought I would verify the following.
> The "harmonic oscillator" equation:
> y''[x] + (2e-x^2) * y[x] =0,
> y[0]=1, y'[0]=0
> has solutions which vanish for large x only if e is one of the
> eigenvalues e= 0.5, 2.5, 4.5, 6.5, etc.. How can I generate this
> series, or the exact function e[n] = 2n+1/2 where n=0,1,2,..?
See the attached Notebook (taken from an exam question from my
computational physics course here at UWA) which addresses a similar
> Focussing on the first eigenvalue e = 0.5, let us try to (i) solve the
> equation with NDSolve,
BTW, here is one way of doing this (assuming that you already know the
NDSolve[{y''[x] + (1 - x^2)*y[x] == 0, y[0] == 1, y'[0] == 0},
y[x], {x, -5, 5}];
Plot and compare with the exact solution:
Plot[Evaluate[{HermiteH[0, x]/E^(x^2/2), y[x] /. First[%]}],
{x, -5, 5}, PlotStyle -> {Hue[1/3], Hue[1]}];
>(ii) evaluate the soln at some large x = L,
> (iii) call the resulting function z[e], and (iv) use FindRoot to solve
> z[e]=0, with a good intitial guess like 0.45. :)
Basically, I think that you would need to use a series solution method,
after factoring off the appropriate asymptotic form, to compute the
eigenvalue in this way. Alternatively, the Notebook demonstrates a
general matrix method for the approximate determination of the
> 3. Assuming I can do this for one eigenvalue, what is the best
> "mathematica way" of iterating the procedure to obtain the first N
> eigenvalues? (I doubt it is Do loop, but I don't know what it is.)
The matrix method with an n x n matrix yields the first n eigenvalues.
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia Nedlands WA 6907
mailto:paul at physics.uwa.edu.au AUSTRALIA
God IS a weakly left-handed dice player | {"url":"http://forums.wolfram.com/mathgroup/archive/1998/Jul/msg00344.html","timestamp":"2014-04-17T04:05:14Z","content_type":null,"content_length":"36859","record_id":"<urn:uuid:582f123c-44dd-4f32-bb48-c68922148c82>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00641-ip-10-147-4-33.ec2.internal.warc.gz"} |
Synthesized Cluster Head Selection and Routing for Two Tier Wireless Sensor Network
Journal of Computer Networks and Communications
Volume 2013 (2013), Article ID 578241, 11 pages
Research Article
Synthesized Cluster Head Selection and Routing for Two Tier Wireless Sensor Network
^1Department of Computer Engineering, Sarvajanik College of Engineering & Technology, Athwalines, Surat 395001, Gujarat, India
^2Department of Computer Engineering, Sardar Vallabhbhai National Institute of Technology, Ichhanath, Surat 395007, Gujarat, India
Received 15 April 2013; Revised 21 June 2013; Accepted 22 June 2013
Academic Editor: Rui Zhang
Copyright © 2013 Keyur Rana and Mukesh Zaveri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Large scale sensor networks can be efficiently managed by dividing them into several clusters. With the help of cluster heads, each cluster communicates using some routing schedule. It is essential
to rotate the role of cluster heads in a cluster to distribute energy consumption if we do not have dedicated high energy cluster heads. Usually routing and cluster head selection for such networks
have been separately solved. If cluster heads are selected with the consideration of routing and routing schedule is prepared with the consideration of selected cluster heads, it can help each other.
We have proposed an integrated approach of cluster head selection and routing in two tier wireless sensor network (WSN) using Genetic Algorithm based cluster head selection with A-Star algorithm
based routing method to extend life of WSN. This approach can lead to significant improvements in the network lifetime over other techniques.
1. Introduction
Wireless Sensor Networks are composed of a large number of sensor nodes with limited resources in terms of energy, memory, and computation. They are operated by a small battery attached to it. This
battery has some initial energy, and in every communication it dissipates a fraction of the energy. Many such communications take place during the network lifetime, and every time sensor node
consumes some energy which makes battery exhaust eventually. When nodes are deployed in hostile environment or in a kind of environments where it is hard to reach, in most of the cases there is no
way to recharge these batteries.
Sensor nodes are used for monitoring physical phenomena like temperature, humidity, acoustic, seismic, video, and so on [1]. For large scale wireless sensor networks, applications exist in a variety
of fields, including medical monitoring [2–4], environmental monitoring [5, 6], surveillance, home security, military operations, and industrial machine monitoring [7]. To fulfill the requirements of
these applications, sensor network should have a lifetime long enough to cater for several months. How to prolong the network lifetime to such a long time is the vital question to design and manage
sensor network systems.
Randomly deployed sensor nodes in the field collect required data and send towards the base station after processing them. If the optimal path (in terms of energy consumption or quality of service)
is chosen for each round of communications, nodes of that particular path may get drained of energy, and network can get partitioned soon. We consider the end of network life as soon as the network
gets partitioned as in [8]. Many approaches have been proposed in the literature for routing and for cluster head selection in WSN to extend the lifetime [9–13].
Data transmission in WSN can be single hop or multihop. In either case, data collected by sensor nodes are sent to the base station. The large and dense network is divided into several clusters, and
each cluster contains one cluster head that is responsible to collect and send data to the base station. The sensor nodes are grouped into clusters geographically and are capable of operating into
two roles. It can work as a sensor node and as the cluster head node [14, 15]. As a sensor node, the node senses the task and sends the sensed data to its cluster head. As a cluster head node, a node
gathers data from its cluster members, performs data fusion, and transmits the data to the base station.
In the single-hop data transmission model [11, 12], the cluster heads send data directly to the base station. The transmission power dissipated by a sender node to transmit each bit of data to a
receiver node is proportionate to the distance between the sender and the receiver [1, 12, 16–18]. Therefore, cluster heads that located farther away from the base station will get drained quickly
due to the large distance for communication. For the large network, multihop data transmission model [19–22] can be used in which such farther located nodes use some intermediate nodes (cluster
heads) to forward the data towards the base station. These cluster heads form a network among themselves to send data to the base station. In this case, the cluster head not only transmit data
gathered from the sensor nodes in their respective clusters but also forward data from other clusters towards the base station. This kind of network is depicted in Figure 1. Entire network with
clusters and cluster head is shown in Figure 1(a). We have spotted cluster head selection and routing using these selected cluster heads as different problems. The network shown in Figure 1(a) is
logically divided into two tiers, namely, lower tier and upper tier. The cluster head selection is to be solved at the lower tier. Selected cluster head forms a network amongst themselves which is
shown in the upper tier of Figure 1(b). Routing using this selected cluster head is to be solved at the upper tier.
In [17, 23–25], authors have focused on the upper tier of the network to find the routing schedule. It is assumed in their approach that the relay nodes (cluster heads) contain relatively high energy
compared to normal sensor nodes, and these nodes are used throughout the network lifetime as cluster heads. But if we do not have such relatively high energized special nodes, we must think about the
changeover of cluster heads with the other member nodes.
By following nonadaptive routing where energy is not taken into account, if a fixed node serves as a cluster head throughout then the battery of this unlucky node will get depleted and will die
quickly. This will result in the end of useful lifetime of all nodes belonging to that cluster. Selection of cluster head periodically and strategically can avoid such circumstances. Low Energy
Adaptive Clustering Hierarchy (LEACH) [11, 12], Hybrid Energy Efficient Distributed Clustering (HEED) [13], Energy Efficient Dynamic Clustering Algorithm (EEDCA) [26], Genetic Algorithm based
Weighted Clustering Algorithm (GA-WCA) [27], and Location Aware 2 Dimensional Genetic Algorithm (LA2D-GA) [28] are algorithms for cluster head selection while GA based routing [23] and Minimum Hop
Routing Model (MHRM) [17, 25] are centralized routing algorithms for WSN available in the literature.
In this paper, we have proposed Genetic Algorithm (GA) based approach to select a cluster head for each cluster optimally for each round of data transmission such that the overall lifetime of a
sensor network is increased. This is done at the lower tier. For the upper tier, we have proposed A-Star algorithm based routing which uses these selected cluster heads to send data to the base
station. Simulation result shows that our synthesized approach of GA based proposed technique to select a cluster head with A-Star algorithm based routing certainly improves the life of sensor
network and outperforms LEACH, HEED, EEDCA, and GA-WCA methods of cluster head selection (at lower tier), GA [23], and MHRM routing algorithms (at upper tier).
Routing and cluster head selection are two important factors which affect life time of two tier WSN, but they have been focused and solved separately traditionally. We have proposed integration of
these two approaches to extend lifetime of WSN in this paper. We assume that the cluster head selection and routing schedule generation are carried out at the base station which is broadcasted
The rest of the paper is organized as follows. In the next section we briefly review correlated matter. In Section 3 GA based cluster head selection is discussed and Section 4 contains A-Star
algorithm based routing. Section 5 discusses synthesized cluster head selection and routing algorithm followed by Section 6 which shows simulation results and analysis.
2. Review
Many protocols have been proposed for the sensor networks in the last few years. Since sensors are typically battery operated with limited energy supply, many researchers have focused on issues like
energy aware routing [29]. Reducing energy consumption due to wasteful sources has been primarily addressed in the context of adaptive MAC protocols, such as EAR [30] and S-MAC [31] which
periodically puts nodes to sleep to avoid idle listening and overhearing. Distributed clustering approaches like Distributed Clustering Algorithm (DCA) [32] assumes quasi-stationary nodes with real
valued weights, and the Weighted Clustering Algorithm (WCA) [33] combines several properties in one parameter called weight, that is used for clustering. Boukerche et al. [34] proposed a clustering
routing protocol which uses the following: the nearest neighbor approach, alternation of nodes responsible for the intercluster communication, and alternation of possible routes to the base station.
In [35], Hao et al. proposed a geographical based multihop clustering algorithm which divides the network area into small regions by adopting multihop links for the intercluster communication.
In the literature, many cluster head selection methods are proposed; LNCA [36], ACE [37], LEACH [11, 12], HEED [13], EEDCA [26], GA-WCA [27], and LA2D-GA [28] are few of them. Local Negotiated
Clustering Algorithm (LNCA) [36] presents a novel clustering algorithm, which employs the similarity of nodes readings as an important criterion in cluster formation. ACE clusters the sensor network
in a constant number of iterations using the node degree as the main parameter. LEACH utilizes a randomized fair rotation of cluster heads to evenly distribute the energy load among the sensor nodes
in the network. HEED periodically selects cluster heads according to a hybrid of sensor node’s residual energy and node proximity, a secondary parameter. EEDCA maintains a neighbor table for the
cluster head which contains information about its member nodes such as ID number, location, residual energy, and so forth. If cluster head’s residual energy falls below threshold energy then the
process of selecting new cluster head starts. In GA-WCA, load balanced factor is considered as one of the weights along with a sum of distance from all neighbor nodes to cluster heads. LA2D-GA takes
only distance as a parameter to calculate fitness function; however, representation of a chromosome is a two-dimensional grid which represents valid statistics of a WSN. Having only chromosome
representation different with a traditional fitness function, it does not help much in improving the lifetime of WSN.
In the upper tier, Bari et al. [23] uses a GA based routing in which a chromosome represents the routing schedule. Every th gene’s value is a node number to whom th node has to forward the packet.
The fitness value is the number of rounds (network lifetime) which is dependent on the maximum dissipated energy of any node of a routing schedule. Each period of data gathering is recognized as a
round [38]. In MHRM routing method, each relay node finds a path to the base station that minimizes the number of hops.
The problem identified is to find the best cluster heads so that using them, routing at upper tier, helps in extending the lifetime of WSN. This is appeared as an optimization and a search problem
having enormous search space. In order to find out a better solution, we need some heuristic search methods; hence, we have proposed GA based and A-Star algorithm based solutions which would work at
lower tier for cluster head selection and at upper tier for routing, respectively.
We have proposed a GA based solution to select cluster head to overcome these limitations. Out of many possible cluster head solutions, we search for a cluster head by which overall communication
cost and the number of weak nodes are minimized. We have discussed these parameters in detail in Section 3.1. In Figure 2, as per our GA approach, node 2 will be selected as a cluster head
irrespective of the current solution and by doing so, overall nodes with enough residual energy level are also maintained.
Out of many possible routing schedules, one typical routing schedule is shown in Figure 1(a) which is represented by dotted arrows. It can be seen that cluster head of cluster 3 sends data to the
base station through cluster 1 and cluster 2. In the next instance, it may be through clusters 4 and 5 or directly forwarded to the base station for the new routing schedule. Hence, cluster heads for
each routing would be different to conserve energy. Considering the current routing schedule, we select cluster heads in our synthesized approach. It helps in improving the network life.
2.1. Overview of GA
GA is modeled after the natural process of survival of the fittest. For most genetic algorithms, the main concept is that the strongest individuals survive a generation and recombine with other
survivors, producing an even stronger child. We have used standard GA terminology as in [39, 40]. The individuals are represented by character strings or an array of genes, often referred to as
chromosomes. From the initial solution, using the principle of survival of the fittest by applying selection, cross-over, and mutation operator of GA, new generation is prepared. This process is
repeated until GA converges or fixed numbers of iterations are performed [39, 41].
2.2. Overview of A-Star Algorithm
A-Star algorithm is used to find a path and to traverse the graph efficiently by using heuristics for the decision making. The A-Star algorithm [42] is a best-first search algorithm that finds the
optimal path from source to the destination.
It uses a distance and a cost heuristic function (usually denoted for node ) to determine the order in which the search visits nodes in the tree. The distance-plus-cost heuristic is a sum of the
following two functions [43]:(1)the path-cost function, which is the cost from the starting node to the current node (usually denoted ) and(2)an admissible "heuristic estimate" of the distance to the
goal (usually denoted ).
Thus for node , is, intuitively, the estimate of the best solution that goes through .
2.3. Network Model
We have considered two tier sensor network model as shown in the Figure 1 with the assumption of the following properties. Nodes always have data to send, and there is only one base station located
far away from the network. All nodes have similar capabilities of processing and communication and are energy constrained. Nodes are randomly deployed having the same initial energy and left
unattended after the deployment; therefore, battery recharge is not possible. All nodes communicate through an ideal shared medium where communication between nodes is handled by the proper MAC
protocol as in [17, 44]. We are focusing on nonflow splitting routing model which avoids many limitations of the flow splitting model [19]. All the sensor nodes are homogeneous and most of the time
As shown in Figure 1, assume that there are total nodes and number of clusters. The value of can be computed as given in [11, 12]. Our goal is to select such cluster heads optimally for each round of
data transmission and using these cluster heads, to search for routing schedule so that the overall life of WSN is improved.
3. GA Based Cluster Head Selection
Dividing network in several clusters helps in the distribution of energy dissipation. Ideal number of clusters in a large scale network supports proper distribution. It has been studied in [12] that
5% of total nodes as cluster heads (i.e., number of clusters) give optimal results. We have kept the number of clusters fixed and worked on assignment of cluster heads per cluster.
3.1. Initial Population and Fitness Function
The chromosome is represented as a string of node numbers. Each number at index represents the cluster head for the cluster . For the network diagram shown in Figure 1(a), chromosome representation
is shown in Figure 2.
This indicates that for cluster 1 node 4, for cluster 2 node 13, for cluster 3 node 7, for the cluster 4 node 16, and for the cluster 5 node 22 are chosen as cluster head. From each cluster, randomly
one node is selected as a cluster head. This process of creating chromosome for initial population is repeated for each chromosome of the first generation.
The fitness of a chromosome represents its credentials of the solution and helps in finding stronger or weaker nodes. Following parameters are considered to calculate fitness value in our approach.
(1)Total Distance as per current Cluster Head selection ().(2) Measure of Weak Nodes (WN[count]).It is represented as a pair of (, WN[count]).
. Total distance as per current cluster head is considered as one of the parameters in the fitness calculation because the energy will be lost in proportion to the distance of communication. It can
be calculated as follows: where is the number of clusters and is the total number of member nodes in a particular cluster. is the distance from node to cluster head of cluster , whereas is the
distance between cluster head of cluster and the base station. For a better solution, this value must be as low as possible.
. Measure of weak nodes is another and prime parameter of fitness calculation. It signifies the health of nodes for a current set of cluster head selection which is measured with respect to the
predefined threshold level of residual energy, TLevel. It is used to explore whether a node is weak or not and if yes then to what extent?
This can be achieved by introducing different levels of energy of node. A node having initial energy , will also have another mark of energy, TLevel of energy (say 40% of ). It has been
experimentally found in [45] that the value of TLevel can be set from 30% to 40% of to achieve better results.
Calculation of Weak Node. WN[count] is calculated by keeping count of the total number of nodes which is on the current selected schedule and is below predefined threshold level of energy TLevel. The
residual energy of the node will be checked, and if it is found less than TLevel energy, then this count will be incremented by Incremental_factor. Value of Incremental_factor is determined as per
the strategy.
Network having sensor nodes with a wide transmission range and located farther away from each other will cause more amount of energy consumption at each transmission. Energy consumption factor will
be more in such case and once residual energy is below TLevel, only for a few more rounds that node can serve as a cluster head and soon it will get exhausted. Thus, strategy for assigning value to
Incremental_factor can be promptly 1. For the dense network having sensor nodes located nearer to each other, energy consumption factor will be less as compared to the previous discussed case. If we
still continue with adding 1 to the Incremental_factor, it will treat those nodes whose residual energy has just entered below TLevel and nodes that were already below TLevel and now about to reach
to zero energy as the same level of weak node. To have precise consideration of energy utilized and available energy (which is required in this type of case), we have divided residual energy from 0
to TLevel further into several subparts. It will differentiate a node having 5% of residual energy left and a node having 25% of residual energy left, quite significantly. The intensity of a node’s
weakness is calculated by observing where the current residual energy of a node falls in this subpart. Incremental_factor value will be incremented more and more as the residual energy (of course
below TLevel) goes nearer to the zero value.
3.2. Selection, Crossover, and Mutation
The selection of individuals is carried out using the roulette wheel, rank based selection, or tournament selection method [39]. We have applied tournament selection method with tournament size = 5
which is 10% of the population size. To produce new offspring from the selected parents, the uniform cross-over or k-point cross-over can be used for each cross-over operation. We know that much of
the power of GA comes from the fact that it contains a rich set of strings of great diversity. Mutation helps to maintain that diversity throughout the GA iterations. According to the mutation
probability, one random gene of a chromosome is replaced by a better node if available, otherwise replaced by a random node.
3.3. Cluster Head Selection
While searching for the cluster head the criteria will be as follows,(1)total energy consumption should be minimum and(2)in a cluster head selection, by selecting any node as a cluster head, if WN
[count] is found with comparatively larger value (it means that some nodes are affected by this selection and become weak node), that node is avoided to be considered as a part of cluster head
selection schedule. Instead, a new node having lesser value of WN[count] is sought although total energy consumption would be higher than the previous selection.
The member nodes of each cluster send data to respective cluster head. Cluster head aggregates data and sends towards the base station by following routing schedule which is searched at the upper
4. A-Star Algorithm Based Routing in Two Tier WSN
Given a collection of cluster heads, numbered from 1 to , and a base station, numbered as , along with their locations, the objective of the A-Star algorithm is to find a schedule for data gathering
in a sensor network, such that the lifetime of the network is maximized. Each sensor node transmits fixed number of packets of data containing a fixed number of bits, in each round. Each period of
data gathering is referred to as a round [38], and the lifetime is measured by the number of rounds until the first relay node runs out of power (N-of-N metric) [46].
Routing schedule is computed at the central entity (base station), and we have assumed that the average amount of data transmitted by each relay node is fixed and is known to the base station. Base
station calculates optimal routing schedule and broadcasts it. Every cluster head follows this schedule. This process of finding optimal path, broadcasting it in the network, and sending data from
all clusters to the base station by following this schedule is repeated in every round. Computation of routing schedule is done dynamically with the consideration of current level of energy of each
cluster head. For this, normally it may require the cluster heads to report their residual energy periodically to the base station to coordinate their status. The base station can then determine the
routing schedule based on this updated information.
4.1. Routing Schedule
A-Star Algorithm can be used to find an efficient path between any sources to destination. In a network if the source node is and the destination node is , then for every intermediate node on the
path, will be the actual cost to reach node from source node , and will be estimated heuristic cost from the current node to the destination node . We have considered actual energy so far expensed to
send packet from the source node to node , to calculate , while is calculated as the estimated energy consumption to send packet from node to the destination node . This can be calculated by
estimating distance between nodes. Energy consumption is calculated as per (2).
A-Star algorithm will be applied for each cluster head. The cluster head where this algorithm is applied will be the source node, and the base station will be destination node. Such different routes
will be created, and these all information is consolidated. This consolidated route information is put in an array. This is shown in Figure 3 for a network shown in Figure 1(a). Note that the base
station is numbered as 6 (i.e., ), and dotted arrow shows the flow of data from one cluster head to the other.
The array has number of indices. Value at th index will represent node number as to where node will be sending data, which in turn, will go to the base station in a same way. For example, in Figure 3
, node 3 will send data to node 1, then node 1 will send data to node 2, and finally node 2 will send it to the node 6 (base station).
4.2. Routing Parameters
4.2.1. Node Strength (Path Cost Count)
In routing, only considering the total amount of energy consumed will not be efficient because it will drain some of the nodes which are on the efficient path. To avoid network partitioning due to
this, energy usage should be balanced and distributed. As discussed earlier, concept of TLevel is used here as well.
While making decision for the routing, we observe the route strength by considering whether a node is a strong node or not and if yes, then to what intensity. In a route, if a node is found having
below TLevel of residual energy, then alternate route is selected with a node having more energy than previous case. Taking this into account, overall energy consumption would be increased more than
the previous case. However, this alternate route will give life extension to those nodes which were selected in the first attempt. This helps in making healthy nodes participate in routing and weak
nodes getting rest, thus overall network lifetime can be extended. Calculating the value of Node Strength (path cost count) is the same as “Calculation of Weak Node” of Section 3.1.
4.2.2. Total Energy Consumed
This is another parameter for searching better routing schedule. As shown in (2), is the total energy consumed in transmitting sensed data by all sensor nodes to their cluster head , energy
dissipated by cluster head to receive such data from all its member nodes (), energy used in the aggregation of data at cluster head side (), and finally energy dissipation by cluster head to send
aggregated data towards the base station (). Here, is the total number of clusters and is the number of member nodes in the particular cluster
A-Star algorithm creates a tree structure in order to search for the optimal route from a given source to the destination. In our approach, in addition to and , we have also taken another parameter
to measure the strength of a route which is the path cost count of weak nodes having less energy.
Thus, for a node , estimated cost function carries two parameters. First parameter is a summation of and , and the other parameter is . can be defined as a two argument function , .
is a primary parameter, and a routing decision will be made based on its value. If is same for multiple routes then only, the value of is checked for further comparisons to search for the better
route. Value of is a path cost count which is calculated as discussed earlier.
5. Synthesized Cluster Head Selection and Routing Algorithm
We are proposing an integrated solution of cluster head selection and routing for two tier sensor network. Pseudocode 1 represents proposed solution. Line number 5 to 11 in the given pseudo code is
for cluster head selection using a GA approach which is integrated with A-Star algorithm based routing [24].
Explanation of functions and variables of Pseudocode 1 is as follows.
INITIALIZE_NETWORK (). This function will initialize the network, in terms of node id, node energy, node coordinates, and so forth. This will also find the estimated distance amongst various nodes
and the base station. Calculation of estimated distance from current node to the base station can be carried out from the methods mentioned in [22, 47] or location can be found using GPS system which
needs to be operated for a very small period of time [25, 44]. It can also be kept fixed by placing sensor nodes at a predetermined location. This distance will be useful to calculate energy
consumption between two nodes and also to get value for heuristic function, .
END_ASTAR (). It will check whether to terminate the process of A-Star based routing or not. This will check the residual energy of each node per cluster of the network. If any cluster is found
totally drained and the cluster is not able to send data, this function will return true, otherwise it will return false.
INITIALIZE_SOL_ARRAY (). It will initialize solution array where routing schedule is to be stored. Routing schedule is an array having number of index values. This routing schedule is discussed and
shown in Figure 3.
INITIALIZE_POPULATION (). It will initialize population of chromosomes (solutions). Each chromosome will carry number of index values. The value at index will contain id no of the node which is
cluster head of cluster . This chromosome representation is shown in Figure 2. This is needed for cluster head selection using GA.
NEXT_GENERATION (). It will perform all operations of GA and generate a new pool of population for the next generation. Parent selection, cross-over, and mutation operators are performed in this
function. The outcome of this function is a new set of chromosomes which would be better than the previous pool.
CALCULATE_FITNESS (). It will calculate the fitness value of each chromosome in the population. Fitness is calculated as discussed in Section 3.1.
CH_DECLARATION (). It will declare selected cluster heads which have been found by GA based approach. Every node will obtain information about their cluster head for the current round. This
information will be taken into account while searching for routing schedule by A-Star algorithm at the upper tier.
CREATE_TREE (i). It will expand the whole tree for a node using A-Star algorithm, to search for the optimal path from source node , to the destination node, the base station. The found solution is
stored in the solution array (routing schedule) as shown in Figure 3. This shall fill solution array partially. For example, for node 3, it will fill indices 3, 1, and 2 only as shown in Figure 3.
PREPARE_SOL_ARRAY(i). It will be called after a tree for a node is created to prepare the solution array which shall get partially filled. Solution array will be filled up in the same manner by every
node, and eventually it will be completely filled. When all nodes are covered, this solution array is ready which we call as a routing schedule.
BROADCAST_SOLUTION (). It will be called after a routing schedule is prepared to broadcast it from the base station. All cluster heads in the network will follow this solution and accordingly send
data towards the base station.
INVITE_STATUS (). This function is to ask each sensor node about their status of residual energy. This is not required to be called after every round, rather called after some fixed interval. Base
station can keep track of the status of all sensor nodes because of two known parameters, namely, location and amount of data to transfer. Using this, energy consumed by each node can be computed and
can be upgraded. But to improve accuracy, this function is called and statuses sent by sensor nodes are matched with the status available at the base station and updated accordingly.
COUNT_ROUND. This is a variable which keeps track of the total number of rounds that the network works for. After every round, the energy of all nodes will be updated, new cluster heads for each
cluster (by GA approach) will be searched, using these new cluster heads, and routing schedule will be searched. This will be broadcasted, and all cluster heads will follow this new schedule for
transferring their data. This variable is incremented by one at every iteration. When any cluster is exhausted by energy depletion, our algorithm will stop. We are measuring lifetime of WSN in terms
of rounds.
6. Experimental Results
We have performed simulation for different size of network to analyze performance of our algorithm. For our experiments, we have used first order radio model for communication energy dissipation of [
12] which has been widely used by many researchers as in [23, 48–52] where is the Euclidian distance between node and , is the transmit energy coefficient, is the amplifier coefficient, is the amount
of data to transmit from node to another node, and is the path loss exponent, . is total transmit energy dissipated.
Similarly, the receive energy, is calculated as follows: where is the number of bits received by node and is the receive energy coefficient.
Hence total energy dissipated by a node for data to receive and then to transmit it further is . Consider the following:
We consider both types of energy in computation of energy consumption. For simulation, the values for the constants are taken same as in [12] as follows:(i)nJ/bit,(ii)pJ/bit/m^2, and(iii)the path
loss exponent, .
The initial energy of each node J as in [22].
We have assumed that the average data generation rate of the sensor nodes and the allocation of sensor nodes to the clusters are known. For GA, the cross-over rate is relatively kept higher than the
mutation rate as it helps to evolve new offspring and takes an entire generation forward. The following is the list of parameters for GA and their values: population size = 50, number of iterations =
200, cross-over probability = 0.8 and mutation probability = 0.2, and selection method is tournament selection with tournament size = 5 and used single point cross-over.
In our simulation, we have assumed that sensor network is spread in 150 × 150 meters. We have experimented and observed several statistics for network sizes of 30, 50, and 80.
Figures 4, 5, and 6 show network lifetime in terms of rounds for several cluster head selections at the lower tier and some routing methods at the upper tier.
For different network size, we have compared proposed GA based cluster head selection with HEED, LEACH, EEDCA, and GA-WCA cluster head selection methods for different routing methods at the upper
tier. For routing, we have used proposed A-Star based routing and compared it with GA based routing of [23] and MHRM.
We have simulated all 5 cluster head selection methods for A-Star algorithm, GA based approach of Bari et al. [23], and MHRM. It is observed that proposed A-Star based routing algorithm outperforms
GA based of [23] and MHRM routing for all cluster head selection methods for different networks. It is also perceived that for different routing methods running at upper tier, proposed GA based
cluster head selection at the lower tier outperforms HEED, LEACH, EEDCA, and GA-WCA. Percentage improvement in the network lifetime for GA based cluster head selection against other cluster head
selection methods is shown in Figure 7. These statistics are observed for different routing methods at the upper tier of the network.
Improvements in the network lifetime using A-Star algorithm based routing as compared to other routing methods are shown in the Figure 8. These routing algorithms take cluster heads selected by
different cluster head selection methods from the lower tier. We have taken average improvement for different network sizes and depicted in these figures. Figures 7 and 8 show significant improvement
in the network lifetime using our proposed solution.
Our proposed algorithm works at the base station, and routing schedule is broadcasted by the base station. Sensor nodes follow this schedule, hence no computation is required at the sensor node
regarding routing. There is an overhead of sending status of sensor nodes to the base station after a fixed interval, but it is quite low as compared to other message handshake and communication
required for clustering and routing by other methods.
7. Conclusion
In this paper, we have proposed synthesized cluster head selection and routing algorithm for two tier WSN to extend life of the network. Cluster head selection and routing are two important aspects
when sensed data is required to send at the base station in a large network. We have proposed GA based cluster head selection with A-Star algorithm based routing mechanism and compared it with
routing algorithms like GA based [23], MHRM, and cluster head selection methods like HEED, LEACH, EEDCA, and GA-WCA. Current cluster head selection according to the current routing and routing
schedule selection with the consideration of cluster head chosen helps in searching for the best combinations. Experimental results clearly exhibit that our proposed approach significantly extends
the network lifetime for different sizes of the networks.
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on Wireless Pervasive Computing, pp. 83–87, February 2007. View at Publisher · View at Google Scholar · View at Scopus | {"url":"http://www.hindawi.com/journals/jcnc/2013/578241/","timestamp":"2014-04-17T17:35:41Z","content_type":null,"content_length":"174348","record_id":"<urn:uuid:0c15cf21-c845-475e-a834-30945b37a493>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00341-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cyclotomic fields and singular moduli
up vote 3 down vote favorite
Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?
elliptic-curves shimura-varieties modular-forms algebraic-number-theory
This is a subcase of: mathoverflow.net/questions/15781/… – Dror Speiser Oct 3 '11 at 16:43
I think it isn't a subcase of that question, unless OP only wants elliptic curves with CM by the maximal order. Also, OP, do you want elliptic curves over Q, or over the algebraic closure of Q? –
Hunter Brooks Oct 3 '11 at 16:50
I think it is a subcase, but I'm not certain: I think the answer is, kind of written in the other thread, that the $j$-invariant is in $\mathbb{Q}^\text{cyc}$ if and only if the class group of the
order is an elementary abelian 2-group. And I think he means over the closure, which is the same as over $\mathbb{C}$. – Dror Speiser Oct 3 '11 at 16:59
I've edited the question to hopefully clear things up – Adam Harris Oct 3 '11 at 17:28
It's not clear to me why this is a subcase. Could anyone please expand a little? – Adam Harris Oct 4 '11 at 23:27
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Patent US7026878 - Flexible synthesizer for multiplying a clock by a rational number
The application claims the benefit of U.S. Provisional Application No. 60/498,697, filed Aug. 29, 2003, and included herein by reference.
1. Field of the Invention
The present invention generally relates to a frequency synthesizer. More specifically, the present invention relates to a frequency synthesizer featuring high precision, wide bandwidth, low jitter, a
broad frequency output range, and an integrated PLL with a limited oscillator frequency range.
2. Description of the Prior Art
Modern multimedia entertainment systems are placing ever increasing demands on the resolution, bandwidth, and switching speed of frequency synthesizers. In the past, these requirements have been
satisfied by the conventional phase-locked loop (PLL) synthesizer. The fundamental advantage of PLLs has been their ability to synthesize an output clock signal of high spectral purity that may be
tuned over a wide bandwidth. However, the switching speed and resolution of synthesizers are becoming critically important, and conventional PLLs are ill-suited to these applications because they
suffer an inability to simultaneously provide fast frequency switching and high resolution without substantial design complexity.
Referring to FIG. 9, the classic analog PLL design comprises a phase detector 30C with two inputs and one output, which is connected to a charge pump 32C, which is in turn connected to a filter 34C,
which in turn is connected to a variable-frequency oscillator 36C, which varies its frequency according to a control input. The oscillator's output is looped back through a divider 24C and into one
input of the phase detector 30C, in addition to being output 62C from the circuit as a whole, optionally through a post divider 28C. The reference clock 60C is connected to the other input of the
phase detector 30C, optionally through a reference clock divider 22C.
This classic design has several limitations when the input and feedback divisors are large values. First, the loop bandwidth must be significantly smaller than the phase detector input frequency in
order to operate stably. Second, as a consequence of this, the filter components must be large, possibly requiring the use of external components. Third, the low bandwidth makes the PLL susceptible
to noise, notably for example the standard 60 Hz power line noise. Fourth, the variable-frequency oscillator frequency limits the possible input and output frequencies of the circuit when the range
of possible divisor values is large. Fifth, such a circuit may have high power consumption. Sixth, the use of external components drives up the cost of production and increases hardware space
It is therefore an objective of the present invention to provide a frequency synthesizer outputting a precision frequency when the input or feedback dividers are large numbers.
It is another objective of the present invention to provide a frequency synthesizer featuring low output clock jitter.
It is another objective of the present invention to provide a frequency synthesizer in which the output frequency range of the frequency synthesizer is maximized while the range of the
variable-frequency oscillator in the PLL is minimized.
To attain these objectives, the claimed invention provides a frequency synthesizer that comprises a phase detector for generating an output according to a difference of a reference input and a
feedback input, an oscillator coupled to the phase detector, the oscillator capable of outputting a variable frequency signal in response to a control input, a first divider module for generating the
feedback input, the first divider module comprising a first fractional divider coupled to the oscillator for dividing a frequency of the variable frequency signal by a first time-varying value, and a
second divider module for generating the reference input, the second divider module comprising a second fractional divider for dividing a frequency of a reference signal by a first time-varying
These and other objectives of the present invention will no doubt become obvious to those of ordinary skill in the art after reading the following detailed description of the preferred embodiment
that is illustrated in the various figures and drawings.
FIG. 1 schematically illustrates a block diagram of a frequency synthesizer in accordance with one preferred embodiment of the present invention.
FIG. 2 illustrates a simplified block diagram of the control circuit, including the noise-shaped quantizers.
FIG. 3 is a diagram of the integer-to-floating-point conversion.
FIG. 4 shows the computation of the floating-point exponent.
FIG. 5 a shows a shift circuit with overflow detection.
FIG. 5 b shows an example one-bit multiplexer.
FIG. 6 shows the exponent update control block.
FIG. 7 is a diagram of the floating-point exponent to divider conversion.
FIG. 8 schematically illustrates a block diagram of a frequency synthesizer in accordance with one preferred embodiment of the present invention as an audio synthesizer.
FIG. 9 schematically illustrates a block diagram of a prior-art frequency synthesizer.
In the following detailed description of the preferred embodiments, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific
preferred embodiments in which the invention may be practiced. The preferred embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to
be understood that other embodiments may be utilized and that logical changes may be made without departing from the spirit and scope of the present invention. The following detailed description is,
therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims.
Refer to FIG. 1, which illustrates a frequency synthesizer in accordance with one preferred embodiment of the present invention. The frequency synthesizer comprises a first divider module 23, a
second divider module 19, a phase detector 30, a charge pump 32, a loop filter 34, a variable-frequency oscillator 36, an output integer divider 28, and a control circuit 8. The first divider module
23 comprises a feedback fractional divider 26 and a feedback integer divider 24. The second divider module 19 comprises a reference clock fractional divider 20 and a reference clock integer divider
A reference clock 60 is coupled to the input of the reference clock fractional divider 20. The reference clock fractional divider 20 outputs a reference clock fractional divider signal 20S to the
input of the reference clock integer divider 22. The output of the reference clock integer divider 22 is connected to the first input of a phase detector 30 for providing a reference input signal 22
S,19S to the phase detector 30. The charge pump 32 generates a charge pump output 32S according to a phase difference or frequency difference of the reference input and a feedback input. The output
of the charge pump 32 is connected to a loop filter 34 which removes high frequency components of the output of the charge pump. The loop filter 34 outputs a control input 34S to the oscillator 36,
which is capable of outputting a variable frequency 36S in response to the control input for generating a clock signal. The oscillator 36 may be a voltage-controlled oscillator, a current-controlled
oscillator, a numerically-controlled oscillator, a digitally controlled oscillator, or other type of oscillator capable of generating a variable frequency output 36S in response to a control input.
The output of the oscillator 36 is connected to both the input of the output integer divider 28 and the input of the feedback fractional divider 26. The feedback fractional divider 26 outputs a
feedback fractional divider output signal 26S to the input of the feedback integer divider 24. The feedback integer divider 24 outputs a feedback integer divider output signal 24S, 23S to the
feedback input of the phase detector 30.
Referring again to FIG. 1, the input to the control circuit 8 comprises a reset CLR 70 which indicates that the synthesizer should be reset to an initial condition, a clock CLK 72 which indicates
when the synthesizer should read a divider control word M 82 and a divider control word N 84, a frequency range indicator exponent value FIN 80 to indicate which frequency range the reference clock
60 falls in, a divider control word M 82, and a divider control word N 84.
The desired output frequency of the synthesizer embodiment is described by the formula
$f out = M N × f in ( eq . 1 )$
where f[out ]is the output frequency, f[in ]is the input frequency, and M 82 and N 84 are the divider control words.
Refer to FIG. 2, which shows the block diagram of the control circuit 8, the main purpose of which is to convert the inputs M 82, N 84, and FIN 80 into the integer divider values for the integer
dividers 22, 24, 28 and quantized divider value sequences for the fractional dividers 20, 26 to achieve the desired function described by (eq. 1). The divider control word M 82 undergoes an
integer-to-floating-point conversion 94 which produces a significand of M M_SIG and an exponent of M M_EXP where the significand is within a preferred range. M_SIG is sent to a noise-shaped quantizer
96 which has a clock input FBCLK 52 which is taken from the output of the feedback fractional divider 26. On each cycle of the clock FBCLK 52, the quantizer 96 outputs a quantized value M_QUANT 46.
The divider control word N 84 undergoes an integer-to-floating-point conversion 90 which produces a significand of N N_SIG and an exponent of N N_EXP, where the significand is within a preferred
range. N_SIG is sent to a noise-shaped quantizer 92 which has a clock input DCLK 50 which is taken from the output of the reference clock fractional divider 20. On each cycle of the clock DCLK 50,
the quantizer 92 outputs a quantized value N_QUANT 40. The preferred ranges for the N significand N_SIG and the M significand M_SIG are not necessarily the same.
M_EXP and N_EXP and FIN 80 are sent to an exponent-to-divider conversion 98. The exponent-to-divider conversion 98, which is illustrated in more detail in FIG. 7, outputs three integer values KM 44,
KP 48, and KN 42.
Revisiting the earlier formula, this embodiment of the control circuit 8 reformulates the divider control words M 82 and N 84 and FIN 80 to produce the desired output of
$f out = M N × f in ( eq . 1 )$
□ by computing values KM 44, KP 48, KN 42, N_sig, and M_sig such that the following equality is met:
$f out = 1 2 KP × 2 KM × M sig × 1 2 KP × 1 N sig × f in = M N × f in ( eq . 2 )$
With simplified logic and less stringent requirements, the equality could be made approximate without departing from the spirit of the invention. The noise-shaped quantizers use a multi-bit 2nd order
delta-sigma algorithm (as described in USPTO application 2004/0036509 by the same inventor, incorporated herein by reference). The noise-shaped quantizer 92 outputs the N_QUANT 40 signal which has an
average value approaching the fixed-point significand N_sig. The noise-shaped quantizer 96 outputs the M_QUANT 46 signal which has an average value approaching the fixed-point significand M_sig.
Therefore the average values of M_QUANT and N_QUANT can be substituted for M_sig and N_sig respectively, giving
$f out = 1 2 KP × 2 KM × M QUANT _ × 1 2 KM × 1 N QUANT _ × f in • M N × f in ( eq . 3 )$
Also note that in the embodiment of FIG. 1, the integer dividers 22, 24, and 28 are power-of-2 integer dividers (i.e., the reference clock integer divider 22 receives divider control signal KN and
divides reference clock fractional divider signal 20S by 2^KN).
Please refer to FIG. 3, the block diagram of the integer-to-floating-point converters 90 and 94. The process is identical for both divider control word M 82 and divider control word N 84, so this
diagram shows the input as a generic value Din. The integer-to-floating-point converter decomposes a numeric input Din into significand and exponent components Sig and Exp, respectively, where
Sig=Din×2^Exp(eq. 4)
Since the typical implementation of a multiply by a power of 2 such as 2^Exp is easily performed by a shift, (eq. 4) can also be computed using the logical shift operation denoted by
Sig=Din×2^Exp=Din<<Exp(eq. 5)
where the ‘A<<B’ operation denotes a bitwise left shift of A by B.
In the preferred embodiment, Din is a 24-bit integer, and Sig has an assumed fixed point format of 4.21 (meaning 4 bits to the left of the decimal point and 21 bits to the right of the decimal
point). This assumed format is a notational convenience that simplifies the formulation of (eq. 2) and (eq. 3). Exp is a 5-bit integer enabling a shift of up to 31 bits.
Din is passed to the compute exponent block 100 and the compute significand block 104. The exponent Exp, computed by the Compute exponent block 100 is passed to the output and to the Significand
conversion computation block 104. The Significand conversion computation block 104 receives the input signal Din and the exponent signal Exp and outputs the significand Sig and the overflow signal
ovfl, where Sig=Din<<Exp, and the overflow signal ovfl is asserted whenever Din<<Exp overflows the internal representation of Sig. The Exponent update control block 102 receives the overflow signal
ovfl and Significand Sig from the Significand conversion computation block, and outputs the control signal RECALC_EXP to the Compute exponent block. The Exponent update control block 102 controls the
update of the exponent Exp such that small deviations of the significand Sig outside the preferred range are allowed when Din changes over time, reducing the occurrence of changes of the exponent
The Compute exponent block is shown in more detail in FIG. 4. The temporary exponent value Exp' is computed using the exp4( ) function 106 with argument Din. The exp4( ) function is calculated by
determining the number of left-shifts to apply to DIN that would be necessary to bring the significand Sig to within a preferred range. Whenever the signal RECALC_EXP 126 is asserted, the temporary
exponent value Exp′ is loaded into the register 108 and output as signal Exp.
The Exponent update control block is shown in FIG. 6. The signal RECALC_EXP is asserted whenever the significand Sig is outside an allowed range, or changes by more than a change tolerance from the
previous cycle, or the ovfl signal is asserted, or the reset signal CLR (not shown) is asserted. The reset signal CLR is asserted whenever the frequency synthesizer is reset to an initial state.
To give a specific example of the integer to floating point conversion, in one embodiment of the invention, the divider control words are input as 24-bit values, and 25-bit registers are used for the
floating-point computation. The upper four bits of the floating-point registers are treated as being to the left of the decimal point. The preferred range is chosen to be [4 . . . 8], and the allowed
range is chosen as [3.5 . . . 8.5]. A divider value 65503 (base 10), 0000000001111111111011111 (base 2), is left-shifted by 8 bits to produce the significand 0111111111101111100000000 (base 2). For
illustration a decimal point is inserted at the assumed point of the 4.21 format resulting in the value 0111.111111101111100000000 (base 2), or 7.9959716796875 (base 10), which is within the
preferred range. The exponent Exp calculated is 8 according to the required left shift amount.
Referring to FIG. 7, N_exp is then subtracted from M_exp, and the frequency range indicator exponent value FIN is added to the result to produce an exponent value K_exp. When said value is negative,
its absolute value is applied to the output integer divider and zero is applied to the feedback integer divider 132,134. When said value is nonnegative, its value is applied to the feedback integer
divider and zero is applied to the output integer divider 132,134. The frequency range indicator exponent value FIN is applied to the input integer divider in all cases. In the preferred embodiment
of FIG. 1, the integer divider control values KN, KM, and KP represent power-of-2 divide values (eg. divide value for reference clock integer divider 22 is 2^KN).
An example shift circuit in FIG. 5 a (simplified to 4-bits input, 5-bits output and left shift from 0–3) shows a circuit for collecting the overflow bits to determine if an overflow has occurred. If
the output of the OR gate is 1, the shifted value is too large for the internal 5-bit representation of Sig. The example multiplexer shown in FIG. 5 b is a two-input one-bit multiplexer as used in
FIG. 5 a.
To give a concrete example, please refer to FIG. 8, which illustrates a frequency synthesizer in accordance with one preferred embodiment of the present invention as an audio clock synthesizer. The
differences between FIG. 8 and FIG. 1 are the addition of a frequency doubler 10, a frequency doubler output signal 10S, a multiplexer 12, a multiplexer output signal 12S, and a multiplier 74. The
RECALC_EXP signal 126 is also coupled to the MUTE signal 76, causing the MUTE signal 76 to assert whenever the RECALC_EXP signal 126 is asserted. Since the re-locking time of the synthesizer after an
exponent change is approximately known, the audio system can be designed to mute for an appropriate period of time whenever the MUTE signal 76 is asserted.
One embodiment of this invention has 24-bit divider control words, 25-bit floating-point registers, 5-bit exponents, a preferred range of [4 . . . 8], an allowed range of [3.5 . . . 8.5], and a
change tolerance of 0.125. The example also has an input clock rate of 27 MHz, a reference frequency divider N of 27000, and a feedback frequency divider M of 6144. In addition the MCLK_MULT 86 value
is set to 2. The required output frequency will be F[out]=(M′/N)*F[in]=(2*M/N)*F[in]=27 MHZ*(2*6144/27000)=12.288 Mhz. As the reference frequency of 27 MHz is less than the preferred embodiment's 50
MHz lower limit, the frequency doubler is used to obtain a higher input frequency, and the FIN frequency range indicator exponent value is set to 0.
The feedback frequency divider control word M is multiplied by MCLK_MULT 86, which in the example is 2, to give a divider control word M′ 182 value of 12288 (base 10) or 0000000000011000000000000
(base 2). When left-shifted by 10 places, the significand value is 6.0.
The reference frequency divider control word N is 27000, or 0000000000110100101111000. Left-shifting 9 places gives 6.591796875.
The exponent calculation is K_exp=exp(M)−exp(N)+(FIN−1)=9 −10+(−1)=−2. Since K_exp is negative, KM=0, KP=abs(K_exp)=+2, and KN=FIN=0.
The phase detector input frequency will be 54 MHz/6.591 796875=8.192 MHz.
The VCO 36 output frequency 36S will be the phase detector input frequency multiplied by 2^KM*average(M_QUANT)=8.192 MHz*2^0*6.0=49.152 MHz.
The synthesizer output frequency will be the VCO frequency divided by 2^KP, or 49.152 MHz/2^2=12.288 MHz, as required.
Those skilled in the art will readily observe that numerous modifications and alterations of the device and method may be made while retaining the teachings of the invention. Accordingly, the above
disclosure should be construed as limited only by the metes and bounds of the appended claims. | {"url":"http://www.google.com/patents/US7026878?ie=ISO-8859-1&dq=7751826","timestamp":"2014-04-20T21:53:03Z","content_type":null,"content_length":"91968","record_id":"<urn:uuid:b58cc5eb-09bc-4399-9863-16b0ab14d080>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00197-ip-10-147-4-33.ec2.internal.warc.gz"} |
Twice as old--again!
Last summer, my older daughter Ann observed that I was at that point twice as old as her younger sister Mary Lynn, again. I was taken aback. I was already 30 at the time of the birth of ML (her
preferred name at present), and I recall making a comment about being twice as old as she was when she turned 30. Then I turned 61 and was no longer (exactly) twice as old. I though that was the end
of it. But Ann's observation was correct, sure enough.
At that point, I decided not to think about it any more because it looked like a good starter question for my calculus course in the fall. On the day of the first class, I gave the problem, without
further elaboration, in the following form. "Last summer, my older daughter, Ann, observed that I was twice as old as her sister Mary Lynn, again. I was thirty when Mary Lynn was born. How old am I
now, and for how many days was I twice as old as Ann?" I then sat back to watch. I was surprised and delighted with what occurred over the next few days.
My first-semester course in honors multivariable calculus was populated almost entirely by first-semester freshmen, all with a 5 on the Advanced Placement BC Calculus test, or the equivalent. The
course is "paperless" in that all assignments are to be handed in over the web. There is a built-in time delay to avoid a competition to see who will be the first to solve the problem, thereby making
it superfluous for anyone to continue. That was good, since the first person who answered, an hour after the end of class, solved the initial problem completely:
Craig Desjardins at Tue Sep 05 18:13:28 EDT 2000
Since you were already 30 years old when Mary Lynn was born, you turned 31 before she turned 1, 32 before she turned 2, and so on including when she turned 30 in the midst of your 60th year. At that
time you were twice as old as Mary Lynn. Then, as time passed, you turned 61, she turned 31, and you turned 62 just as before. At this time, when you are 62 and she 31, you are once again twice her
age. You are 62. Since Ann is older, you have already been twice her age, twice for that matter (unless you have the same birthday, a case in which the answer to this question is only too obvious).
The number of days that you were twice her age the first time is the number of days from her birthday and your birthday chronologically. The second time it was the number of days from your birthday
to her birthday, also chronologically. Since this comprises an entire year, the number of days during which you were twice Ann's age is either 365 or 366, depending on leap year. -------
Later that evening, another student, Josh Butler, offered a geometric illustration:
If you graph the days of a year as equal-length arcs on a circle, your birthday and her birthday will be two arcs. Shading each day/arc from her birthday to your birthday the first time you were
twice her age, then shading from your birthday to her birthday the second time you were twice her age, will effectively shade the entire circle exactly once. Thus, depending on whether a leap-day
fell while you were twice her age you were twice her age for either 365 or 366 days.
The surprising thing is that the question generated so many other general questions over the course of the next several days. The first observation is that the analysis above shows that it does not
make any difference what the actual birthdates are, as long as those birtdays are different. If my daughter had been born on my birthday, I still would have been exactly twice as old for a calendar
year, but I would not have been twice as old "again", i.e. on two separate intervals. In particular, the analysis shows that there is nothing special about the number 30 when it comes to being twice
as old. The total number of days I was twice as old as Ann is a full year, independently of my age when she was born (assuming that she is not just born ahead of her sister in a multiple birth, in
which case the full year would not be up at the time of the observation.)
Twice As Old: Graph and Close-Up
The thing that makes the problem interesting in a calculus course is the relationship with the Intermediate Value Property. In our classes we stress the fact that if one continuous function starts
out below another and surpasses it as some future time, then for some intermediate point, the two functions must be equal. (Actually we stress this theorem in a sligthly different form but the
concepts are equivalent, for continuous functions in any case.)
In the case at hand, the problem is of the sort "A train leaves Philadelphia 30 minutes after another has left on a parallel track, but the second train goes twice as fast. Where do the trains meet?"
For linear functions, the answer is that the two trains arrive at the same point exactly once, and just for an instant.
Here, however, where age is measured in full years, changing only at birthdays, we have a pair of step functions, clearly discontinuous. Why should they have any overlaps at all, and what will the
nature of those overlaps be?
Students are familiar with a number on instances where discontinuous functions do not have to meet in order to cross, even in the case where both are increasing functions. Sports scores provide the
most accessible examples. If one American football team is ahead at the half but the other wins, then there did not have to be any time when the scores were the same, since the lead could change with
a touchdown. European football, or soccer, has a different scoring structure, and there has to be a time when both scores are the same if one team is to overtake another. Basketball is of the first
type and ice or field hockey is of the second. Baseball, with its very specific and sometimes arcane rules, is a special case. Can the score of a game ever go up instantaneously by two runs? That
does not happen if there is a home run with one person on base since the run is scored only when a runner crosses home plate, so two runs are scored in quick succession, but not at the same time.
There are other more obscure and somewhat contrived scenarios that cause greater difficulties, although whether or not two or more runs appear simultaneously will not affect who wins the game so the
problem is of "academic interest".
We can distinguish two kind of overlap intervals, where two functions have the same value. If f(t) = g(t) for an entire interval a < t < b, then we say that the interval is a crossing interval if f
(t) < g(t) for a - h< t < a and f(t) > g(t) for b < t < b+h, for some positive h, in which case f is said to overtake g, or the opposite happens, inwhich case g overtakes f over the interval. If f
and g are both step functions that can jump exactly one unit, then the Intermediate Value theorem holds, namely the result that if one overtakes the other, there must be a crossing interval sometime
in between. Moreover, there must be an odd number of crossing intervals under these circumstances, since only at such intervals can the lead change hands, and it must end up at a different place than
it started.
Although the function g(t) = "my age plus 30" jumps by one unit every time I have a birthday, the function f(t) = "twice my daughter's age" jumps by two units every birthday. However the analysis
above indicates that f and g can be equal on two separate intervals. By now, f(t) > g(t) and it will stay that way. So which of the intervals is a crossing interval? Must there be an odd number of
crossing intervals? Where did f overtake g? One surprise is that neither of the intervals where f was equal to g was a crossing invertal, and the overtaking happened instantaneously at my daughter's
birthday that occurred between the two intervals when my age was twice hers. This becomes clear if we graph the two functions on the same diagram. Twice her age starts out below, then catches up at
the first coincidence interval, then drops below then jumps above at one point, then drops back for another interval, then goes ahead for good.
The Generalization to n Times as Old.
At this point in the class, I recounted a famous Abbott and Costello dialogue that I remembered from my youth, from the movie "Buck Privates":
Abbott: You're 40 years old, and you're in love with a little girl, say 10 years old. You're four times as old as that girl. You couldn't marry that girl, could you?
Costello: No.
Abbott: So you wait 5 years. Now the little girl is 15, and you're 45. You're only three times as old as that girl. So you wait 15 years more. Now the little girl is 30, and you're 60. You're only
twice as old as that little girl.
Costello: She's catching up?
Abbott: Here's the question. How long do you have to wait before you and that little girl are the same age?
Costello: What kind of question is that? That's ridiculous. If I keep waiting for that girl, she'll pass me up. She'll wind up older than I am. Then she'll have to wait for me!
(I am indebted to Ivars Peterson for providing the reference to this dialogue in his weekly online column for the MAA.)
It appears that 30 years of separation provides a good basis for the mathematical joke above, but is there something special about 30? And how many times was I n times as old as Mary Lynn, where n is
an integer? This refined question was answered by Ryan Roth in his posting two days later:
Ryan Roth at Thu Sep 07 19:29:48 EDT 2000 There are quite a few times when Professor Banchoff was n times older than his daughter. (here n is an integer.) These values occur at n=
2,2,3,4,6,7,11,16,31,32 n=2 is the only to occur twice.
Ryan then went on to write a Basic program to figure how many distinct numbers n would show up for various starting ages. In all of the cases he checked, the answer came out to be even. However
subsequent investigation showed that 15 (an early age for parenting but still possible) yielded 1,2,3,4,5,8,15, and 16. He asked the conditions for this number to be even. That question was answered
only after the general situation was analyzed.
The most extended investigation took place as a dialogue between Gregory Balthazar, and Steve Canon, a senior who acted as an assistant in the course. The interchange is reproduced here in its
Gregory Baltazar at Thu Sep 07 23:33:16 EDT 2000 If n is an integer, then n can equal 2, 3, 4, 6, 7, 11, 16, 31, 32 times as old as ML [ Expatiate ]
Stephen Canon at Fri Sep 08 14:02:53 EDT 2000 What if Professor Banchoff wasn't 30 when she was born? If he was 31 years old, then which n work? If he was 32? 33? etc.? Can you find any interesting
patterns? Can you prove them?
Gregory Baltazar at Sat Sep 09 02:28:32 EDT 2000 Well, let's say that professor Banchoff's age is represented by the letter X. Then 1 + any divisor of X is an n.
Proof: X=professor Banchoff's age when ML was born Y=years that have elapsed since the birth
(X+Y)/Y = n So, if you want n=2, let X=Y (X must be divisible by one) *therefore n can always be 2 if you want n=3, let X=2Y (X must be divisible by two) if you want n=35, let X=34Y (X must be
divisible by 34) and so on . . .
N.B. I don't know why I'm writing this at 2 in the morning. For reasons beyond my control, I've become nocturnal.
Stephen Canon at Sat Sep 09 19:07:56 EDT 2000 Yes, 1 plus any divisor of X is such an n, but there are more n's besides. Can you find some? Can you describe all of the n's? Keep working!
Gregory Baltazar at Sat Sep 09 23:17:02 EDT 2000 Are you referring to the intance in the "twice as old" problem where it was possible for Professor Banchoff to be 30 years older than ML as well as 31
years older than ML?
Stephen Canon at Sun Sep 10 00:20:10 EDT 2000 It's related, but no. For example, if X=11, then n=4 is a solution, even though 3 doesn't divide 11. (Specifically, it happens when prof. B is 16).
Gregory Baltazar at Mon Sep 11 12:38:09 EDT 2000 Ok then, The set of all n consists of 1+the divisors of X and 1+the divisors of X+1.
Stephen Canon at Mon Sep 11 22:55:58 EDT 2000 exactly. good work. You've shown that any divisor of X or (X+1) will yeild an n. Now can you prove that every n is of the form (1+d), where d is a
divisor of X or (X+1)?
Gregory Baltazar at Tue Sep 12 09:05:39 EDT 2000 Same proof for X+1 as in the proof for X above. Just substitute X+1 in for X.
Stephen Canon at Sun Sep 17 14:23:42 EDT 2000 Well, that's not what I asked. You proved that every n of the form (d+1) where d divides X or X+1 works. But you are yet to prove that these are the only
n's that will work. i.e. you haven't yet proved that there can't be some n that isn't of the form (d+1). Do you understand?
Gregory Baltazar at Mon Sep 18 18:52:53 EDT 2000 Well, all n are of the form: n=(X+Y)/(Y), where X is the professor's age when ML is zero years old and Y is the number of years that have elapsed
since ML's birth. This simplifies to n= X/Y + 1 Now n doesn't have to be of the form d+1 Professor Banchoff could be 2.5 times as old, if he was 30 at her birth and he is now 50. But, if you want n
to be an integer, then n-1 is also an integer. This means that the term X/Y is an integer or to put it in technical terms: X/Y=k, where k is a nonnegative integer. (it wouldn't make sense for Prof.
B. to be (-2+1) times as old as ML for example) Then X=kY, which means that X is a multiple of Y or Y is a divisor of X. Since X/(X/Y)=Y, (X/Y) must also be a divisor of X. So all n are of the form
(d + 1) where d is a divisor of X.
Stephen Canon at Tue Sep 19 13:55:47 EDT 2000 well done. Sorry to pester you so much, but we often like to write things down really precisely in mathematics, just to make sure that we aren't missing
Perhaps the most succinct treatment of the topic was by Ju Dee Ang, nearly a week after the problem was first introduced:
Ju Dee Ang at Mon Sep 11 00:31:23 EDT 2000 I would assume that you are referring to integer values of n (or perhaps not?). Let m be Mary Lynn's age at any time, in which case n=(30+m)/m or (31+m)/m.
So n could only be an integer if m is a factor of 30 or 31. For Mary Lynn, the possible values of m when n would be an integer would be when she is 1, 2, 3, 5, 6, 10, 15, 30 or 31 years old, when n
would be 31, 16, 11, 7, 6, 4, 3 or 2 (twice). ----
Ju Dee Ang at Tue Sep 12 00:55:27 EDT 2000 Yes, and I forgot about n=32 for Mary Lynn, given by(31+1)/1.
Following the lead given by Gregory Baltazar and Ju Dee Ang, we can answer Ryan Roth's question. The number of distinct n that will appear in the "n times as old" problem is the number of divisors of
m plus the number of divisors of m + 1, minus 1, where m is the age of the father at the birth of the daughter. Thus that number will be odd unless either m or m+1 is a square.
It was a very complete answer to a problem that began with a chanceobservation, a great way to get a class involved in developing amathematical idea and seeing where it goes. Now I just have to
findanother equally stimulating problem for my next class.
Three Times as Old: Graph and Close-Up
Addendum, Concerning Horses
According to The University of Georgia College of Agriculture and Environmental Sciences, Cooperative Extension Services, a stallion can sire a filly who is born while he is officially still one year
old. He will then turn two years old on January 1, the same day his daughter turns one. He will be twice as old as she is for exactly one year, and there will never be another time when he is an
integer times as old as she is. This is the absolute minimum number of times that a father can be an integer number of times older than his daughter.
For human beings, the minimum number of times is two, which occurs when the father and daughter share the same birthday and the father's age is a prime number when his daughter is born.
Additional Addendum: Following my J. Sutherland Frame lectureincluding this topic at the MAA MathFest, I received a note fromJerrold Grossman from Oakland University pointing out a similarobservation
made by the young Dan Kleitman, as reported in the Journalof Discrete Mathematics 257 (2002) 193-224: "Kleitman's brother Davidtells us when he realized that his brother would probably become
amathematician. 'When Danny was 3 years old, we were playing in asandbox. The parent of another child in the park asked us how old wewere. I said I was 6, and my little brother was 3. "You are twice
asold as he is," the parent replied. Danny apparently though aboutthat, because he said, "You know, after my birthday next month, thatwon't be true anymore. It will never be true again."'"
Grossmanobserves, "Aha! He is right, but it WAS true 22 months previous tothat, when David was 4 and Dan was true. Maybe you can work that intoyour article on birthday multiples." Thank you, Jerry.
Thanks to David Eigen for the computer graphics illustrations.
Thomas BanchoffLast modified: Fri Jun 27 15:04:53 EDT 2003 | {"url":"http://www.math.brown.edu/~banchoff/twiceasold/","timestamp":"2014-04-17T06:56:04Z","content_type":null,"content_length":"20650","record_id":"<urn:uuid:145c8484-0a3c-4991-9c77-c6b2485e5b0c>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00557-ip-10-147-4-33.ec2.internal.warc.gz"} |
About Basic Geometry
Date: 10/14/98 at 21:14:34
From: Tony Santiago
Subject: Basic geometry
Basic geometry --
1. Who developed the course?
2. What is it used for?
3. Who uses it
Thanks for your time,
Tony Santiago
Date: 10/15/98 at 18:24:27
From: Doctor Rick
Subject: Re: Basic geometry
Hello, Tony. I don't know if you have to write a report or you're just
curious about geometry, but I'll try to answer your questions briefly.
Who developed the course?
In the narrowest sense, look in the front of your textbook. Each
geometry course is organized a little differently, and the authors of
the book developed this particular course. But all geometry courses
more or less follow the trail blazed by Euclid, before 300 BC. There
were others who contributed to geometry centuries earlier in Greece,
and even farther back, the Babylonians and Egyptians had some practical
geometrical knowledge. But Euclid is the one who systematized geometry
- set it up as a collection of definitions, postulates, and theorems,
all logically following from one another.
Look here for information on the history of geometry:
What is it used for? Who uses it?
The very word "geometry" points to its practical origins: it means
"measurement of the earth." A Greek named Eratosthenes, among others,
used it (and its relative, trigonometry, which means "measurement of
triangles") to find the circumference of the earth. (This was crucial
for mapmaking, and it's even more relevant today, with all our
Geometry is still used in its original sense by surveyors. It's used
every second by computers - those GPS devices you may have heard of,
which can pinpoint where you are by triangulation from several
People of all sorts use geometry. Just one example: I do some artwork,
and a few months ago I was making a poster that had 3 circles on it.
I had to use high school geometry to figure out just how big to make
the circles and where to place them so it would look right. If geometry
can be used in art, surely it can be used just about anywhere.
But you know, the greatest value of geometry has nothing to do with
what people use it for. What Euclid did 2300 years ago was
revolutionary because it got people thinking logically and reasoning
things out - thinking about why something is true. In geometry,
something is true or it isn't, and you don't prove something by yelling
loud enough to intimidate people, or by being persuasive and winning in
the polls.
Euclid really set the stage for science, for careful examination of the
world, of cause and effect. And every year, when students study
geometry, it once again sets the stage for some of those students to
head into sciences and technical fields that require careful thought.
So I would say that all science and technology is a "use" of geometry,
whether or not people ever think about circles and triangles. The arts
aren't unaffected, either. Visual arts are clearly influenced by
geometry, whether the study of perspective that came in with the
Renaissance (believe me, that's hard core geometry) or cubism.
Geometry and logic aren't everything. People have reasoned logically
from wrong postulates and come to horrible conclusions, in politics and
personal life. But you can't get away from the effects of geometry in
the modern world, and I wouldn't want to.
- Doctor Rick, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/52279.html","timestamp":"2014-04-20T10:12:23Z","content_type":null,"content_length":"9066","record_id":"<urn:uuid:affb4720-82dd-488b-afbd-1a0d7bf6af8a>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00104-ip-10-147-4-33.ec2.internal.warc.gz"} |
area under curve help
April 26th 2009, 07:40 AM #1
Super Member
Sep 2008
area under curve help
A and B are two points which lie on the curve C, with equation $y = -x^2 + 5x +6$ The diagram shows C and the line l passing through A and B.
(a) Calculate the gradient of C at the point where x=2.
The line l passes through the point with coordinates (2, 3) and is parallel to the tangent to C at the point where x=2.
(b) Find an equation of l.
(c) Find the coordinates of A and B.
The point D is the foot of the perpendicular from B on to the x-axis.
(d) Find the area of the region bounded by C, the x-axis, the y-axis and BD.
(e) Hence find the area of the shaded region.
The equation of L is $y = x+1$
the coordinates of A (-1,0) B (5,6)
for question 'd' I got the area = $\int_{0}^5 [-x^2 + 5x + 6] dx$ = $50\frac{5}{6}$ which is correct.
I am stuck on question 'e',
If I work out the whole area between point D and A would that be the $\int_{-1}^5 [-x^2 +5x+6 ]$ ?
and than I minus the area of the triangle and the little area above the triangle on the left hand side of the y-axis ?
$\int_{-1}^5 [-x^2 +5x+6 ]$
$(-\frac{x^3}{3} + \frac{5x^2}{2} + 6x )$
$[ -\frac{125}{3} + \frac{125}{2} + 30]$ - $[\frac{1}{3} + \frac{5}{2} - 6]$ = 54 .
$\int_{-1}^0 (-x^2+5x+6)dx$ = $3\frac{1}{6}$
Area of triangle = $6 \times 6 \times \frac{1}{2} = 18$
$18+ 3\frac{1}{6} = 21\frac{1}{6}$
$54 - 21\frac{1}{6} = 32\frac{5}{6}$
The correct answer is $33\frac{1}{3}$
Can someone show me where I have gone wrong or whats the correct method?
The equation of L is $y = x+1$
the coordinates of A (-1,0) B (5,6)
for question 'd' I got the area = $\int_{0}^5 [-x^2 + 5x + 6] dx$ = $50\frac{5}{6}$ which is correct.
I am stuck on question 'e',
If I work out the whole area between point D and A would that be the $\int_{-1}^5 [-x^2 +5x+6 ]$ ?
and than I minus the area of the triangle and the little area above the triangle on the left hand side of the y-axis ?
$\int_{-1}^5 [-x^2 +5x+6 ]$
$(-\frac{x^3}{3} + \frac{5x^2}{2} + 6x )$
$[ -\frac{125}{3} + \frac{125}{2} + 30]$ - $[\frac{1}{3} + \frac{5}{2} - 6]$ = 54 .
$\int_{-1}^0 (-x^2+5x+6)dx$ = $3\frac{1}{6}$
Area of triangle = $6 \times 6 \times \frac{1}{2} = 18$
$18+ 3\frac{1}{6} = 21\frac{1}{6}$
$54 - 21\frac{1}{6} = 32\frac{5}{6}$
The correct answer is $33\frac{1}{3}$
Can someone show me where I have gone wrong or whats the correct method?
Let the "big function" minus the "small one", and integrate from 0 to 5.
Area= $\int_{0}^5 ((-x^2+5x+6)-(x+1))dx$ = $\frac{100}{3} =33\frac{1}{3}$
I dont understand how you got that, becasue although it gives you the right answer is that not just the curve-line expression?
How does that give you the shaded region, as that expression also includes the area thats not shaded?
Actually ignore my other post I get what you did.
Always let the "big function" minus the "small one", and integrate from the left to the right, and you get the area between two curves.
April 26th 2009, 08:41 AM #2
May 2007
April 26th 2009, 08:53 AM #3
Super Member
Sep 2008
April 26th 2009, 08:55 AM #4
Super Member
Sep 2008
April 26th 2009, 09:16 AM #5
May 2007 | {"url":"http://mathhelpforum.com/calculus/85729-area-under-curve-help.html","timestamp":"2014-04-18T08:56:57Z","content_type":null,"content_length":"49398","record_id":"<urn:uuid:80375758-4a7d-4c9a-b3b8-9054e1932b59>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00537-ip-10-147-4-33.ec2.internal.warc.gz"} |
need to develop a furmula for a string
Hi all,
I have following string, and need to develop a single formula for all terms.
n, (n-1), (n-2)!, (n-3)!+(n-4)!+(n-5)!+...(n-(n-1))!, ((n-4)!+(n-5)!+(n-6)!+...(n-(n-1))!)+((n-5)!+(n-6)!+(n-7)!+...(n-(n-1))!)+...+(n-(n-1))!, ((n-5)!+(n-6)!+(n-7)!+...(n-(n-1))!)+((n-6)!+(n-7)!+
as illustrated after 3rd term, each sub-term (n-m)! in next term turn to (n-(m-1))!+ (n-(m-2))!+ (n-(m-3))!+....+(n-(n-1))!
any idea is appreciated
thank you:) | {"url":"http://www.mathisfunforum.com/viewtopic.php?id=20397","timestamp":"2014-04-20T23:31:00Z","content_type":null,"content_length":"17491","record_id":"<urn:uuid:744f61de-8343-47c9-b808-44715ab82773>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00388-ip-10-147-4-33.ec2.internal.warc.gz"} |
e co
Results 1 - 10 of 13
- Math
"... The problem to decide whether a given algebraic variety defined over the rational numbers has rational points is fundamental in Arithmetic Geometry. Abstracting from concrete examples, this
leads to the question whether there exists an algorithm that is able to perform this task for any given variet ..."
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The problem to decide whether a given algebraic variety defined over the rational numbers has rational points is fundamental in Arithmetic Geometry. Abstracting from concrete examples, this leads to
the question whether there exists an algorithm that is able to perform this task for any given variety. This is probably
- Algorithmic number theory. 5th international symposium. ANTS-II , 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of
possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
Cited by 14 (3 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of
possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a
perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime
algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work,
it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI,
MAGMA, SIMATH, apec...
- Math. Comp , 2001
"... Abstract. This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities
associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevic ..."
Cited by 14 (9 self)
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Abstract. This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities
associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for
the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we
could compute. 1.
- Math. Comp , 1999
"... Abstract. We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π: X1(N) → C defined over Q and the jacobian of C is Q-isogenous to
the abelian variety Af attached by Shimura to a newform f ∈ S2(Γ1(N)). We determine the corresponding new ..."
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Abstract. We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π: X1(N) → C defined over Q and the jacobian of C is Q-isogenous to the
abelian variety Af attached by Shimura to a newform f ∈ S2(Γ1(N)). We determine the corresponding newforms and present equations for all these curves. 1.
- J. Math. Kyoto Univ , 1996
"... Abstract. In this note, we will show that Bogomolov conjecture holds for a non-isotrivial curve of genus 2 over a function field. 1. ..."
Cited by 9 (3 self)
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Abstract. In this note, we will show that Bogomolov conjecture holds for a non-isotrivial curve of genus 2 over a function field. 1.
- COMPOSITIO MATHEMATICA , 1999
"... Let K be a discrete valuation field with ring of integers OK.Letf: X → Y beafinite morphism of curves over K. In this article, we study some possible relationships between the models over OK of
X and of Y. Three such relationships are listed below. Consider a Galois cover f: X → Y of degree prime t ..."
Cited by 8 (1 self)
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Let K be a discrete valuation field with ring of integers OK.Letf: X → Y beafinite morphism of curves over K. In this article, we study some possible relationships between the models over OK of X and
of Y. Three such relationships are listed below. Consider a Galois cover f: X → Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable
reduction over K, thenX achieves semi-stable reduction over some explicit tame extension of K(B). When K is strictly henselian, we determine the minimal extension L/K with the property that XL has
semi-stable reduction. Let f: X → Y be a finite morphism, with g(Y) � 2. We show that if X has a stable model X over OK, then Y has a stable model Y over OK, and the morphism f extends to a morphism
X → Y. Finally, given any finite morphism f: X → Y, is it possible to choose suitable regular models X and Y of X and Y over OK such that f extends to a finite morphism X → Y? As was shown by
Abhyankar, the answer is negative in general. We present counterexamples in rather general situations, with f a cyclic cover of any order � 4. On the other hand, we prove, without any hypotheses on
the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.
- Ann. Inst. Fourier (Grenoble , 2009
"... Abstract. For a smooth and proper curve XK over the fraction field K of a discrete valuation ring R, we explain (under very mild hypotheses) how to equip the de Rham cohomology H 1 dR(XK/K) with
a canonical integral structure: i.e. an R-lattice which is functorial in finite (generically étale) K-mor ..."
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Abstract. For a smooth and proper curve XK over the fraction field K of a discrete valuation ring R, we explain (under very mild hypotheses) how to equip the de Rham cohomology H 1 dR(XK/K) with a
canonical integral structure: i.e. an R-lattice which is functorial in finite (generically étale) K-morphisms of XK and which is preserved by the cup-product auto-duality on H 1 dR(XK/K). Our
construction of this lattice uses a certain class of normal proper models of XK and relative dualizing sheaves. We show that our lattice naturally contains the lattice furnished by the (truncated) de
Rham complex of a regular proper R-model of XK and that the index for this inclusion of lattices is a numerical invariant of XK (we call it the de Rham conductor). Using work of Bloch and of
Liu-Saito, we prove that the de Rham conductor of XK is bounded above by the Artin conductor, and bounded below by the Efficient conductor. We then study how the position of our canonical lattice
inside the de Rham cohomology of XK is affected by finite extension of scalars. 1.
"... Abstract. S.-W. Zhang recently introduced a new adelic invariant ϕ for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to
2. 1. ..."
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Abstract. S.-W. Zhang recently introduced a new adelic invariant ϕ for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2. 1.
"... Abstract. Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] has a natural tree structure. Using this
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Abstract. Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] has a natural tree structure. Using this data, we
construct an “arboreal ” Galois representation ω whose image combines that of the usual ℓ-adic representation and the Galois group of a certain Kummer-type extension. For several classes of A, we
give a simple characterization of when ω is surjective. The image of ω also encodes information about the density of primes p in K such that the order of the reduction mod p of α is prime to ℓ. We
compute this density in the general case for several A of interest. For example, if F is a number field, A/F is an elliptic curve with surjective 2-adic representation and α ∈ A(F), with α ̸∈ 2A(F(A
[4])), then the density of primes p with α mod p having odd order is 11/21. 1.
"... Abstract. Consider the smooth projective models C of curves y 2 = f(x) with f(x) ∈ Z[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one
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Abstract. Consider the smooth projective models C of curves y 2 = f(x) with f(x) ∈ Z[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one
rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g → ∞. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves,
via Chabauty’s method at the prime 2. 1. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=714780","timestamp":"2014-04-21T17:04:51Z","content_type":null,"content_length":"36045","record_id":"<urn:uuid:aaad6407-0cd5-4135-9290-f6a7f7bbc6cd>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00388-ip-10-147-4-33.ec2.internal.warc.gz"} |
Current Research Topics
UCL Centre for Nonlinear Dynamics and its Applications
CNDA >> Research Index >> Postgraduate Information > Research Descriptions
Current Research Topics
Engineering Applications
One of the main motivations for the establishment of the Centre was the desire to apply the emerging ideas of nonlinear dynamics to practical problems in engineering. Today this area of research
continues to form the core of the Centre's activities. Current projects in this area are described below, but it should be noted that new research topics are continually being developed.
Escape Phenomena, Capsize and Basin Erosion
The escape from a potential well is a universal problem in science and engineering. It arises in the buckling and post-buckling of elastic structures; in the study of the capsize of ships; in solid
state physics, including the response of Josephson junctions; in electrical power engineering, where the loss of synchronization of generators can lead to the black-out of a power grid (see below);
and in physical chemistry, for instance when modelling activation energies of molecular reactions. Under periodic excitation, escape under steady state conditions is governed by a complex web of
regular and chaotic bifurcations; while escape under transient conditions is governed by the basins of attraction which can undergo elaborate metamorphoses. The onset of a fractal basin boundary can
be predicted by Melnikov estimations of the first homoclinic tangency (see below), whilst the subsequent basin erosion process is governed by a later heteroclinic connection. Current effort is
directed towards understanding these global bifurcation phenomena and developing methods for their prediction. The engineering relevance of the dramatic basin erosion that is observed under
increasing excitation has been highlighted by the definition of various engineering integrity measures. This work has important implications for the wave-tank testing of ship hulls, where our new
concept of a transient capsize diagram seems to offer a more rational approach to ship safety in waves.
Contact: J.M.T. Thompson.
Control of Chaos
Since chaotic systems exhibit extreme sensitivity to initial conditions it has for many years been generally thought that chaotic systems are neither predictable nor controllable. Thus chaos is often
regarded as an annoying property to be avoided in the design of engineering systems. More recently, however, studies have shown that chaotic dynamics are not only controllable but can also be
exploited to achieve useful goals. Utilisation of the properties of chaotic systems can achieve special advantages: a) small perturbations can lead to large effects; b) flexible switching between
many different periodic orbits without changing the global configuration of the system. These benefits cannot be achieved in non-chaotic systems in which a large effect typically requires a large
control. The great potential of the control of chaos has been demonstrated in applications in electronics, lasers, chemical reactions, communications and biological systems.
The aim of this research topic is to investigate basic concepts of chaotic control, develop new theoretical algorithms, and consider their implementation in a variety of systems. To date, three new
methods have been introduced for controlling chaotic systems. The first approach is a parametric control scheme based on a linear approximation using a one step optimal process. The method does not
require analysis of eigenvalues and eigenvectors and can be applied without using explicit knowledge of the system dynamics - an aspect which is particularly useful for experiments. The method is
very simple especially when applied to Hamiltonian systems. To enhance the effectiveness for stabilising highly unstable orbits in a very noisy environment, the method is extended to a variational
algorithm for flows based on multiple control surfaces. The second method is based on the contraction mapping theorem in the form of state variable feedbacks. The control strategy does not rely on
linear approximation, thus control can be achieved regardless of whether the system state is close to the desired target point or not. To achieve a stabilisation of orbits with high accuracy, a third
method with a self-locating function has been developed which can automatically detect the location of a desired orbit without having any explicit knowledge of the system dynamics or the exact
location of the orbit. The robustness of each control scheme is also discussed including the effects of noise and overall stability of the control algorithms.
Contact: S.R. Bishop.
Sigma-Delta Modulators
Sigma-delta modulation is an analogue to digital conversion technique used in many telecommunication, radar, sonar and audio applications. It employs high over-sampling rates to achieve a good signal
to noise ratio by exchanging temporal resolution for amplitude resolution. Due to the negative feedback structure employed, this results in circuits which are especially insensitive to device
imperfections and mismatches. However, sigma-delta modulators can exhibit complex dynamic behaviour, which has a direct bearing on their performance. In collaboration with Coventry University and the
University of Manchester Institute of Science and Technology we are therefore undertaking a study of the dynamics of such circuits. First order modulators can be readily analysed using well-known
results for circle maps, but unfortunately real-world circuits are usually second order or higher. Very little is understood about this case, and very few mathematical techniques exist to analyse it.
The project is therefore proceeding with a mixture of experimental work and numerical simulation in an attempt to obtain better insight into the qualitative properties of higher order modulators.
Contact: S.R. Bishop, J. Stark.
Impacting and Constrained Engineering Systems
Dynamical systems which involve impacts frequently arise in engineering. The resonant phenomenon of rattling, caused by repeated impacts at one or two motion limiting stops, often needs to be
suppressed. In mechanical systems such rattling can lead to noise and wear. Even more seriously, in some offshore situations such as those motivating this work, excessive loads at the constraints can
lead to complete and expensive failure.
The impact process introduces a strong nonlinearity into the dynamics, and hence impacting oscillators can exhibit the typical types of complicated behaviour associated with nonlinear dynamical
systems. One of the main results of our research is the identification of bifurcational behaviour associated with part of an orbit of the dynamical system just touching a stop under the change of
some system parameter. After this type of bifurcation the system can suddenly begin undergoing much more severe impacts, or can undergo chaotic motion. Experimental work has been carried out which
clearly demonstrates the qualitative behaviour caused when these types of bifurcation occur. It is important to be aware of the possibility of such sudden jumps to what could be "dangerous" states in
the design of structures which may undergo repeated impacts, and thus to avoid them. The dynamical behaviour of more general constrained dynamical systems is now also under investigation. As with
impacting systems, the constrained nature of these systems ensures that their responses can be highly nonlinear. The effects of constraint-induced nonlinearities on the overall dynamical responses of
this class of system is being studied both theoretically and experimentally.
Contact: S.R. Bishop.
Instabilities In Building Fires
Flashover is a phenomenon describing the transition by which a small localized fire with a relatively low burning rate suddenly undergoes a rapid increase in both size and intensity. In the evolution
of a fire in a confined space the occurrence of flashover signals a highly dangerous transition in the nature and intensity of the fire and its potential effect on building and occupants. It is thus
of paramount importance that an understanding of this critical change in the character of the fire be attained.
A recent collaborative research project with the Unit of Fire Safety Engineering at the University of Edinburgh applied the techniques of nonlinear dynamics to this problem. It examined a variety of
current fire growth models and identified their most significant nonlinear features. This has led to the development of archetypal models incorporating such features. These models were subjected to
in-depth analysis to determine mechanisms leading to instability. New computational techniques were devised to display fire growth within a building, permitting increasingly sophisticated interactive
and automated computer experiments to be carried out.
Contact: S.R. Bishop.
Climate Modelling
Circulation within the ocean dramatically affects, and often determines the climate of many regions of the globe. For example, the Gulf Stream, which manifests itself as a warm surface current
between the Gulf of Mexico and the North Atlantic, has a significant influence upon the climate of Northern Europe. Surface currents of this type are the result of circulation deep in the ocean and
originate from differences in temperature between equatorial and polar regions, effectively creating a single large convective cell. However, the situation is complicated by the effect of evaporation
at the equator and precipitation in polar regions, which lead to a gradient in ocean salinity (salt content). Ignoring the thermal circulation, this alone would produce a pressure gradient and hence
a circulation, which would in fact be in the opposite direction to that produced by thermal forcing. When combined, the opposing thermal and saline driving forces are referred to as thermohaline
circulation, which may produce more complex behaviour than that produced by either effect alone. Thermohaline circulation has been investigated using simple box models, in which a cross-section of
the ocean is divided into a finite number of boxes which possess the average properties of those regions of the ocean. This system of boxes may be described by a set of simple nonlinear differential
equations (based upon differences in temperature and salinity), which govern the transport of fluid between the regions. Such models demonstrate certain instabilities depending upon the choice of
parameters used to describe the diffusion of fluid between the boxes. A collaborative project with the UK Meteorological Office has investigated the dynamics of such systems, and the instabilities
which occur, by applying techniques developed in the study of other nonlinear dynamical systems.
Contact: S.R. Bishop.
Phase Locked Loops
A phase-locked loop is an electronic circuit designed to demodulate frequency-modulated signals. Although widely used in a variety of communications systems ranging from satellites to portable
telephones there are many aspects of its general behaviour which are not well understood. In particular current linear models of phase-locked loops do not explain much of the behaviour seen in actual
working systems. This research project is therefore studying the full non-linear equations describing a phase-locked loop's dynamics, employing both analytical and computer simulation techniques .
The equation describing the phase-locked loop can be simplified to an equation of a pendulum with constant torque and forcing applied to it. This equation is also used to model other systems like the
Josephson junction. The possible steady states of the equation and the robustness of these steady states have been analysed. These are related to the parameter regime in which the phase-locked loop
is operating. As the values of the operating parameters change, the observed steady states change in magnitude or bifurcate into other types of behaviour. It is hoped that by understanding the
possible behaviour and the parameter regimes of the interesting behaviour, the design of phase-locked loop circuits will have clearer guidelines.
Contact: J.M.T. Thompson.
Chaotic Time Series
Perhaps the single most important lesson to be drawn from the study of non-linear dynamical systems over the last few decades is that even simple deterministic dynamical systems can give rise to
complex behaviour which is statistically indistinguishable from that produced by a completely random process. One obvious consequence of this is that it may be possible to describe apparently complex
signals using simple non-linear models. This has led to the development of a variety of novel techniques for the manipulation of such "chaotic" time series. In appropriate circumstances, such
algorithms are capable of achieving levels of performance which are far superior to those obtained using classical linear signal processing techniques. The Centre has wide interests in this area,
with projects focusing on both applications, and the development of new theory and algorithms.
State-Space Reconstruction
Virtually all approaches to the analysis of time series generated by nonlinear deterministic dynamical systems are based on the Takens Embedding Theorem. This typically allows us to reconstruct an
unknown dynamical system which gave rise to a given observed scalar time series simply by constructing a new state space out of successive values of the time series. This provides the theoretical
foundation for many popular techniques, including those for the measurement of fractal dimensions and Liapunov exponents, for the prediction of future behaviour, for noise reduction and signal
separation, and most recently for control and targeting.
Recent work in the Centre has been concerned with a number of generalizations to Takens Theorem. In particular, existing versions assume that the underlying system is autonomous. Unfortunately this
is not the case for many real systems; in the laboratory we often force an experimental system in order for it to exhibit interesting behaviour, whilst in the case of naturally occurring systems it
is very rare for us to be able to isolate the system to ensure that there are no external influences. We have recently proved two versions of Takens Theorem which are relevant to forced systems: one
is applicable to the case where the forcing is unknown, and the other to the situation where we are able to determine the state of the forcing system independently (usually because we are responsible
for the forcing ourselves). In collaboration with the University of Manchester Institute of Science and Technology, we are extending these results to encompass certain types of random forcing, and
investigating new algorithms for the analysis of nonlinear stochastic time series based on these ideas. Finally a related project is attempting to develop a framework for the analysis and processing
of irregularly-sampled time series.
Contact: J.P.M. Heald, J. Stark.
One of the main consequences of Takens Theorem is that time series generated by a nonlinear dynamical system are predictable, and this has given rise to a wide range of prediction algorithms.
However, hitherto these have all suffered from the disadvantage that they use batch processing. In other words they build a model of the dynamics of the time series from a pre-determined block of
observations x[1], ... , x[n], but then have no means of updating the model from further measurements x[n][+1], x[n][+2] ... as they are made. This severely limits the usefulness of such schemes in
many real time signal processing applications. To overcome this difficulty we have recently developed a continuous update prediction scheme for chaotic time series. This is based on a combination of
a standard radial basis function algorithm already widely used in chaotic time series prediction with an advanced recursive least squares method originally developed for use in (linear) adaptive
filtering. The resulting algorithm can also be made adaptive and hence can process chaotic time series whose underlying dynamics slowly vary with time. It has also proved possible to incorporate
model selection techniques, so that the choice of radial basis centres used in the model of the dynamics can be adaptively updated.
A different problem with most current prediction schemes is that they make no attempt to estimate the error inherent in any given prediction. In practical applications this is a serious drawback,
since an estimate of the quality of a prediction is often as important as the actual predicted value. Based on ideas of Bayesian estimation we have developed an algorithm for estimating
state-dependent noise levels in a time series and hence calculating error bars for predicted values. This has been found to work well for a single state space dimension, and efforts are currently
being made to extend it to higher dimensions.
Contact: J.P.M. Heald, J. Stark.
Recursive Filters
A recursive filter, broadly speaking, is one whose output is fed back into the filter and allowed to affect its future operation. Within the context of conventional digital filtering such filters
range from simple infinite impulse response (IIR) filters, used for a variety of signal processing tasks, including echo cancellation, signal equalisation and speech encoding, to recursive least
squares (RLS) filters such as the well known Kalman filter.
Traditionally, the mathematical analysis of filters assumes that the input signal is given by a stochastic process. However there is increasing interest in using recursive filters in so called
"chaotic" time series analysis. In such applications the input signal to the filter is then derived from a chaotic deterministic dynamical system. Obviously, a stochastic analysis is then
inappropriate and new methods have to be developed to understand the filter's behaviour. Our approach has been largely through the theory of invariant manifolds for skew products, and this has led to
both a new insight into existing results about IIR filters, and the development of a new framework for the analysis of the stability of RLS filters; the latter in particular has been a long standing
problem within the signal processing community.
Contact: J. Stark.
Noise Reduction
In most applications, we are unlikely to be given a pure chaotic time series. Instead, typically we will be asked to manipulate a mixture of a chaotic time series and some other signal. The latter
may represent noise, in which case we want to remove it, or it may be a signal that we wish to detect, in which case we want to extract it from our source time series and discard the chaotic part. An
example of the latter might be a faint speech signal masked by deterministic "noise" coming from some kind of vibrating machinery, such as an air conditioner. In both cases, the mathematical problem
that we face amounts to separating the original time series into its two components. A variety of schemes have been developed in the last few years to perform this task.
Our work has concentrated on improving the efficiency and stability of such algorithms. Thus for instance many noise reduction schemes suffer from instability in the vicinity of homoclinic
tangencies. Such tangencies are common in real chaotic systems, severely limiting the usefulness of these techniques. We have therefore investigated more sophisticated approaches to noise reduction,
culminating in a robust scheme based upon the Levenberg-Marquardt algorithm, which appears to be insensitive to the presence of homoclinic tangencies. Further improvements can be achieved if extra
information, either about the noise or about the deterministic signal is available; for example the noise may be slowly varying. Generalizations of this are under investigation, including
applications to under-sampled time series.
Similarly, all noise reduction algorithms involve the choice of a number of parameters; often these have to be set essentially arbitrarily. We are therefore attempting develop a Bayesian framework
for noise reduction which will hopefully allow a more rational selection of these parameters, as well as an objective evaluation of the performance of the resulting algorithms.
Contact: J.P.M. Heald, J. Stark.
Spatio-Temporal Time Series
Virtually all of the attention in the area of chaotic time series has been on the case of a single scalar signal. Many important applications, however, involve the simultaneous measurement of
observables at different spatial locations, leading to multivariate spatio-temporal time series. This raises a variety of interesting questions, of both a theoretical and a practical nature. We are
just beginning to address some of these, in collaboration with the University of Manchester Institute of Science and Technology.
One particular application of these ideas is the analysis of road traffic congestion, in collaboration with the London Centre for Transport Studies at UCL. Many motorways are now being fitted with
sensors which can collect large volumes of detailed information about traffic flow. Typically such sensors are placed at regular intervals along a motorway (say every 500m), and collect traffic
information at regular intervals in time (say every minute), giving rise to a spatio-temporal time series. The analysis of such data is obviously important from the point of view of both planning
future roads and managing traffic flow on existing roads, but is currently lagging far behind our ability to collect the data. We are therefore intending to investigate the relevance of techniques
from nonlinear dynamics to the characterisation and prediction of such time series, with a particular emphasis on the detection of incipient congestion.
Contact: J. Stark.
Applications to Biology
Biology and medicine is an area rich with potential applications of nonlinear dynamics, many of which have hitherto had relatively little attention. The Centre has begun to develop this area
relatively recently, but a number of projects, described below, have already become established, and several more are under consideration. To enhance collaboration in the field of mathematical
biology at UCL, the Centre has recently started a regular seminar series on this topic.
Developmental Biology
Animals of different species are both very similar and very different. The cell types found in mice, frogs, flies and worms are very similar, but the way in which they are organized to generate the
whole animal is very different. The cells are spatially arranged into a pattern, whose form represents an animal. Since all animals derive from a single cell, the fertilized egg, the pattern
characteristic of any individual must be generated during embryonic development - this process is called pattern formation. Understanding the mechanisms underlying the way in which pattern is
generated during embryonic development is an important biological problem that remains to be resolved. One of the main approaches to this problem has been through the building of mathematical models
of the developing embryo, with the aim of duplicating the qualitative behaviour observed in real organisms. Unfortunately, whilst current models can often reproduce some of the properties of real
embryos, they in general do not incorporate realistic biological information, nor do they lead to experimentally verifiable predictions.
In an attempt to overcome these objections we have embarked on a collaborative research project with the Department of Anatomy and Developmental Biology at UCL to devise models of the early embryo
which both make use of its known biological properties and give rise to testable predictions. These focus on cell-cell signalling via the exchange of small molecules through a specialized
intercellular structure, the gap junction. The embryo is thus modelled as a simple circuit or network consisting of the cells and the junctions between them. In mathematical terms, this leads to
systems of coupled ordinary differential equations, in contrast to the partial differential equations usually used to model pattern formation. We are currently investigating the dynamics of such
networks, with the aim of understanding which kinds of patterns they can give rise to.
Contact: S. Baigent, J. Stark.
Control of Ovulation
In collaboration with Kate Hardy and Steve Franks of the Imperial College School of Medicinewe are trying to understand the mechanisms that control mammalian ovulation and its failure. The ovary
contains a large number of immature dormant follicles, each one of which contains an egg. Such follicles continuously leave the dormant state and begin to develop. The vast number of these atrophy
and die before they reach maturity and usually only a single follicle ovulates in each reproductive cycle. It is important both medically and scientifically to understand how this follicle is
selected, and why such selection fails in PCOS, which is the single largest cause of anovulatory infertility. We were able to take an existing model and extend it so that it exhibited behaviour
characteristic of PCOS, and hence drew some tentative conclusions about the causes of PCOS which are consistent with current biological thinking. Unfortunately, the important feedback loop in the
model is an abstract one, and hence cannot be related to biological parameters, and we are therefore currently working towards more biologically realistic versions.
Contact: J. Stark.
Initiation of Follicles in the Human Ovary
Between puberty and the menopause, a woman will normally ovulate a single egg during each menstrual cycle. Such eggs are stored in specialized structures called follicles, in her two ovaries. All the
follicles and eggs that a woman ever possesses are laid down in a dormant state in her ovaries approximately half way through her foetal development. Throughout her life there is a steady stream of
follicles that initiate growth. It takes approximately 6 months for them to reach a stage where they can ovulate, and during this period the vast majority (at least 99.9%) of follicles die. When the
stock of follicles is exhausted, a woman enters the menopause. The mechanisms controlling initiation are largely unknown, and difficult to study experimentally. A understanding of these is both of
fundamental scientific interest, and of great significance to the many women in which the process fails to function properly, leading to infertility and other adverse consequences (eg increased risk
of diabetes, obesity and cardiovascular disease).
Some data is available on how the number of follicles in the ovary changes with time. This suggests a process analogous to radioactive decay though clearly the detailed mechanisms must be very
different. In collaboration with Kate Hardy and Steve Franks of the Imperial College School of Medicinewe propose to construct and analyse models of follicle initiation. By comparing these models to
observed experimental data we shall propose plausible mechanisms for initiation. Such models in turn will suggest new experiments, and by iterating this process we intend to gain a better
understanding of follicle development.
Contact: J. Stark.
Cytokine Networks
Cytokines are small soluble proteins that convey information between cells. To date more than one hundred have been identified and cloned and the literature is full of descriptions of their
biological activities. Interactions between cytokines and their target cells are extremely complex, and attempts to understand how they regulate cellular growth and differentiation have depended
mostly on the reductionist approach of identifying the many activities each has on a cell or cellular system of interest, and then trying to fit them together in some sort of coherent biological
model. Although enormous progress has been made in cataloguing the many cytokine activities, the sheer complexity of the system has so far defeated any attempt to understand how they work together in
a co-ordinated fashion to control cellular function.
In collaboration with the Institute of Child Health we are beginning to develop and analyse models of the cytokine network. It is initially unrealistic to attempt to model the whole network, and
hence we are initially focusing on relatively simple models of only a part of the system, namely the cytokine interactions that regulate human TH1 and TH2 T cell subset growth and differentiation and
the information flow required for IgE antibody production. Most of the interacting cytokines involved in this sub-system have been identified and characterized, and in vitro culture techniques in
which predictions made by non-linear dynamical modelling can be tested are well established. As experience with the model grows, it will gradually be refined and additional interactions will be
incorporated. The tools of modern non-linear dynamics will be used to investigate a number of fundamental issues: the number and properties of the equilibria of the system, the possibility of
oscillations or more complex "chaotic" dynamics, the robustness and stability of the system to various classes of perturbations, and the effect of structural changes such as the removal of cytokines
and/or receptors from the system.
Contact: J. Stark.
Modelling T Cell Activation
Together with immunologist Andrew George at Imperial College School of Medicine, we aim to develop a fuller understanding of how engagement of the T cell receptor activates the T cell. This event
involves a number of stochastic processes, such as ligand dissociation, and so it is difficult to understand how the cell can respond in a specific and sensitive manner to foreign antigen. Recently
we have used an interdisciplinary approach of combining mathematical and computer modelling with basic biochemical data to address this problem. We have made a stochastic cellular automata model and
have shown, for the first time in the context of eukaryotic cells, that cross talk between receptors can be used to enhance the specificity of ligand recognition with little loss of sensitivity. We
aim to go on and use this approach to more fully explore the following aspects of T cell recognition
• Is the action of altered ligands (antagonists and partial agonists) explained by cross talk between receptors?
• Does the presence of MHC bearing the incorrect ligand (>99.9% of MHC) alter the sensitivity and specificity of the signalling event?
• What role does thymic selection have on the nature of signalling events?
• How can alloreactivity be accommodated in relationship to serial triggering and receptor cooperativity?
Contact: J. Stark, Cliburn Chan
Applications of Inertial Manifolds to Biology
This has developed out of the work on modelling gap junctions described above. See abstract for a description of completed research. We have now turned our attention towards stochastic models of
parasite dynamics developed by Prof. V. Isham in the Department of Statistical Science. Many of the mathematical tools used for this work are closely related to those for the study of invariant
graphs for skew product systems below.
Contact: S. Baigent, J. Stark.
Cell Death in Pre-implantation Embryos
Together with Kate Hardy and Prof. Lord Winston of the Imperial College School of Medicine we have recently begun to develop models of programmed cell death in pre-implantation human embryos. Such
embryos exhibit surprisingly high levels of cell death, and high rates of developmental arrest during the first week in vitro. The relation between the two is unclear and difficult to determine by
conventional experimental approaches, partly due to limited numbers of embryos. We have there-fore applied a mixture of experiment and mathematical modelling to show that observed levels of cell
death can only be reconciled with the high levels of embryo arrest seen in the human if the developmental competence of embryos is already established at the zygote stage, and environmental factors
merely modulate this. We also predicted, and subsequently verified experimentally, that cell death did not occur during the first few cell divisions.
Contact: J. Stark.
Theoretical Dynamics
Many of the above applications require the development of new theoretical concepts. The Centre therefore maintains a vigorous research programme in several barnches of more abstract dynamical
Forced Nonlinear Oscillators
Many problems in engineering and the applied sciences are modelled by low dimensional forced oscillators. Examples include vibrating mechanical and structural (e.g. beams, bridges) engineering
systems, marine systems (e.g. ships, oil platforms), electronic circuits (see below) and devices (e.g. Josephson junctions), as well as biological oscillators. Whilst the linear behaviour of such
systems has long been well understood, the potential dynamics of nonlinear oscillators is far more complex and many open problems remain. The Centre has a long history of research in this area.
Current effort has concentrated on the particular topics of Melnikov's method.
Melnikov's method is a perturbation technique widely used to estimate the parameter location at which a homoclinic tangency occurs. This is an indicator of complex motion and, loosely speaking,
implies the existence of "chaos" in the system. However, it is not always appreciated that this particular behaviour is often a comparatively insignificant feature of the dynamics. Although it exists
as a solution of the system equations, it can be extremely difficult to excite a physical system into the associated "chaotic" motions. Other regimes of complex motion occur which are much more
readily encountered but for which there exist no comparable mathematical techniques to estimate their locations. This has motivated recent work in the Centre on the application of topological methods
such as lobe dynamics and symbolic dynamics to the study of homoclinic tangles. This in turn has demonstrated the importance of higher order tangencies (referred to as Birkhoff signature changes)
which are associated with these more "accessible" occurrences of complex response and chaotic instability. Using a simple energy interpretation of Melnikov's method, it has been shown how this could
be extended to detect these higher order, but more important, phenomena.
Contact: J.M.T. Thompson.
The Parametrically Excited Pendulum
The parametrically excited pendulum can exhibit a variety of dynamical behaviour including stable equilibria, periodic oscillations, continuous rotations, and motions which include rotations and
oscillations: called tumbling solutions. Chaotic behaviour which is either oscillatory, rotating or tumbling, can also be viewed. Research over recent years has used numerical simulations and
theoretical treatments to identify possible motions and provide an understanding of the bifurcational structure. A topological analysis using knot and braid theory has revealed much about the
underlying dynamics.
Contact: S.R. Bishop.
Phenomenological Studies of Bifurcations
Using a wide variety of computational and geometrical techniques we are exploring the bifurcational behaviour of a number of carefully chosen archetypal models of dynamical systems drawn from
engineering and the applied sciences. Of particular interest at the moment are the indeterminate jumps to resonance that have been identified as robust and very typical events in a variety of
softening oscillators under both direct and parametric excitation. They arise, for example, when the saddle of a saddle-node fold bifurcation is located on a fractal basin boundary, and give rise to
a totally unpredictable jump to two or more disparate solutions, one of which might signal the failure of an engineering system.
Contact:J.M.T. Thompson.
Invariant Manifolds for Skew Product Systems
Many dynamical systems of both practical and theoretical importance consist of one system driven by another. The simplest, best known, and most widely studied case of this is that of periodic
forcing, but a variety of other examples, involving more complex forms of forcing, can be found throughout engineering and the applied sciences. The standard approach to studying the dynamics of
driven systems is to extend the state space of the driven system to include the driving system; systems of this kind are often called skew products. Where such systems are designed to perform some
useful task, the forced system will usually be stable in some sense or another, at least in the absence of any forcing; this is for instance the situation in certain classes of filters, and in the
study of synchronization. When this contraction is uniform, it can easily be shown that there exists a globally attracting invariant set which is the graph of a function from the driving state space
to the driven state space; this is analogous to the well known concept of an inertial manifold. If the driving state space is a manifold and the contraction is sufficiently strong this invariant set
is a normally hyperbolic manifold, and hence smooth. Unfortunately, in many applications uniform estimates of contraction rates are not available. The aim of this project is therefore to generalize
such results to non-uniform contraction rates. The invariant graph is then only defined over a set of full measure of input conditions and hence we need to use an appropriate notion of its
smoothness. This turns out to be that given by the Whitney Extension Theorem. Using standard ideas from Pesin theory, we have been able to show that the invariant graph is indeed smooth in this
sense, with the smoothness depending on average contraction rates. We are currently investigating the structural stability and other properties of such invariant graphs.
Contact: J. Stark.
Quasiperiodically Forced Systems
A different class of forced systems arises if we restrict the type of forcing we wish to consider, as opposed to restricting the class of forced system, as above. After periodic, the next simplest
class of forcing to consider is that of two forcing frequencies whose periods are not rationally related. Such systems are called quasiperiodic, and appear to exhibit interesting new phenomena not
found in periodically forced systems. In collaboration with Queen Mary and Westfield College and the University of Potsdam, Germany, we are trying to understand several aspects of such systems,
partly using the tools of invariant manifolds.
Contact: J. Stark.
UCL >> CNDA >> Research Index >> Postgraduate Information > Research Descriptions
Original by J.Stark 1993; Converted by rtftohtml 16/3/95 jpmh; Last modified 17/6/1996 js.
UCL Centre for Nonlinear Dynamics and its Applications,
University College London, Gower Street, London, WC1E 6BT, UK. | {"url":"http://www.ucl.ac.uk/CNDA/about/research/outline.html","timestamp":"2014-04-19T19:45:18Z","content_type":null,"content_length":"45846","record_id":"<urn:uuid:60c26126-658c-4f23-84e4-bc7971d7bc5d>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00399-ip-10-147-4-33.ec2.internal.warc.gz"} |
Motivation/interpretation for Quillen's Q-construction?
up vote 20 down vote favorite
This question has been on my mind for a while. As I understand it, the Q-construction was the first definition for higher algebraic K-theory. Some details can be found here.
I have always wondered what train of thought led Quillen to come up with this definition. Does anyone know an interpretation of the Q-construction that makes it seem natural?
5 For the development and underlying intuitions of the several K-theory constructions see Grayson's recent artcle: math.uiuc.edu/K-theory/1018 – Peter Arndt Dec 10 '11 at 16:35
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4 Answers
active oldest votes
Quillen's $Q$-construction naturally arises as a cleaned-up version of Segal's edgewise subdivision of Waldhausen's $s_\bullet$-construction.
Waldhausen's $s_\bullet$-construction gives a rather natural definition of the algebraic $K$-theory space of an exact category $C$. It is a simplicial set $s_\bullet C$, whose elements in
degree $q\ge0$ are ``length $q$ filtrations'' $0 \rightarrowtail M_1 \rightarrowtail \dots \rightarrowtail M_q$ in $C$, plus some choices. The algebraic $K$-theory space $K(C)$ is defined
as the loop space $\Omega |s_\bullet C|$ of the geometric realization of this simplicial set.
My understanding is that Quillen knew of this construction, but was unhappy with the passage from categories to simplicial sets. At some point he realized that even if $s_\bullet C$ is not
the nerve of a category, its Segal subdivision $Sd(s_\bullet C)$ is (basically) the nerve of a category. That category is the $Q$-construction, $Q(C)$.
up vote 19 Since Segal subdivision does not change the homotopy type, the loop space $\Omega |Q(C)|$ of the classifying space of $Q(C)$ is as good a definition for $K(C)$ as the one I first stated.
down vote
Some people would write $BQ(C)$ where I write $|Q(C)|$. See Section 1.9 of
Waldhausen, Friedhelm, Algebraic $K$-theory of spaces. Algebraic and geometric topology (New Brunswick, N.J., 1983), 318--419, Lecture Notes in Math., 1126, Springer, Berlin, 1985.
for details.
• John
1 I'd never heard that Quillen was aware of the S.-construction. On the other hand, it's very similar to Segal's construction of the Gamma-space associated to a category with sums, which
Segal almost certainly learned from Quillen. So maybe in the end this isn't a huge surprise. But it does go a long way towards explaining where the Q-construction came from! – Dan
Ramras Sep 30 '10 at 19:52
4 Waldhausen's construction came after Quillen's. As it happens, in one of the few conversations I ever had with Quillen (this was in the mid-90s, in the course of a car ride from
Cambridge to Providence) he asked me to tell him something about Waldhausen K-theory, and I remember telling him that the S. construction is related to his Q-construction by edgewise
subdivision. – Tom Goodwillie Dec 11 '11 at 16:26
I seem to recall that in Segal's paper about Gamma-spaces he modestly describes what he is doing as developing some of Quillen's ideas about algebraic K-theory. – Tom Goodwillie Dec 11
'11 at 16:28
add comment
I've always liked the interpretation Quillen gave in his "On the group completion of a simplicial monoid" paper (Appendix Q in Friedlander-Mazur's "Filtrations on the homology of algebraic
varieties"). Here is a somewhat revisionist version.
Associated to a monoidal category C, you can take its nerve NC, and the monoidal structure gives rise to a coherent multiplication (an A[∞]-space structure) on NC. (If you work a little
harder you can actually convert it into a topological monoid.)
May showed in his paper "The geometry of iterated loop spaces" that an A[∞]-space structure on X is exactly the structure you need to produce a classifying space BX, and there is a natural
map from X to the loop space Omega(BX) that is a map of A[∞]-spaces, and is a weak equivalence if and only if π[0](X) was a group rather than a monoid using the A[∞]-monoid structure. In
fact, Omega(BX) satisfies this property, and so you can think of it as a "homotopy theoretic" group completion of the coherent monoid X.
What Quillen showed was that you can recognize the homotopy theoretic group completion in the following way: the homotopy group completion of X has homology which is the localization of
the homology ring of X by inverting the images of π[0](X) in H[0](X). Moreover, the connected component of the identity in the homotopy group completion is a connected H-space, so its
fundamental group is abelian and acts trivially on the higher homotopy groups.
up vote 19 In particular, if X is the nerve of the category of finitely generated free modules over a ring R, then X is homotopy equivalent to a disjoint union of the classifying spaces BGL[n](R),
down vote with monoidal structure induced by block sum. The monoid π[0](R) is the natural numbers N, and so you can consider the map
X = coprod_(n∈N) BGL_n(R) → coprod_(n∈Z) BGL(R)
to a union of copies of the infinite classifying space. This map induces the localization of H[*](X), so the space on the right has to have the same homology as the homotopy group
completion, but the problem is that the connected component of the identity on the right (BGL(R)) doesn't have an abelian fundamental group that acts trivially on the higher homotopy
groups, so this can't be the homotopy group completion yet.
So this leads to the plus-construction: to find the homotopy group completion you're supposed to take BGL(R) and produce a new space, which has to have the same homology as BGL(R), and
which has an abelian fundamental group (plus stuff on higher homotopy groups). This is what the plus-construction does for you.
Quillen's Q-construction contains within it the symmetric monoidal nerve construction (you can consider just the special exact sequences that involve direct sum inclusions and
projections), but it's got the added structure that it "breaks" exact sequences for you. I wish I could tell you how Quillen came up with this, but this is the best I can do.
man. I wish underscores didn't try to italicize things. – Tyler Lawson Oct 18 '09 at 12:41
they don't inside ` . Also, you can use <sub>` – Ben Webster♦ Oct 18 '09 at 14:15
Whoops, stupid lack of comment in previews. If you put things inside a pair of ``` they won't italicize. – Ben Webster♦ Oct 18 '09 at 14:16
@Tyler: I took the liberty of replacing symbols like oo, pi, and in with their Unicode analogs, if you don't mind. – Dmitri Pavlov Dec 10 '11 at 14:20
add comment
Expanding on Tyler Lawson's comment on the Q-construction, I would say the following. The K[0] functor involves two processes, a group completion (of the monoid structure given by direct
sum) and an identification of all the extensions of any two objects. That is, K_0 of an exact category E is the free abelian group generated by the objects of E (group completion) together
with the relations [B] = [A] + [C] for every exact sequence
A >---> B --->> C
Now, the higher K-theory is a sort of categorification, making both processes above remember higher homotopical data. Then we make the definition
up vote 9
down vote K[i](E) = pi[i] Omega BQE
Here Omega B corresponds to a homotopical group completion as explained by Tyler. Quillen's Q-construction changes the morphisms of the category E in a way that when group-completed, "the
extensions become split" (giving the relation [B] = [A] + [C] above). Strictly speaking, the Q-construction does not make exact sequences split in QE, as QE has the same isomorphisms as E,
only the non-isos change: a morphism A ---> B in QE correspond to an identification of A with a subquotient of B.
add comment
There is an altogether different motivation different from the ones discussed above that appears in a paper by Graeme Segal ("K-homology and algebraic K-theory," LNM 575 K-theory and
Operator Algebras, Athens Georgia 1975, pp. 113–127).
The $Q$-construction there is motivated by considering self-adjoint Fredholm operators on Hilbert space.
More, precisely Segal shows that the homotopy type of the classifying space of Q-construction of the category of finite dimensional vector spaces over the reals or complex numbers is the
same as that of the space $Saf(H)$ consisting of self-adjoint Fredholm operators on infinite dimensional Hilbert space $H$: $$ BQC \simeq Saf(H) . $$ A map $V\to V'$ in the $Q$-construction
on the category $C =$ Vect of finite dimensional vector spaces is represented as a triple $(W_+,W_-;\alpha)$ in which $\alpha: W_+\oplus V\oplus W_- \to V'$ is an isomorphism of vector
According to Segal, the idea is supposed to be that a Fredholm operator is determined up to contractible choice by its kernel and cokernel, which are a pair of finite dimensional vector
up vote 9 spaces. When a Fredholm operator is deformed continously, its kernel and cokernel can jump but only by adding isomorphic pieces to each.
down vote
In the self adjoint case, the operator is determined by its kernel up to contractible choice. The kernel then corresponds to an object of the $Q$-construction. When the operator is
deformed, the kernel jumps in such a way that the part added to it is a direct sum of the part on which the operator was positive and a part on which it was negative. These correspond to
the terms $W_\pm$ appearing above.
So the heuristic motivation in a nutshell is this: the objects of the $Q$-construction correspond to self-adjoint Fredholm operators and the morphisms correspond to deformations of such
operators. The passage is given by taking operator kernels.
(note: I think Segal wants to consider the above $C$ as a topological category).
add comment
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Displacement (net change)
April 11th 2010, 06:42 PM #1
Apr 2010
Displacement (net change)
The velocity function is v(t)= -t^2 +4t -3 for a particle moving along a line. Find the displacement (net distance covered) of the particle during the time interval [-1,6].
I've tried integrating the function and then plugging in the values.
Also I tried factorizing the function and dividing the integral based on the negative and positive sign and then integrating and plugging the values.
But none of the methods give the right answer. Anybody got other ideas?
Don't leave us hanging man! What is the right answer?
The velocity function is v(t)= -t^2 +4t -3 for a particle moving along a line. Find the displacement (net distance covered) of the particle during the time interval [-1,6].
I've tried integrating the function and then plugging in the values.
Also I tried factorizing the function and dividing the integral based on the negative and positive sign and then integrating and plugging the values.
But none of the methods give the right answer. Anybody got other ideas?
Have you tried integrating the velocity to find displacement as a function of time without bounds.
If you don't know the right answer, how do you know that the integral of v from -1 to 6 is wrong?
+70/3 or -70/3?
Oh sign errors, you all-too-familiar seductress you!
oh yea.....thats me.....lol
anyways here's another one if anyone's up for it.....I'm at a total loss here.
Two people, Jaime and Tyler, are racing each other. Assume that both their accelerations are constant, Jaime covers the last 1/10 of the race in 7 seconds, and Tyler covers the last 1/7 of the
race in 10 seconds. Who wins, and by how much?
April 11th 2010, 07:03 PM #2
Senior Member
Feb 2010
April 11th 2010, 07:09 PM #3
Apr 2010
April 11th 2010, 07:10 PM #4
April 11th 2010, 07:11 PM #5
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April 11th 2010, 07:13 PM #6
Apr 2010
April 11th 2010, 07:14 PM #7
Apr 2010
April 11th 2010, 07:14 PM #8
April 11th 2010, 07:20 PM #9
Apr 2010
April 11th 2010, 07:20 PM #10
Senior Member
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April 11th 2010, 07:23 PM #11
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April 11th 2010, 07:24 PM #12
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April 11th 2010, 07:27 PM #13
Apr 2010 | {"url":"http://mathhelpforum.com/calculus/138577-displacement-net-change.html","timestamp":"2014-04-21T15:13:18Z","content_type":null,"content_length":"61541","record_id":"<urn:uuid:25027f44-c601-4d94-8bee-5d5dad4110a6>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00209-ip-10-147-4-33.ec2.internal.warc.gz"} |
Cavendish Spheres, 5B30.20
In 1772 Cavendish used an apparatus consisting of two concentric spheres that are insulated from each other and from ground to prove the inverse square law of electrostatic force. This elegant
experiment requires some investment of time by the demonstrator beforehand to track down extraneous effects that could dominate the desired effect.
Equipment Location ID Number
Cavendish Spheres E&M (RHS), Bay B1, Shelf 1 5B30.20
Faraday Cage E&M (RHS), Bay B1, Shelf 2 5B.EQ21.a
Keithley Electrometer E&M (RHS), Bay B1, Shelf 2 5B.EQ20.a
1000V DC Power Supply, Pasco E&M (RHS), Bay B2, Shelf 2 5B.EQ23.a
Proof Plane E&M (RHS), Bay A1, Shelf 1 5A.EQ13.a
• RHS -- Means the Right-Hand-Side of the Electricity & Magnetism Cabinet Peninsula
1. Connect the Faraday cage to the Keithley Electrometer by connecting the ground to the outside of cage and the positive lead to the inside. Make sure the two leads are twisted around each other to
help cancel external charges. The Keithley Electrometer is sensitive to charging due to the wires rubbing against each other.
2. Connect a short lead (red wire in picture) to the outside cage. This wire can be used as a (momentary) grounding wire for the inside mesh in the event it accidentally gets charged or you can't
remove the charge by touching both meshes with your fingers.
3. Set the Keithley Electrometer voltage range knob to 30V full-range and the scaling knob to 0V on center.
4. Remove the top hemisphere.
5. Connect the outer hemisphere to the outside of the cage (blue wire in photo).
6. Connect the positive terminal of the Pasco 1000V DC Power Supply to the inner sphere (yellow wire in photo).
7. Do not connect the ground terminal of the Pasco 1000V DC Power Supply to anything. The Power supply and Electrometer share a common ground through the power cord.
8. Raise the inner sphere to a potential of 1000V.
9. How to ground the Proof Plane!
□ The Proof Plane and handle must be grounded
□ Best way to do this is with a burner or open flame by moving the Proof Plate through the open flame and then testing it in the Faraday Cage. Repeat if needed.
□ Another way is to simultaneously tough both meshes of the Faraday Cage with your fingers and touch or rub the Proof Plate to the inside of the Faraday Cage. Repeat if needed.
At this point the bottom hemisphere is grounded and the inner sphere is at 1000V. Make some measurements to gain confidence:
1. Touch the proof plane to the inner sphere
2. Bring the proof plane inside the inner cage. Electrometer needle should deflect positive by about 6V. This indicates there is positive charge on the inner sphere
3. Ground the proof plane on the outside cage. Touch the proof plane to the outside of the bottom hemisphere. Bring the proof plane inside the inside cage. The needle should not deflect. This
indicates there is no charge on the outside of the bottom hemisphere.
4. Ground the proof plane on the outside cage. Touch the proof plane to the inside of the bottom hemisphere. Bring the proof plane inside the inside cage. The needle should deflect negative almost
6V. This indicates there is negative charge on the inside of the outer hemisphere.
5. Disconnect the yellow wire from the inner sphere connector. Turn off the power supply.
6. Ground the proof plane on the outside cage. Touch the proof plane to the inner sphere. Bring the proof plane inside the inner cage. The needle should deflect almost as much as before.
At this point the bottom hemisphere is grounded and the inner sphere is at 1000V. Make some measurements to gain confidence:
1. Touch the proof plane to the inner sphere
2. Bring the proof plane inside the inner cage. Electrometer needle should deflect positive by about 6V. This indicates there is positive charge on the inner sphere
3. Ground the proof plane on the outside cage. Touch the proof plane to the outside of the bottom hemisphere. Bring the proof plane inside the inner cage. The needle should not deflect. This
indicates there is no charge on the outside of the bottom hemisphere.
4. Ground the proof plane on the outside cage. Touch the proof plane to the inside of the bottom hemisphere. Bring the proof plane inside the inner cage. The needle should deflect negative almost
6V. This indicates there is negative charge on the inside of the outer hemisphere.
5. Disconnect the yellow wire from the inner sphere connector. Turn off the power supply.
6. Ground the proof plane on the outside cage. Touch the proof plane to the inner sphere. Bring the proof plane inside the inner cage. Needle should deflect almost as much as before.
1. Replace the outer upper hemisphere.
2. Disconnect the grounding wire (blue wire in photo) from the outside cage and touch it momentarily to the inner sphere connector. This brings the two spheres to the same potential.
3. Replace the grounding wire (blue wire in photo) on the outside cage. This grounds the outer sphere.
4. Remove the outer top hemisphere.
5. Ground the proof plane on the outside cage. Touch the proof plane to the inner sphere. Bring the proof plane inside the inside cage. The needle should not deflect!
Discussion: The fact that the needle does not deflect on the last measurement shows that the inner sphere has been left neutral. This was proposed by Cavendish as a consequence of the inverse square
law of the electrostatic force. I.e. if the electrostatic force dropped off faster than inverse square, the inner and outer spheres would end up with charges of the same sign. Alternatively, if the
electrostatic force dropped off slower than inverse square, the inner and outer spheres would end up with charges of opposite sign. Only if the electrostatic force drops off as inverse square can the
inner sphere end up with no charge.
Cautions, Warnings, or Safety Concerns:
1. This demonstration requires practice
2. Setup time is at least 10 minutes
3. Do not wear wool or a sweater
attachment other photos attachment other photos
attachment other photos attachment other photos
1. "The Electrical Researches of the Honourable Henry Cavendish", edited by James Clerk Maxwell, pages 104-113. UW-Copy
2. "The Electrical Researches of the Honourable Henry Cavendish", edited by James Clerk Maxwell, Cambridge: University Press (1879) Pages 104-113
3. "Classical Electrodynamics", 2nd Edition J.D.Jackson, Pages 5-7
Electricity & Magnetism Cabinets | {"url":"https://wiki.physics.wisc.edu/facultywiki/CavendishSpheres","timestamp":"2014-04-16T20:27:51Z","content_type":null,"content_length":"23087","record_id":"<urn:uuid:87e70c62-f85b-4cdc-9c72-7df79d49a554>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00526-ip-10-147-4-33.ec2.internal.warc.gz"} |
September 2002 - June 2003
Mathematical Optimization is experiencing substantial advances. There have been remarkable developments in the underlying methods, for example, the emergence of interior methods for both linear and
nonlinear optimization problems, the rediscovery of cuts as an effective tool for solving general integer programs, as well as for structured problems, and the emergence of new disciplines such as
positive semidefinite programming which attack a broad variety of discrete and continuous problems. The problems being studied are much larger than before and the computing platforms are orders of
magnitude more powerful. We believe that the 2002-2003 academic year will be an excellent time for the consolidation and advancement in this field that would be made possible by the focus of an IMA
Special year. In addition, the next SIAM meeting on Optimization is scheduled for 2002, which should have a significant amount of synergy with an IMA Special Year on Optimization.
We propose to break the year into three semesters, each focusing on a different topic. The first will focus on the rapidly evolving area of supply chain, transportation and logistics optimization, as
well as advances in integer programming. These are some of the fields placing increasingly large demands on the mathematical methods, as well as driving a focus on stochastic optimization and the
notion of robustness.
The second semester will focus on advances in the underlying methods dealing with both nonlinear and linear optimization. Specific focus areas will probably include semidefinite programming,
computational differentiation and nonconvex optimization.
The third will deal with the connections between optimization and information technology, an area that is proving to be increasingly important, and which would substantially benefit from a period of
focus. This includes areas of discrete mathematics as well as areas such as network design and optimization.
We plan to hold a one-week introductory workshop at the beginning of the year, for the particular benefit of the optimization year postdocs. This will provide an overview of the state of the art in
the underlying mathematical disciplines, and should provide a basis for much of the years activities. In addition, we will hold shorter tutorial workshops as needed during the three semesters.
Annual Program Workshops and Tutorials
9/9-13/02 Tutorial: Supply Chain and Logistics Optimization
9/23-27/02 Workshop: The Role of Optimization in Supply Chain Management
10/14-19/02 Workshop: Computational Methods for Large Scale Integer Programs
11/11-15/02 Workshop: Travel and Transportation
1/8/03 Tutorial: Optimization in Simulation-Based Models
1/9-16/03 Workshop: Optimization in Simulation-Based Models
3/11/03 Tutorial: Semidefinite Programming and Robust Optimization
3/12-19/03 Workshop: Semidefinite Programming and Robust Optimization
4/6/03 Tutorial: Network Management and Design
4/7/-11/03 Workshop: Network Management and Design
5/5/03 Tutorial: Data Analysis and Optimization
5/6-9/03 Workshop: Data Analysis and Optimization | {"url":"https://www.ima.umn.edu/2002-2003/","timestamp":"2014-04-17T06:43:07Z","content_type":null,"content_length":"57299","record_id":"<urn:uuid:7ef515fd-58b9-4392-9dbc-67660db15de3>","cc-path":"CC-MAIN-2014-15/segments/1398223201753.19/warc/CC-MAIN-20140423032001-00226-ip-10-147-4-33.ec2.internal.warc.gz"} |
Jacobson-Bourbaki correspondence
up vote 3 down vote favorite
The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications
of this correspondence known ? (It only seems to lead an existence as an exercise in algebra textbooks, and never appears in lectures.) 2. Does there exist an infinite(-dimensional) version,
exploiting something akin to the Krull topology ? Thanks in advance for any helpful remarks or insights ! Kind regards, Stephan.
ac.commutative-algebra linear-algebra ra.rings-and-algebras
If you can access MathSciNet, you might get some clues from a simple 'Anywhere' search of their database for 'Jacobson-Bourbaki'. This returns 31 items, ranging from a paper by Henri Cartan (1947)
to one by Lars Kadison (2008). Of course, that kind of search doesn't get into the texts of articles. My point (as a non-specialist) is that further developments of mathematical ideas almost
always occur; but whether they are 'interesting' is another question. – Jim Humphreys Apr 23 '11 at 16:42
(No, I'm afraid I can't get into MathSciNet.) Hopefully nobody misunderstands the intention of my question; I most certainly did not want to detract from the merit of the Jacobson-Bourbaki
correspondence as such, quite the contrary ! I just find it a pity that such a fine instrument is left in the toolkit unused (seemingly) ... Kind regrads, Stephan. – Stephan F. Kroneck Apr 23 '11
at 17:06
What is the Jacobson-Bourbaki correspondence? – Qiaochu Yuan Apr 23 '11 at 17:56
1 @ Qiaochu Yuan: probably easiest if I just give you this link: eom.springer.de/J/j110010.htm Kind regards, Stephan. – Stephan F. Kroneck Apr 23 '11 at 18:12
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Kolmogorov Algorithms are Stronger than Turing
- Bulletin of the European Association for Theoretical Computer Science , 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware
systems. We describe the quest to understand and define the notion of algorithm. We start with the Church-Turin ..."
Cited by 19 (9 self)
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems.
We describe the quest to understand and define the notion of algorithm. We start with the Church-Turing thesis and contrast Church's and Turing's approaches, and we finish with some recent
- Bull. of Euro. Assoc. for Theor. Computer Science , 1988
"... I felt honored and uncertain when Grzegorsz Rozenberg, the president of EATCS, proposed that I write a continuing column on logic in computer science in this Bulletin. Writing essays wasn’t my
favorite subject in high school. After some hesitation, I decided to give it a try. I’ll need all the help ..."
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I felt honored and uncertain when Grzegorsz Rozenberg, the president of EATCS, proposed that I write a continuing column on logic in computer science in this Bulletin. Writing essays wasn’t my
favorite subject in high school. After some hesitation, I decided to give it a try. I’ll need all the help I can get from you: criticism, comments, queries, suggestions, etc. Andrei Nikolayevich
Kolmogorov died a few months ago. In recent years he chaired the Department of Mathematical Logic at the Moscow State University. In a later article or articles, I hope to discuss Kolmogorov’s ideas
on randomness and information complexity; here let me take up the issue of Kolmogorov machines and their close relatives, Schönhage machines. I believe, we are a bit too faithful to the Turing model.
It is often easier to explain oneself in a dialog. To this end, allow me to introduce my imaginary student Quizani. • Quizani: I think you should introduce yourself too. Don’t assume everyone knows
you. • Author: All right. I grew up in the Soviet Union and started my career in the Ural University as an algebraist and self-taught logician. In 1973, I emigrated to Israel where I did logic and
taught at Ben-Gurion
"... In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NP-hard problems in
polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved s ..."
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In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NP-hard problems in
polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. 1 Introduction and Formulation of the Problem Many practical problems are NP-hard.
It is well known that many important practical problems are NP-hard; see, e.g., [11, 14, 27]. Under the usual hypothesis that P̸=NP, NP-hardness has the following intuitive meaning: every algorithm
which solves all instances of the corresponding problem requires, for
"... In late 1970s and early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NP-hard problems in polynomial
time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. 1 ..."
Add to MetaCart
In late 1970s and early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NP-hard problems in polynomial
time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. 1 Introduction and Formulation of the Problem Many practical problems are NP-hard. It is well
known that many important practical problems are NP-hard; see, e.g., [7, 9, 22]. Under the usual hypothesis that P̸=NP, NP-hardness has the following intuitive meaning: every algorithm which solves
all the instances of the corresponding problem requires, | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=4310921","timestamp":"2014-04-18T12:15:13Z","content_type":null,"content_length":"20926","record_id":"<urn:uuid:c63036f1-10d3-44e6-97c3-10e6b556e475>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00563-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Help Ms.Sue!!! Social Studies
Yes, it is B. Most people there are Sunni, Arab, and do live in urban areas.
Help Ms.Sue!!! Social Studies
You're welcome.
Help Ms.Sue!!! Social Studies
I'm so sorry...I meant to say that it is not C. It is actually B.
Help Ms.Sue!!! Social Studies
It's not B.
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adoption-the ethical dilemmas the experience, hrs wages and the importance role of their...
have to point out the ethical dilemmas(so I think problems when adapting and what people in social work adaption receive in terms of wages and hours in different states and the job outlook in
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exactly 3 out of the first 5 questions (d) at least 3 of the first 5 questions EXCE...
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Shania is having trouble understanding her algebra assignment. To help her understand, she asks Mark why is x^2*y^3 not equal to (x,y)^5 Which of the following responses would be best for Mark to
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Suppose you make an electromagnet with a copper wire and a battery by putting the two ends of the wire on the two ends of the battery. If you switch the side of the battery, are you changing the
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I mean how do you simplify that?
-sqrt60 How to do this?
Calculus Help Please!!!
The anwser is B
SCIENCE PLEASE HELP ME!!!!!!!! ASAP PLEASE PLEASE
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FIGURE IT OUT YOURSELF!!!!!!!!! it is really easy lol
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(Justify your answer.) show steps! Thanks!
MATH VERY URRRGENT!!!!!!!!!!!!!
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English (essay help)
All I needed were 2-3 sentences that's all, I'm not asking you to write my paragraph, I've written the entire essay but if you can't help with 2 sentences then there was no point in posting my
question cause it was a waste of my time and I didn't receive help
English (essay help)
Like can you give me an example of an intro about confederation including my ideas and that will help me because I already have my body paragraphs done and all I need is the intro
English (essay help)
And that's y I posted the question because I don't get any help from websites, I need someone to show me
English (essay help)
I don't understand, my teacher said my intro sentence was good and my thesis was good, all I need is 2-3 sentences leading from general to specific
English (essay help)
I have to write a essay about the confederation of Canada in 1867 and why it was a good choice. I have all of my body paragraphs done but I'm having trouble thinking of a proper introduction. My
first sentence for my intro is: confederation was the best choice for the Brit...
no one's here to help us!!! -.-
well what if my made-up name is anonymous? :P
no one's here to help us!!! -.-
oh i have to create a user or something?
no one's here to help us!!! -.-
help us!!! plzz btw this is not a question :P
What higher education institution were established to assist blacks during reconstruction? is it college?
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Possible Answer
Sparking curiosity in students plays a vital role in education. When children develop an interest in a topic, they dedicate more time to exploring and learning about it. Math ... - read more
There are several different techniques and instructional strategies that teachers can use to encourage students. ... A second strategy or activity that can be used to encouraging curiosity
Share your answer: strategies for encouraging curiosity in mathematics?
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Sufficiency in quantum statistical inference
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Coarse-grining is a basic concept in mathematical statistics and its analogue may be defined in the formalism of quantum mechanics. In the lecture sufficiency of a coarse-graining is discussed with
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Respected Sir/Madam;
Please help me with Q.2 in the attached image....please...!!!!
Answer of Q.2 is given in the... - Homework Help - eNotes.com
Respected Sir/Madam;
Please help me with Q.2 in the attached image....please...!!!!
Answer of Q.2 is given in the page only.
For a projectile thrown with initial velocity u, and at an angle `theta` with the horizontal, the average velocity for motion between its starting point and the highest point of its trajectory, the
average velocity has to be found.
We know, velocity is displacement /time. So, average velocity is the total displacement /total time.
The initial velocity, u can be resolved into two components: one horizontal, `ucostheta` , the other vertical, `usintheta` .
At the highest point, the vertical velocity is zero. Applying (v=u+at),
`rArr t = (usintheta)/g`
Time to reach the highest point is `(u*sin theta) /g`
Horizontal displacement = `(u*costheta*u*sintheta)/g` = `(u^2sinthetacostheta)/g` (as horizontal velocity remains constant throughout)
Vertical displacement =`((usintheta)/g)*((usintheta)/2)`
= `(u^2sin^2theta)/(2g)` (as vertical velocity is a constantly decreasing one, and is zero at the top, the average during rise is half its initial value).
As displacement is a vector quantity, resultant can be obtained by squaring and adding these two displacements, and finally taking the square root.
`d = sqrt (((u^2sinthetacostheta)/g)^2 + ((u^2sin^2theta)/(2g))^2)`
= `sqrt ((u^4sin^2thetacos^2theta)/g^2 + (u^4sin^4theta)/(4g^2))`
= `sqrt((u^4sin^2theta)/(4g^2)(4cos^2theta + sin^2theta))`
= `(u^2sintheta)/(2g) sqrt(cos^2theta + sin^2theta+3cos^2theta)`
= `(u^2sintheta)/(2g) sqrt(1+3cos^2theta)`
Average Velocity =` ((u^2sintheta)/(2g) sqrt(1+3cos^2theta))/((usintheta)/g) `
= `u/2 sqrt(1+3cos^2theta)`
Hence correct answer is option B).
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Theorems and Postulates
Date: 12/02/2006 at 04:05:45
From: Alona
Subject: difference between a theorem and a postulate
If SSS, SAS, and AAS are theorems, why do other books still use them
as postulates? And can you show me the PROOFS that were used for
these theorems? :)
Sorry, I know you've answered questions about the topic many times but
as I was reading the answers I realized that you were trying to say
that SSS, SAS, and AAS are really theorems because they were proved
(theorems need proofs). But, why do books and even teachers still
teach students like us SSS, SAS, and AAS "POSTULATES". Do the words
"theorem" or "postulate" really matter? I am a high school student
studying geometry right now and your answers to my questions would be
a great help for me.
Thank you very much.
Date: 12/02/2006 at 23:29:48
From: Doctor Peterson
Subject: Re: difference between a theorem and a postulate
Hi, Alona.
These facts CAN be postulates, but they don't have to be. It's a
matter of how an author chooses to present geometry to his audience.
Different geometry texts choose different starting points. The best
way to do geometry is to start with as few assumptions as possible,
and prove everything from those. Many texts "cheat" a bit by using as
postulates anything they don't want to bother proving (probably
because the proofs are difficult and wouldn't really help their
students understand the subject). Others use a good, small set of
postulates, but state some theorems without proof, explaining that the
proof is beyond the level of the text. I prefer the latter approach,
but I can understand the "cheating".
It is possible to take ONE of these congruence facts as a postulate
and prove the others from it (so they become theorems). It is also
possible to define congruence in such a way that all three can be
proved from some more basic postulate about congruence.
You can take them however your own text presents them; but be aware
that they are all really equivalent facts, and which you take as
postulates doesn't affect how you use them, which is what really
matters. In other words, in answer to your question as to whether
"theorem" or "postulate" really matters: it matters in presenting a
specific systematic treatment of geometry, but not in USING the facts
you learn, which are true one way or another regardless.
See this page for more discussion:
Theorem or Postulate?
See also:
The Role of Postulates
If you have any further questions, feel free to write back.
- Doctor Peterson, The Math Forum
Date: 12/10/2006 at 23:07:42
From: Alona
Subject: Proving theorems from theorems
Dear Dr. Math, thank you very much for answering my past question
about "theorems" and "postulates". I now know that one of the SSS,
SAS and ASA theorems can be considered as a postulate. It just
depends on the starting point of the discussion. But they are really
theorems (I hope I understood it the way you want me to understand
My question is, is it really possible to prove theorems from theorems?
What I mean is, is it possible to call all the three congruency
theorems "theorems" and still prove each one using each theorem?
In our geometry class, it is possible to prove theorems from previous
theorems. Then why do we need to assume one of the congruency
theorems as "postulate" when we could really prove it using "theorems".
Thank you very much for your previous reply. It answered 70% of my
questions. These questions are the remaining 30%. :)
Date: 12/11/2006 at 12:49:58
From: Doctor Peterson
Subject: Re: Proving theorems from theorems
Hi, Alona.
Certainly you can prove a theorem from a theorem; you do it all the
time, I would think. You can use any known fact, whether theorem or
postulate, as the basis for a proof.
What you CAN'T do is prove A from B, and B from C, and C from A!
Such circular reasoning is not allowed, because you have to start
with something that is known to be true. So if you call ALL THREE of
these "theorems", then at least one of them has to be proved on the
basis of something else (such as a definition of congruency that is
more powerful than what elementary texts usually use).
That's why the best approach is to take one of them as a postulate,
and then prove the others as theorems. It doesn't matter which one
you start with, but you have to start with one without assuming
another is already true.
If you do merely prove each from another of them, then what you have
done is to show that they are all EQUIVALENT--that is, IF one is true,
then they all are. But then either they are all true, OR they are all
false. You don't know which!
I think the pages I referred you to answer this question, by
explaining the role of postulates as starting points. You may want
to reread them with this new perspective in mind.
- Doctor Peterson, The Math Forum
Date: 12/20/2006 at 01:11:37
From: Alona
Subject: Thank you (Proving theorems from theorems)
Thank you very very very much!! I totally understood the topic now! :) | {"url":"http://mathforum.org/library/drmath/view/70279.html","timestamp":"2014-04-19T12:53:46Z","content_type":null,"content_length":"10431","record_id":"<urn:uuid:a6cb8d2a-65df-434f-8e9a-f530bed75a14>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00476-ip-10-147-4-33.ec2.internal.warc.gz"} |
Difference between Compute and estimate statistics
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Thread: Difference between Compute and estimate statistics
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1. 04-23-2001, 12:14 PM #1 Expand Forum to
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Join Date
Jun 2000
chennai,tamil nadu,india
Can anyone reply me with the actual difference between compute statistics and estimate statistics of analyze command.
2. 04-23-2001, 12:18 PM #2
Senior Member
Join Date
Nov 2000
Compute uses all rows in the table/index/cluster to determine the statistics.
Estimate uses only a portion of the rows (you specify how many rows or what percent of rows). If you are using, say , 50 % of the rows, I don't know if Oracle uses every
other row, or the first or last half, though... does anyone know which portion of the rows Oracle uses?
3. 04-23-2001, 01:22 PM #3
Join Date
Nov 2000
Baltimore, MD USA
I'm not actually sure which records it reads. However, what I *have* noticed:
- The stats *will not* change if neither the number of rows nor the data changed.
This is important. This means that the algorithm they use is deterministic and based on the number of rows in the table together with the % to estimate.
- If you add a single row, which may be less than 1/1000th of 1%, the stats can change by up to 5%. This is affected significantly by how 'skewed' your data is.
- If you have any 'extreme' data values, do *not* use estimation! For example, say 99% of the data in a given, indexed date field are within the past 5 years. *However*,
this one rogue entry exists with a year of 1897. With Estimation, this value may or may not be found each time. As you can imagine, such a value will significantly change
the high and low values in your stats, and therefore the applicability of the index in ranged selects. Therefore, with estimation, you never know what kind of plans you
will get.
Of course, the solution is to fix the data, but this is sometimes not possible. Another solution is to create a histogram, but that only works if your date values are
'hard-coded' in the queries. The final choice is to always COMPUTE that table and optimize to a consistent set of 'bad' statistics, possibly hinting the index where
- So, basically, estimation is best for tables with relatively even distribution and no 'extreme' values. If you *do* use estimation, Oracle suggests using at least 30%.
Further, any value of 60% or greater is the same as using 100%.
Hope that all made sense,
- Chris
Can anyone reply me with the actual difference between compute statistics and estimate statistics of analyze command.
Compute uses all rows in the table/index/cluster to determine the statistics. Estimate uses only a portion of the rows (you specify how many rows or what percent of rows). If you are using, say , 50
% of the rows, I don't know if Oracle uses every other row, or the first or last half, though... does anyone know which portion of the rows Oracle uses? -John
I'm not actually sure which records it reads. However, what I *have* noticed: - The stats *will not* change if neither the number of rows nor the data changed. This is important. This means that the
algorithm they use is deterministic and based on the number of rows in the table together with the % to estimate. - If you add a single row, which may be less than 1/1000th of 1%, the stats can
change by up to 5%. This is affected significantly by how 'skewed' your data is. - If you have any 'extreme' data values, do *not* use estimation! For example, say 99% of the data in a given, indexed
date field are within the past 5 years. *However*, this one rogue entry exists with a year of 1897. With Estimation, this value may or may not be found each time. As you can imagine, such a value
will significantly change the high and low values in your stats, and therefore the applicability of the index in ranged selects. Therefore, with estimation, you never know what kind of plans you will
get. Of course, the solution is to fix the data, but this is sometimes not possible. Another solution is to create a histogram, but that only works if your date values are 'hard-coded' in the
queries. The final choice is to always COMPUTE that table and optimize to a consistent set of 'bad' statistics, possibly hinting the index where necessary. - So, basically, estimation is best for
tables with relatively even distribution and no 'extreme' values. If you *do* use estimation, Oracle suggests using at least 30%. Further, any value of 60% or greater is the same as using 100%. Hope
that all made sense, - Chris | {"url":"http://www.dbasupport.com/forums/showthread.php?9732-Difference-between-Compute-and-estimate-statistics&p=38512","timestamp":"2014-04-19T19:35:17Z","content_type":null,"content_length":"58533","record_id":"<urn:uuid:1e76cd7f-9942-4ee7-a606-866af6440500>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00452-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ars Mathematica
Alg-top in CS
In a post below, I mentioned algebraic topology in computer science. A nice application of alg-top is for study of concurrency in distributed systems. For instance, one approach is to consider
execution traces of a computational system being represented by time-directed paths through a space, and then to use alg-top methods to ask and answer questions about the structure of this space.
This approach leads naturally to a concept of homotopies of paths, equivalence classes of paths which may be transformed into one another via other paths in the space. What is different from
traditional homotopy theory is that the paths are directed, and so these are referred to as directed homotopies or dihomotopies. Paths which are not dihomotopically equivalent represent execution
traces on which there are events not reachable from one to another. For more on this, start with the GETCO conference pages. | {"url":"http://www.arsmathematica.net/2005/12/15/188/","timestamp":"2014-04-16T17:06:36Z","content_type":null,"content_length":"9442","record_id":"<urn:uuid:3d9bc8f3-a974-42d2-901c-de61ca9de1e3>","cc-path":"CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00457-ip-10-147-4-33.ec2.internal.warc.gz"} |
4.7 The Binary Black Hole
It is clear that the 3-dimensional inspiral and coalescence of black holes challenges the limits of present computational know-how. CCM offers a new approach for excising an interior trapped region
which might provide the enhanced flexibility to solve this problem. In a binary system, there are major computational advantages in posing the Cauchy evolution in a frame which is co-rotating with
the orbiting black holes. Such a description seems necessary in order to keep the numerical grid from being intrinsically twisted and strangled. In this co-orbiting description, the Cauchy evolution
requires an inner boundary condition inside the black holes and also an outer boundary condition on a worldtube outside of which the grid rotation is likely to be superluminal. An outgoing
characteristic code can routinely handle such superluminal gauge flows in the exterior [60]. Thus, successful implementation of CCM would solve the exterior boundary problem for this co-orbiting
CCM also has the potential to handle the inner black hole boundaries of the Cauchy region. As described earlier, an ingoing characteristic code can evolve a moving black hole with long term stability
[61, 62]. This means that CCM would also be able to provide the inner boundary condition for Cauchy evolution once stable matching has been accomplished. In this approach, the interior boundary of
the Cauchy evolution is located outside the apparent horizon and matched to a characteristic evolution based upon ingoing null cones. The inner boundary for the characteristic evolution is a trapped
or marginally trapped surface, whose interior is excised from the evolution.
In addition to restricting the Cauchy evolution to the region outside the black holes, this strategy offers several other advantages. Although, finding a marginally trapped surface on the ingoing
null hypersurfaces remains an elliptic problem, there is a natural radial coordinate system
This global strategy is tailor-made to treat two black holes in the co-orbiting gauge. Two disjoint characteristic evolutions based upon ingoing null cones would be matched across worldtubes to a
central Cauchy region. The interior boundary of each of these interior characteristic regions would border a trapped surface. At the outer boundary of the Cauchy region, a matched characteristic
evolution based upon outgoing null hypersurfaces would propagate the radiation to infinity.
Present characteristic and Cauchy codes can handle the individual pieces of this problem. Their unification appears to offer the best chance for simulating the inspiral and merger of two black holes.
The CCM module is in place and calibrated for accuracy. The one missing ingredient is the long term stability of CCM, which would make future reviews of this subject very exciting.
Characteristic Evolution and Matching
Jeffrey Winicour
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de | {"url":"http://relativity.livingreviews.org/Articles/lrr-1998-5/node16.html","timestamp":"2014-04-17T12:28:53Z","content_type":null,"content_length":"7090","record_id":"<urn:uuid:f6a36d8f-2fec-4be8-8a31-28d1b44161e1>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00646-ip-10-147-4-33.ec2.internal.warc.gz"} |
Tangent of 90 Degrees
Date: 5/24/96 at 13:28:26
From: Anonymous
Subject: Trignometry
Why is the tangent of 90 degrees "no solution"?
Date: 5/28/96 at 20:24:37
From: Doctor Pete
Subject: Re:Trignometry
There are lots of ways to think about why tan(90) is not defined. You
can go by the book: tan(x)=sin(x)/cos(x), but since sin(90)=1 and
cos(90)=0, and division by 0 is not allowed, tan(90) has no meaning.
Or think about it geometrically: say we have a right triangle ABC,
with angle ACB=90. Then look at, say, angle BAC. tan(BAC) = BC/AC,
but BAC is less than 90 degrees. We can make angle BAC closer and
closer to 90 degrees by increasing the length of BC, but as we keep
doing this, AC stays the same so the ratio BC/AC gets infinitely
large. So tan(90) has no geometric meaning. (Draw a picture, it'll
help a lot.)
-Doctor Pete, The Math Forum
Check out our web site! http://mathforum.org/dr.math/ | {"url":"http://mathforum.org/library/drmath/view/53988.html","timestamp":"2014-04-20T11:14:23Z","content_type":null,"content_length":"5912","record_id":"<urn:uuid:3051d638-2646-413d-8d7a-aedcd76a245d>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00556-ip-10-147-4-33.ec2.internal.warc.gz"} |
galois fields-conversion between polynomial and power represenation for extension
July 6th 2012, 08:06 AM #1
Junior Member
Feb 2009
galois fields-conversion between polynomial and power represenation for extension
Hey All,
In an extended galois field of size p^m we can have basically two representations of the elements...one based on all polynomials over GF(p) of degree m-1 or less...and second powers of the
primitive element. now given a power of primitive element u can find the polynomial representation by taking modulo by primitive polynomial ..but i cant figure out how to go back ..please kindly
explain the other way round i.e given polynomial representation how to find which power of the primitive polynomial it corresponds too...
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