## Feb 16, 2010

### Key Stage 3 National Strategy

http://www.counton.org/resources/ks3framework/

### Key Stage 3 National Strategy

This area contains the complete text of the Framework for teaching mathematics: Years 7, 8 and 9 in pdf format.

You will need to have Adobe Acrobat Reader installed in order to open these files. This is available to download free from Adobe. Framework Management Summary
Guide to the Framework
Key objectives
Yearly teaching programmes
Supplement of examples Years 7, 8 and 9
Using and applying mathematics to solve problems:
Applying mathematics and solving problems
Numbers and the number system:
Place value ordering and rounding Integers, powers and roots
Fractions, decimals, percentages, ratio and proportion
Calculations:
Number operations and the relationships between them: Mental methods and rapid recall of number facts and more
Algebra:
Equations, formulae and identities
Sequences and functions
Graphs of functions
Shape, space and measures:
Geometrical reasoning: lines, angles and shapes
Transformations
Coordinates Construction and loci
Measures and mensuration
Handling Data:
Specifying a problem, planning and collecting data Processing and representing data Interpreting and discussing results
Probability
Vocabulary checklist

## Feb 13, 2010

### Rooted product of graphs - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Rooted_product_of_graphs

In mathematical graph theory, the rooted product of a graph G and a rooted graph H is defined as follows: take |V(G)| copies of H, and for every vertex vi of G, identify vi with the root node of the i-th copy of H.

More formally, assuming that V(G) = {g1, ..., gn}, V(H) = {h1, ..., hm} and that the root node of H is h1, define $G \circ H := (V, E)$

where $V = \left\{(g_i, h_j): 1\leq i\leq n, 1\leq j\leq m\right\}$

and $E = \left\{((g_i, h_1), (g_k, h_1)): (g_i, g_k) \in E(G)\right\} \cup \bigcup_{i=1}^n \left\{((g_i, h_j), (g_i, h_k)): (h_j, h_k) \in E(H)\right\}$

If G is also rooted at g1, one can view the product itself as rooted, at (g1, h1). The rooted product is a subgraph of the cartesian product of the same two graphs.

The rooted product is especially relevant for trees, as the rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find graceful numberings for a wide family of trees.

## 

### AKS primality test - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/AKS_primality_test

The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by three Indian Institute of Technology Kanpur computer scientists, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena on August 6, 2002 in a paper titled PRIMES is in P. The authors received many accolades, including the 2006 Gödel Prize and the 2006 Fulkerson Prize for this work.

The algorithm determines whether a number is prime or composite within polynomial time, and was soon improved by others. In 2005, Carl Pomerance and H. W. Lenstra, Jr. demonstrated a variant of AKS that runs in O(log6+ε(n)) operations where n is the number to be tested, a marked improvement over the initial O(log12+ε(n)) bound in the original algorithm.

## References

1. ^ a b c Manindra Agrawal, Neeraj Kayal, Nitin Saxena, "PRIMES is in P", Annals of Mathematics 160 (2004), no. 2, pp. 781–793.
2. ^ a b H. W. Lenstra, Jr. and Carl Pomerance, "Primality Testing with Gaussian Periods", preliminary version July 20, 2005.

## CS369E: Expanders in Computer Science - Spring 2005

### Expanders, constructions and their applications

Time: Mon 2:15-4:05pm
Location: Educ 130 (Stanford)
Instructors : Cynthia Dwork (dwork AT microsoft DOT com) and Prahladh Harsha (<firstname> AT tti-c DOT org)
Homepage: http://cs369e.stanford.edu

#### Lectures

Complete Lectures Notes (82 pages) [ ps | gzipped-ps | pdf | gzipped-pdf ]
Individual Lectures (below)

1. Lecture 1 (Apr 4): Introduction, motivation and some applications
Definition; applications: time-space tradeoffs (super-concentrators), (almost everywhere) Byzantive agreement, Amplification with no extra random bits (Karp Pippenger Sipser), Expander Codes, use of expanders in metric embeddings
Ref: [Lin, Rob].
Lecture Notes (by Arpita Ghosh): [ ps | pdf ]

2. Lecture 2 (Apr 11): Expanders and eigen-values
Probabilistic method - existence of expanders (Pinsker);
eigen-value connection, spectral gap implies expansion, expander mixing lemma, expanders have large spectral gap (Alon), Statements w/o proof: Alon's 2nd eigen-value conjecture (Friedman), Ramanujan graphs, converse to expander mixing lemma (Bilu and Linial)
Ref: [Alo, BL, Fri].
Lecture Notes (by Geir Helleloid): [ ps | pdf ]

3. Lecture 3 (Apr 18): Random Walks
Random walks in (general) undirected graphs, undirected connectivity is in randomized logspace (Aleliunas et. al.); Universal Traversal Sequences (UTS) - definition and existence; Random walks on expanders
Ref: [Lov]
Lecture Notes (by David Arthur): [ ps | pdf ]

4. Lecture 4 (Apr 25): Derandomization
Pseudo-random generators: AKS generator, INW construction of Nisan's Generator.
Ref: [AKS, INW, IZ, Nis]
Lecture Notes (by Adam Barth and Prahladh Harsha): [ ps | pdf ]

5. Lecture 5 (May 2): Derandomized Linearity Testing & Error Correcting Codes
Part 1: Derandomized Linearity Testing (Shpilka-Wigderson); Lecture Notes (by Adam Barth): [ ps | pdf ]
Part 2: Introduction to Error-Correcting Codes and basic construction of linear decodable codes (Sipser-Spielman, Zemor)
Ref: [Gur2, SW, SS, Zem]

6. Lecture 6 (May 9): Error Correcting Codes and Expander Constructions
Part 1: Linear time decodable codes (Sipser-Spielman, Guruswami-Indyk, Zemor); Lecture Notes (by Hovav Shacham): [ ps | pdf ]
Part 2: Explicit Construction of Expanders: Margulis, Gaber-Galil, LPS (all without proof); Zig-Zag expanders (Reingold-Vadhan-Wigderson)
Ref: [Gur2, GI, RVW, SS, Zem]

7. Lecture 7 (May 16): Zig-Zag Product and Undirected Connectivity in Logspace
Part 1: Zig-Zag Product and Expander Construction (Reingold-Vadhan-Wigderson); Lecture Notes (by Geir Helleloid): [ ps | pdf ]
Part 2: Reingold's proof of SL=L; Lecture Notes (by Cynthia Dwork and Prahladh Harsha): [ ps | pdf ]
Ref: [RVW, Rei]

8. Lecture 8 (May 31): The PCP Theorem
Dinur's Proof of The PCP Theorem
Lecture Notes (by Krishnaram Kenthapadi): [ ps | pdf ]
Ref: [Din]
Note: Lecture on May 31 (Tuesday) in Gates 200

Style files for scribes: sample.tex and preamble.tex .

#### References

We will be adding more references as we go along. For the present, below are some sources of material on expanders:

##### Other References:

The links below point to the actual location of papers at the author's homepages or other electronic repositories and the papers are not reposted locally.

1. [AKS] Miklos Ajtai, Janos Komlos, Endre Szemeredi: "Deterministic Simulation in LOGSPACE", STOC 1987: 132-140.
2. [Alo] Noga Alon, "Eigenvalues and expanders", Combinatorica 6(2): 83-96 (1986).
A writeup of the proof that "expansion implies spectral gap" from Alon's paper can be found in pages 14-16 of the following thesis:
David Xiao, "The Evolution of Expander Graphs", AB Thesis, Harvard University, 2003.
3. [BL] Yonatan Bilu, and Nati Linial, "Lifts, discrepancy and nearly optimal spectral gaps", To appear in Combinatorica, (A preliminary version appeared in FOCS 2004).
4. [Din] Irit Dinur, "The PCP Theorem by Gap Amplification", ECCC Technical Report TR05-046, 2005.
5. [Fri] Joel Friedman, "A proof of Alon's second eigenvalue conjecture and related problems", CoRR cs.DM/0405020: (2004) (A preliminary version appeared in STOC 2003).
6. [Gur2] Venkatesan Guruswami. Error-correcting codes and Expander graphs SIGACT News, 35(3): 25-41, September 2004.
7. [GI] Venkatesan Guruswami, Piotr Indyk: "Linear time encodable/decodable codes with near-optimal rate", To appear in IEEE Transactions on Information Theory. (Preliminary Version in STOC'02).
8. [INW] Russell Impagliazzo, Noam Nisan, Avi Wigderson: "Pseudorandomness for network algorithms", STOC 1994: 356-364
9. [IZ] Russell Impagliazzo, David Zuckerman: "How to Recycle Random Bits", FOCS 1989: 248-253.
10. [Lov] Lazlo Lovasz: Random Walks on Graphs: A Survey'', Combinatorics, Paul Erdos is Eighty, Vol. 2, Janos Bolyai Mathematical Society, Budapest, 1996, 353--398
11. [Lin] Nati Linial, "Expanders, eigenvalues and all that", Invited Talk, NIPS, Dec 2004.
(A nice introduction to expanders).
12. [Nis] Noam Nisan: "Pseudorandom generators for space-bounded computation", Combinatorica 12(4): 449-461 (1992).
13. [Rei] Omer Reingold: "Undirected ST-Connectivity in Logspace", ECCC Tech Report TR04-094, 2004.
14. [RVW] Omer Reingold, Salil Vadhan, and Avi Wigderson: "Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors", Annals of Math 02.
15. [Rob] Sara Robinson, "Computer Scientist Finds Small-Memory Algorithm for Fundamental Graph Problem", SIAM News, Volume 38, Number 1, January/February 2005.
(A news report on Reingold's result, also covers some history on expanders)
16. [SW] Amir Shpilka, Avi Wigderson: "Derandomizing homomorphism testing in general groups". STOC 2004: 427-435
17. [SS] Michael Sipser, Daniel A. Spielman: Expander codes. IEEE Transactions on Information Theory 42(6): 1710-1722 (1996)
18. [Zem] Gilles Zemor: "On Expander Codes", IEEE Transactions on Information Theory 47(2):835-837 (2001).

#### Requirements

Students taking the course for credit will be expected to:

• Attend lectures
• Scribe notes for a lecture or two (rare) in LATEX.
• Present to Cynthia and/or Prahladh (not necessarily in class) the proofs of some of the theorems presented without proof in lecture.

## Search Results

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10. ### The Pythagorean Theorem - Math Forum - Ask Dr. Math

2 posts - Last post: 19 Sep 1997
Date: 07/07/97 at 13:57:09 From: Noelle Subject: Geometry I'm having problems with the Pythagorean Theorem! My teacher has explained it but ...
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11. ### Math Forum: Ask Dr. Math FAQ: Pythagorean Triples

What do Pythagorean triples have to do with Fermat's Last Theorem? ... Formulas for Primitive Pythagorean Triples and Their Derivation ...
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13. ### Pythagorean Theorem Derivation/Proof? - Yahoo! Answers

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15. [PDF]

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How are the Pythagorean theorem and the distance formula related? ..... Derivation of Distance. Formula. NS 2.5. AF 4.1. MG 3.2. 2.0, 4.0, 5.0 ...
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18. ### How to derive pythagorean identity?

6 posts - 3 authors
How to derive pythagorean identity? Precalculus Mathematics discussion. ... Pythagorean Theorem, Lou1024, Precalculus Mathematics, 6, Mar3-06 12:21 AM ...
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19. ### Geometric theorems [Archive] - Physics Forums

12 posts - 6 authors
Can someone tell me (or help me find) the derivation of the pythagorean theorem, and the laws of sin,cos, and tangent. I know the first is a derivation of ...
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20. ### Heron's Formula For Tetrahedrons

The article Heron's Formula For Triangle Area gives a very direct derivation of Heron's formula based on Pythagoras's Theorem for right triangles. ...
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24. ### Reading of Pythagorean Triangle to Derive Destiny - Video  2 min 5 sec - 16 Dec 2009
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### A Geometrical Derivation of the Three Trigonometric Identities ...

File Format: Microsoft Powerpoint
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27. ### Understanding the Pythagorean theorem - Mathematics - Helium

This article expands the previously written article, Understanding the Pythagorean theorem. It discusses the derivation the law of cosines, ...
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28. ### A geometrical derivation of the excess mean square error for ...

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The next section describes the geometrical interpretation and applies the Pythagoras theorem to the right triangles. Then the derivation of the EMSE in both ...
29. ### Law Of Cosines Sides Theorem Pythagorean Angle Triangle Line

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31. ### The computation of traveltimes from a velocity function The ...

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equation, whose derivation goes back to the work of. Pythagoras about 2500 years ago. ... Pythagorean theorem in geometry—one of the most famous ...
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32. ### The eikonal equation and the secret Pythagorean theorem ...

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... whose derivation goes back to the work of Pythagoras about 2500 years ago. ... The famous Pythagorean theorem has a close relative, here termed the ...
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33. ### WikiAnswers - History of pythagoras theorem

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36. ### THE PYTHAGOREAN THEOREM

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37. [PDF]

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39. [PDF]

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42. ### curvature of space and lorentz transformation - SciForums.com

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44. [PDF]

### Pythagorean theorems in the alpha plane 1. Introduction

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by HB Colakoglu - 2009
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45. ### Deflationism and the Derivation Game

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The following derivation helps make this point. 1. The Pythagorean theorem is true, if and only if 2+2 = 4. 2. 2+2 = 4. 3. The Pythagorean theorem is true. ...
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### The Law of Cosines

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53. ### A proof of the Pythagorean theorem by Liu Hui

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55. ### The Pythagorean Theory | Scienceray

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56. ### Pythagoras' theorem « What's new

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57. [PDF]

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62. ### Trigonometry--The derivation of Sin 18°

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63. [PDF]

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The Pythagorean theorem is a celebrity: if an equation can make it into the ... I would like to see a derivation of mccoyn's result of c = a * sqrt(1 + b^2 ...
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65. ### The Pythagorean Theorem and Physics

The Pythagorean Theorem and Physics. Donald G. Burkhard .... Alternate Derivation of the Angular Momentum Equation for Interacting Particles ...
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66. ### Pythagorean Triples: A New, Easy-to-Derive Formula With Some ...

EJ093537 - Pythagorean Triples: A New, Easy-to-Derive Formula With Some ... Identifiers: Geometric Constructions; Proof (Mathematics); Pythagorean Theorem ...
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67. [DOC]

### Cosine Law

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Derive the cosine law using the primary trigonometric ratios and the Pythagorean Theorem to set up one of the equations for the cosine law. ...
www.mr-marshall.ca/slis/Handouts/TICTACTOE-CosineLaw.doc -
68. ### Mathematical papers by Andrew Jobbings

A derivation of the quadratic formula from the relationships between the roots and ... A generalisation of Pythagoras' theorem to squares of areas in three ...
www.arbelos.co.uk/papers.html -
69. ### Teaching the Distance Formula

19 Feb 1997 ... use the Pythagorean theorem to find the length of each hypotenuse. ... then use the Pythagorean Theorem to derive the Distance Formula: ...
www.ed.uiuc.edu/pie21/best-math-sp96/zehr.html -
70. ### R. M. R. Proof of Pythagerean Theorm

Willis Yocom and I collaborated on the following "derivation" which was summarily .... After the project of finding a proof of the Pythagorean Theorem was ...
www.smecc.org/r__m__r__proof_of_pythagerean_theorm.htm -
71. [PDF]

### Week Two, Day Nine UNIT: RIGHT TRIANGLES & THE PYTHAGOREAN THEOREM ...

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The students will be able to derive the distance formula using the. Pythagorean Theorem. MATERIALS NEEDED: ▪ Review sheet for exam. ▪ Lecture notes ...
www.dustinjpurdy.com/ePortfolio/pdf/instructional_unit/lesson.../Day9.pdf -
72. ### PlanetMath: Pythagorean triplet

See Also: Pythagorean theorem, incircle radius determined by Pythagorean triple, ... rational sine and cosine (Theorem) by pahio derivation of Pythagorean ...
planetmath.org/encyclopedia/PythagoreanTriple.html -
73. ### Pythagorean theorem: Definition from Answers.com

(Click to enlarge) Pythagorean theorem The Pythagorean theorem is: (Academy ... that the theorem is due to relative changes and its derivation is nearly ...
74. ### A DERIVATION OF THE PYTHAGOREAN WON-LOSS FORMULA IN BASEBALL 1 ...

by SJ MILLER - Cited by 6 - Related articles - All 6 versions
The goal of this paper is to derive Bill James' Pythagorean Formula (see [Ja], .... this to prove our main result, the Pythagorean Formula (Theorem 2.2). ...
www.williams.edu/go/math/sjmiller/public.../PythagWonLoss_Paper.pdf -
75. ### ClassOf1_geometry_derivation_distance_9

The role of the Pythagorean theorem in the distance formula is of utmost importance. As shown in the derivation, the Pythagorean theorem is the basis of the ...
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76. ### theorem "recursion theorem freely generated - half angle theorem"

transportation pythagorean theorem pythagorean theorem derivation fundamental theorem of linear programming central limit theorem erf ...
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77. ### Pythagorean theorem | Flash simulation, Animation, Illustration ...

Derivation and integration ... Summary. Here you have one of the numerous geometric demonstrations of the Pythagorean Theorem. eduMedia ...
www.edumedia-sciences.com/en/a368-pythagorean-theorem -
78. ### Quadratic Polynomials: The Quadratic Formula

The derivation of the quadratic formula. ... This formula is as important and widely used as the Pythagorean Theorem. Teachers do not have mercy on students ...
www.sosmath.com/algebra/factor/fac08/fac08.html -
79. ### » The Scientific and Medical Network

18 Nov 2009 ... Then, from Pythagoras' Theorem: (moc)2 + (mv)2 = (mc)2 ... The above derivation was published by me in 1968 and again in 2007  along with ...
www.scimednet.org/relativity-equations-derived-in-one-line-and-without-assuming-the-two-relativity-postulates/ -
80. [DOC]

### BIG IDEA #2:

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Simplify a radical expression; Explain the derivation of the Pythagorean Theorem; Define and identify hypotenuse and leg in a right triangle ...
81. ### Math Lesson: Geometry, Distance Formula

So mathematicians have developed a formula using the Pythagorean theorem to ... In order to understand the derivation, you need to change the points into ...
www.hsunlimited.com/math-lessons/topic/Distance-Formula -
82. ### A heuristic derivation of Minkowski distance and Lorentz ...

by S Hassani - 2008 - Related articles
We present the derivation in such a way that the transition to spacetime becomes ..... E1 to E2—to be C. Then the Pythagorean theorem for the right triangle ...
www.iop.org/EJ/article/0143-0807/29/1/010/ejp8_1_010.pdf -
83. ### The Sine and Cosine Functions - Properties

From the above cosine equation, we can derive that. cos(2x) = cos2(x) - sin2(x) ... It can be thought of as a generalized form of the pythagorean theorem. ...
math2.org/math/algebra/functions/sincos/properties.htm -
84. ### Theorem - e-Math Tutor, A Disproof of Pythagoras' Theorem?, Math ...

An interesting method of disproving the Theorem of Pythagoras (intended as a ... Rigorous derivation of sound wave equations from a molecular model of an ...
www.sciencecentral.com/product/32231 -
85. ### Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation ..... that the theorem is due to relative changes and its derivation is nearly ...
www.cs.mcgill.ca/~rwest/wikispeedia/.../Pythagorean_theorem.htm -
86. ### Ask NRICH: pythagorus continued...

10 posts - 7 authors - Last post: 3 Nov 2005
Are you trying to derive the sin and cosine rules from Pythagoras' theorem? If so, start with a diagram of a triangle, not a right angled ...
nrich.maths.org › Ask NRICHArchive 2005-2006Please Explain -
87. [PDF]

### Flight: Testing Newton's Laws

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M.O.G.3.12 apply the Pythagorean Theorem and its converse to solve real-world problems and derive the special right triangle relationships (i.e. 30-60-90, ...
www.hq.nasa.gov/office/aero/edu_standards/wv/nl/math/math.pdf -
88. [PDF]

### Chinese Checker Versions of the Pythagorean Theorem 1 Introduction

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by HB Colakoglu - 2009 - Related articles
of the Pythagorean Theorem are not true. Finally, we give a necessary ..... Remark 2.1 We note that when AB is parallel to the x-axis, our derivation of ...
www.m-hikari.com/ijcms.../1-4.../colakogluIJCMS1-4-2009.pdf -
89. [PDF]

### 'What Time is It?': An Algebra Based Derivation of Time Dilation

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Thank you for showing me the beauty of Relativity with this derivation! ... Apply the Pythagorean theorem in both abstract and real-life settings (G-2-H) ...
https://dcc.ligo.org/public/0000/T080159/001/T080159-v1-G.pdf -
90. [PDF]

### A Note on the Pythagorean Theorem of Baseball Production

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by J Ruggiero - 1997 - Cited by 8 - Related articles - All 4 versions
conclude that application of the Pythagorean Theorem' for manager evaluation is ... and he uses the theorem to derive a measure of managerial performance. ...
www.a1wrestling.com/Pythagoras.pdf -
91. ### [theorem] pythagoras theorem calculator, pythagorean theorem hands ...

derivation of apollonius theorem first fundamental theorem of calculus examples. impulse-momentum theorem. discuss ken french theorem pythagorean theorem ...
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92. ### Reference for Pythagorean theorem - Search.com

The Pythagorean theorem: The sum of the areas of the two squares on the legs .... the elements found in the Shulba Sutras may be of Mesopotamian derivation. ...
www.search.com/reference/Pythagorean_theorem -
93. ### Pythagoras%27 Theorem

... that the theorem is due to relative changes and its derivation is nearly ..... The coin's tail has an image of Pythagoras and the Pythagorean theorem, ...
ovis.ui.ac.id/wiki/Pythagoras'_theorem -
94. ### Adherents.com - Religious Groups in Literature

'I figured out how to derive the distance formula. It's the Pythagorean theorem, isn't it?' " Pythagorean, Roman Empire, 284 C.E., Bradley, Marion Zimmer ...
95. [DOC]

### Chapter 4 Euclidean Geometry

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Enrichment 1 page 240 (1) – (5) in the Pythagorean Theorem derivation. See the second Moment for Discovery on page 247 to get started. ...
www.math.uh.edu/.../Chapter%204%20Euclidean%20Geometry.doc -
96. [DOC]

### Geometry.doc - Welcome to MPS Online

File Format: Microsoft Word
Discover the relationship between squares and right triangles to derive the Pythagorean Theorem: a2 + b2 = c2; Use the Pythagorean Theorem to find the ...
97. ### section 2.1: Distance Formula

In what follows, we will derive a formula for the distance between points on the Cartesian plane using the Pythagorean theorem. Example 6 ...
www.maa.org/projectwelcome/ca_preview/txt.../sec21b.htm -
98. [PDF]

### Unit Template for CFB District

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Pythagorean Theorem orally or in writing. LEARNING PLAN. It is assumed that students are ..... (C) derive, extend, and use the Pythagorean Theorem.