I have adjusted my study plans so that I am on a oneTMApermonth schedule as follows:
Feb  TMA01 M373
Mar  TMA01 M381
Apr  TMA02 M373
May  TMA02 M381
Jun  TMA03 M373
Jul  TMA03 M381
Aug  TMA04 M373
Sep  TMA04 M381
Okt  Exam M373, Exam M381
( This represents a 60point workload. )
Sunday, January 30, 2011
Number Theory in Mathematica
Euler's totient function as a Dirichlet Product
$$\varphi(n) = \mu * N$$
or in Mathematica:
$$\varphi(n) = \mu * N$$
or in Mathematica:
In[3]:= DirichletConvolve[MoebiusMu[n],n,n,n] Out[3]= EulerPhi[n]Link: Multiplicative Number Theory functions in Mathematica
Saturday, January 29, 2011
Google Logic
I found an excellent summary of the Google Search Logic and tons of other study tips at the site of McGrawHill.
Link: Power Google
Link: Power Google
17. Be Competent.
171 Look,
172 Learn,
173 Practice.
The Way to Happiness, LRH
Friday, January 28, 2011
How to learn from failure
January 2011 is almost history. Last year around this time I was quite excited about M208 and MT365. Not knowing that MT365 would end in failure and that ( the tutor of ) M208 had some horror in stock for me that made me almost end my Open University studies. Failure. There is no way to undo or repair failure. It has been said that one 'learns from failure'. Failure is something different than losing a game. If you lose a game you are simply beaten by a stronger opponent. When you fail you were beaten by yourself. Is failure then a sign of a weak character? Could be, but not necessarily so. It could simply be a matter of not having the right selfmanagement and planning tools. Focus, dedication, ambition are cool words to say. "I am dedicated" to finish my mathematics study. But what do these words mean in your daytoday life? How can you implement 'dedication'? How can you maintain 'focus'?  Last year I sort of drifted away from my goals set early in the year to the point in August where I basically had it with mathematics at the Open University. Such a thing does not happen overnight.
(1)
Implement statistics by monitoring your results. Can you see in an instant what you have learned this month? The previous month? How many hours have you studied. You know what statistic is best for you.
Periodically ( weekly, even better daily ) chart your statistics. Do you see trends? If so take corrective actions. With the purpose to stay focused and keep the results coming.
(2)
In future posts.
P.S.
Almost forgot why I started this post. Selfstudy is much, much harder than doing a programmed course. This year I hope to study "Analytic Number Theory" by Apostol. Of the 14 chapters I have done the first chapter. The odd thing is that chapter 1 covers about the same as 8 modules ( the entire course ) of number theory in M381. The book is hard and dense. It covers two modules from the Master program as well. Completion of this selfstudy project is how I measure my study performance in 2011, not my grades on M381 or M373.
(1)
Implement statistics by monitoring your results. Can you see in an instant what you have learned this month? The previous month? How many hours have you studied. You know what statistic is best for you.
Periodically ( weekly, even better daily ) chart your statistics. Do you see trends? If so take corrective actions. With the purpose to stay focused and keep the results coming.
(2)
In future posts.
P.S.
Almost forgot why I started this post. Selfstudy is much, much harder than doing a programmed course. This year I hope to study "Analytic Number Theory" by Apostol. Of the 14 chapters I have done the first chapter. The odd thing is that chapter 1 covers about the same as 8 modules ( the entire course ) of number theory in M381. The book is hard and dense. It covers two modules from the Master program as well. Completion of this selfstudy project is how I measure my study performance in 2011, not my grades on M381 or M373.
[News]  Formula for partition number discovered
As I blogged about earlier Ono and colleagues have developed a formula that spits out the partition number of any integer. This news thrills me. It motivates me to get faster to that edge of the field. Below you'll find a link to the paper by Ono et al. Like me, you may not be ready yet to fully understand such a paper. I will use it as a benchmark to measure my skills by and as a guide for selfstudy. Come to think of it I need to add more papers to the benchmark. One about the Riemann hypothesis and one about Fermat's Last Theorem.
Links:
 Article in NewScientist
 Paper "lAdic properties of the partition function." ( pdf )
Links:
 Article in NewScientist
 Paper "lAdic properties of the partition function." ( pdf )
Thursday, January 27, 2011
Fascination of Pi
The string 123456789 did not occur in the first 200000000 digits of pi after position 0.
Search for strings in the first 200000000 digits of pi.
Source: The PiSearch Page
Wednesday, January 26, 2011
M381ML#3
M381Logic
Worked on the URMemulator today. I have added function composition.
Example:
The following URM code represents the function f: n>n^2
{
{j, 1, 4, 10},
{c, 1, 4},
{s, 2},
{j, 1, 2, 10},
{z, 3},
{s, 3},
{s, 4},
{j, 1, 3, 3},
{j, 1, 1, 6},
{c, 4, 1}
}
The following URM code represents the function g: n>3n
{
{c, 1, 3},
{j, 2, 3, 10},
{s, 2},
{s, 1},
{s, 1},
{j, 1, 1, 2}
}
The URM then generates ( i.e. my Mathematica code ) for (f*g): n>9n^2
{
{c, 1, 3},
{j, 2, 3, 7},
{s, 2},
{s, 1},
{s, 1},
{j, 1, 1, 2},
{z,2},
{z, 3},
{z, 4},
{j, 1, 4, 19},
{c, 1, 4},
{s, 2},
{j, 1, 2, 19},
{z, 3},
{s, 3},
{s, 4},
{j, 1, 3, 12},
{j, 1, 1, 15},
{c, 4, 1}
}
and for (g*f) n:=n>3n^2
{
{j, 1, 4, 10},
{c, 1, 4},
{s, 2},
{j, 1, 2, 10},
{z, 3},
{s, 3},
{s, 4},
{j, 1, 3, 3},
{j, 1, 1, 6},
{c, 4, 1},
{z, 2},
{z, 3},
{z, 4},
{c, 1, 3},
{j, 2, 3, 23},
{s, 2},
{s, 1},
{s, 1},
{j, 1, 1, 15}
}
Low level hacking indeed.
Worked on the URMemulator today. I have added function composition.
Example:
The following URM code represents the function f: n>n^2
{
{j, 1, 4, 10},
{c, 1, 4},
{s, 2},
{j, 1, 2, 10},
{z, 3},
{s, 3},
{s, 4},
{j, 1, 3, 3},
{j, 1, 1, 6},
{c, 4, 1}
}
The following URM code represents the function g: n>3n
{
{c, 1, 3},
{j, 2, 3, 10},
{s, 2},
{s, 1},
{s, 1},
{j, 1, 1, 2}
}
The URM then generates ( i.e. my Mathematica code ) for (f*g): n>9n^2
{
{c, 1, 3},
{j, 2, 3, 7},
{s, 2},
{s, 1},
{s, 1},
{j, 1, 1, 2},
{z,2},
{z, 3},
{z, 4},
{j, 1, 4, 19},
{c, 1, 4},
{s, 2},
{j, 1, 2, 19},
{z, 3},
{s, 3},
{s, 4},
{j, 1, 3, 12},
{j, 1, 1, 15},
{c, 4, 1}
}
and for (g*f) n:=n>3n^2
{
{j, 1, 4, 10},
{c, 1, 4},
{s, 2},
{j, 1, 2, 10},
{z, 3},
{s, 3},
{s, 4},
{j, 1, 3, 3},
{j, 1, 1, 6},
{c, 4, 1},
{z, 2},
{z, 3},
{z, 4},
{c, 1, 3},
{j, 2, 3, 23},
{s, 2},
{s, 1},
{s, 1},
{j, 1, 1, 15}
}
Low level hacking indeed.
[Sign of the times]  Breakthrough in partition theory
Partitions are fractal.Read this and remember the name Ken Ono.
Tuesday, January 25, 2011
Tons of mathematics tutorials on video
Literally tons of mathematics tutorials on video of about 5 minutes each on various topics like trigonometry, calculus, linear algebra, discrete mathematics and differential equations. Interesting for MST121, MS221, M208 and MST209.
Link: Patrick: Just Math Tutorials  ( Thank you, Patrick! )
Link: Patrick: Just Math Tutorials  ( Thank you, Patrick! )
Think about learning math in the same way you would learn to play piano or learn another language: it takes time, patience, and LOTS of practice.  Patrick
Monday, January 24, 2011
How to remember 1000 digits of Pi
"How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!"  Can you remember this sentence?
Remembering a sentence of fifteen words is, for most of us, easier then a string of fifteen digits. Remembering a string of 1000 digits is impossible, unless you belong to the class of savants of prodigies. Remembering a text of 1000 words is possible though, not easy, possible.
There are however some limitations, you can only use words of 1 to 9 letters. The number of letters of a word represent the digit.
"How(3) I(1) want(4) a(1) drink(5), alcoholic(9) of(2) course(6), after(5) the(3) heavy(5) lectures(8) involving(9) quantum(7) mechanics(9)!"
3.14159265358979
Any suggestions for a better strategy to remember 1000 digits of Pi?
Remembering a sentence of fifteen words is, for most of us, easier then a string of fifteen digits. Remembering a string of 1000 digits is impossible, unless you belong to the class of savants of prodigies. Remembering a text of 1000 words is possible though, not easy, possible.
There are however some limitations, you can only use words of 1 to 9 letters. The number of letters of a word represent the digit.
"How(3) I(1) want(4) a(1) drink(5), alcoholic(9) of(2) course(6), after(5) the(3) heavy(5) lectures(8) involving(9) quantum(7) mechanics(9)!"
3.14159265358979
Any suggestions for a better strategy to remember 1000 digits of Pi?
M373 site opens
Confirmation of the first cutoff date: it is indeed as soon as 15 Feb for TMA01. I better start on it asap, as I have to go through quite some revisions and new stuff as well.
Looked briefly through course book of Block 1/Unit 2: ( again ) about solving systems of linear equations. But this time enough tools are supplied to solve systems of zillions equations ( if necessary ). Lots of linear algebra and matrix stuff. Looks cool to me.
No details yet on tutors and tutorials. As far as I am concerned Edinburgh is fine.
More about M373 as I go through this course.
Looked briefly through course book of Block 1/Unit 2: ( again ) about solving systems of linear equations. But this time enough tools are supplied to solve systems of zillions equations ( if necessary ). Lots of linear algebra and matrix stuff. Looks cool to me.
No details yet on tutors and tutorials. As far as I am concerned Edinburgh is fine.
More about M373 as I go through this course.
Sunday, January 23, 2011
Proof by contradiction ( M381NT#2 )
M381 Unit 1 Foundations is about
 number patterns,
 proof by mathematical induction,
 divisibility and the division algorithm
 GCD and LCM ( greatest common divisor and least common multiple )
 the Euclidean algorithm
 solving linear Diophantine equations
Unit 1 also contains the proof of the method of mathematical induction. 'The proof of proof by induction'. This post is part 1 of a forthcoming series with an indepth explanation of this proof. We begin with the concept of proof by contradiction.
We assume that $\sqrt{2}$ is rational ( not irrational ) and can thus write it as follows: $$\sqrt{2} = \frac{p}{q}$$ where $(p,q)=1$ ( have no common divisors, are relatively prime ).
Then:
$\sqrt{2} = \frac{p}{q} $
$2 = \frac{p^2}{q^2} $
$p^2 = 2q^2 $
So $p^2$ is even. Since the square of an odd number is always odd and the square of an even number is always even, we know that $p$ must be even and can thus be factorized to $2r$.
Then:
$\sqrt{2} = \frac{2r}{q} $
$2 = \frac{4r^2}{q^2} $
$q^2 = 2r^2 $
So q can be factorized further as well to $2s$.
Then:
$(p,q) = (2r,2s) = 2(r,s) > 1$.
This is clearly a contradiction and thus proves that $\sqrt{2}$ must be irrational.
 number patterns,
 proof by mathematical induction,
 divisibility and the division algorithm
 GCD and LCM ( greatest common divisor and least common multiple )
 the Euclidean algorithm
 solving linear Diophantine equations
Unit 1 also contains the proof of the method of mathematical induction. 'The proof of proof by induction'. This post is part 1 of a forthcoming series with an indepth explanation of this proof. We begin with the concept of proof by contradiction.
Proof by contradiction
If finding a direct proof fails we can try proving by contradiction. If we have to prove a proposition P we then assume ~P and show that this assumption implies a contradiction and thus ~P is false or P is true.Example
Show that $\sqrt{2}$ is irrational ( can not be expressed as a fraction ).We assume that $\sqrt{2}$ is rational ( not irrational ) and can thus write it as follows: $$\sqrt{2} = \frac{p}{q}$$ where $(p,q)=1$ ( have no common divisors, are relatively prime ).
Then:
$\sqrt{2} = \frac{p}{q} $
$2 = \frac{p^2}{q^2} $
$p^2 = 2q^2 $
So $p^2$ is even. Since the square of an odd number is always odd and the square of an even number is always even, we know that $p$ must be even and can thus be factorized to $2r$.
Then:
$\sqrt{2} = \frac{2r}{q} $
$2 = \frac{4r^2}{q^2} $
$q^2 = 2r^2 $
So q can be factorized further as well to $2s$.
Then:
$(p,q) = (2r,2s) = 2(r,s) > 1$.
This is clearly a contradiction and thus proves that $\sqrt{2}$ must be irrational.
Saturday, January 22, 2011
M373 TMA01 is due soon
M373 of which the website opens on Monday published the TMA cutoff schedule. The course officially starts February 5th and has a cutoff date for TMA01 on 15 feb. I would think that this was a typo but since the first TMA only counts for 10 points it may very well be possible. I suppose this means that I have to start working on this TMA immediately.
Relation between Phi and Pi
$$\phi = \frac{1+\sqrt{5}}{2}= 2\cos {\frac{ \pi}{5} }$$
$$\pi = 5 \arccos \frac{\phi}{2}$$
( Who can improve on Euler's identity by adding $\phi$ to it in an elegant fashion? )
I watched the BBC Horizon documentary "What is Reality?" The constants in physics seem nothing more than carpets to stash away the dust, i.e. stuff we don't understand yet. It looks as though there are no beautiful equations in physics: physicists make them look beautiful by creating all sorts of constants.  Forgive my ignorance, my knowledge of physics is limited. But when I heard the lead scientist of Fermilab explaining that they don't know what mass is I was flabbergasted. They "need to find the Higgsboson particle" first. Then he talked about the pure ecstasy and euphoria he experienced when they found the last quark. They are completely obsessed by a particle that may not exist, they look and live like heroinaddicts, caring about one thing only: Higgsboson.  ( Forgive me, I am jealous! )
Back to mathematics. What are the fundamental constants in mathematics? I am not sure. I suppose Euler's Identity is an excellent start with 1, 0, i, $e$ and $\pi$. Given a URM, then $e$ and $\pi$ become 'computable' to any decimal precision. So in that sense one might argue that $e$ and $\pi$ are not fundamental constants. 0 and 1 are, of course. Because they are part of the definition of a URM, think of the zero and successor instructions. But what about geometry? In geometry $\pi$ is a constant: the ratio of a circle's circumference to its diameter.
$$\pi = 5 \arccos \frac{\phi}{2}$$
( Who can improve on Euler's identity by adding $\phi$ to it in an elegant fashion? )
I watched the BBC Horizon documentary "What is Reality?" The constants in physics seem nothing more than carpets to stash away the dust, i.e. stuff we don't understand yet. It looks as though there are no beautiful equations in physics: physicists make them look beautiful by creating all sorts of constants.  Forgive my ignorance, my knowledge of physics is limited. But when I heard the lead scientist of Fermilab explaining that they don't know what mass is I was flabbergasted. They "need to find the Higgsboson particle" first. Then he talked about the pure ecstasy and euphoria he experienced when they found the last quark. They are completely obsessed by a particle that may not exist, they look and live like heroinaddicts, caring about one thing only: Higgsboson.  ( Forgive me, I am jealous! )
Back to mathematics. What are the fundamental constants in mathematics? I am not sure. I suppose Euler's Identity is an excellent start with 1, 0, i, $e$ and $\pi$. Given a URM, then $e$ and $\pi$ become 'computable' to any decimal precision. So in that sense one might argue that $e$ and $\pi$ are not fundamental constants. 0 and 1 are, of course. Because they are part of the definition of a URM, think of the zero and successor instructions. But what about geometry? In geometry $\pi$ is a constant: the ratio of a circle's circumference to its diameter.
Friday, January 21, 2011
An excellent textbook about Mathematica
If you want to learn Mathematica thorough and fast then this book will help. This book will not turn you into a Mathematica Guru but it will set you on the track of becoming one, some day.
Mathematica  A problemcentered approach
by Roozbeh Hazrat
Springer 2010
Mathematica  A problemcentered approach
by Roozbeh Hazrat
Springer 2010
Thursday, January 20, 2011
GeomagicSquares
Magic squares + Geometry = GeomagicSquares, a concept created by Lee Sallows. Fascinating stuff.
I watched the BBC Horizon documentary "What is Reality?". Lots of interesting food for thought. About the race that is going on between CERN and Fermilab for finding the HiggsBoson particle for example. A fundamental particle that supposedly explains the creation of mass. Therefore alone the particle is badly needed as it's a bit silly that almost every physics formula has mass in it but mass itself can't be explained.
The laws of physics can be described best with mathematics. Deep physics however can only be described with mathematics which leads to the thought that mathematics is not an invention but a discovery. In that sense research mathematicians are much like archaeologists.
I watched the BBC Horizon documentary "What is Reality?". Lots of interesting food for thought. About the race that is going on between CERN and Fermilab for finding the HiggsBoson particle for example. A fundamental particle that supposedly explains the creation of mass. Therefore alone the particle is badly needed as it's a bit silly that almost every physics formula has mass in it but mass itself can't be explained.
The laws of physics can be described best with mathematics. Deep physics however can only be described with mathematics which leads to the thought that mathematics is not an invention but a discovery. In that sense research mathematicians are much like archaeologists.
Result POLL on Library Service usage
This blog receives between 60 and 80 unique visitors per day, viewing up to 200 posts. Over a period of one week five visitors participated in the poll. Only 2 Open University students use the Library Services at least once a week. The other three don't use the Library Services very often. The Open University is organizing free online courses in using the Library so I think they came to a similar conclusion. Everyone can startup a browser and 'search' Internet. In order to differentiate yourself from the masses you need good ( or excellent ) search skills. There is a huge difference between passive surfing and active searching.
Based on the response of this poll I looked more closely at the visitor statistics and what have been, and still are, popular posts. My estimate is that less than 5% of the visitors are Open University students. The purpose of this blog remains creating a logbook of my mathematics study and not trying to get more visitors. Therefore I won't change much on this blog except maybe explaining more Open University jargon.
Based on the response of this poll I looked more closely at the visitor statistics and what have been, and still are, popular posts. My estimate is that less than 5% of the visitors are Open University students. The purpose of this blog remains creating a logbook of my mathematics study and not trying to get more visitors. Therefore I won't change much on this blog except maybe explaining more Open University jargon.
Tuesday, January 18, 2011
Tutor assigned for M381
My M381 tutor has sung leading tenor roles in operas ranging from Poulenc to Puccini and is now in great demand as a professional director. :) I have seen him in several Open University videos. He is from Edinburgh.
The Open University assigns The Netherlands to the Open University in Scotland / Edinburgh area while we live much closer to London. Closer even than many in the UK. "It's an organizational issue.", so they say at the OU. Well, I intend to go to at least some tutorials this year, I hope I am allowed to go somewhere in the South of England. For us, Scotland is more like a holiday destination anyway.
View Larger Map
The Open University assigns The Netherlands to the Open University in Scotland / Edinburgh area while we live much closer to London. Closer even than many in the UK. "It's an organizational issue.", so they say at the OU. Well, I intend to go to at least some tutorials this year, I hope I am allowed to go somewhere in the South of England. For us, Scotland is more like a holiday destination anyway.
View Larger Map
The Art of Proof
Seems like an excellent companion to either M208 or M381 to me. Check out the site of Springer if you are interested in books like this.
Reading protocols and satisfaction surveys
Reading mathematics for leisure. Is that possible? Of course. If you follow protocol. Compare holiday literature. Thriller novels are popular with those enjoying their holidays. Others however prefer books full of Sudoku puzzles or other games. Both are reading a book, but follow a different reading protocol. This article explains the reading protocol for books about mathematics. So a lot depends on the 'readingprotocol'.
The Open University (mathematics) course books have a protocol of their own. Understanding this, and applying it well to your preferred study method, is a success factor for passing a course. I expect many different study methods though. Until now I have always added supplementary textbooks. It gives a different perspective on the topic.
And by comparing the two I have learned to appreciate the quality of the Open University course books. An Open University course is more than a set of course books and an exam though. I really like the overall course plan, on paper, and the different views of the plan on the Student Home Page. You are assigned a tutor, which you can reach via email or by phone, there are the student forums and there are various services on the website.
I read that 97% of the Open University students are satisfied with their course(s). I believe that ( although 3% counts for a lot of courses / students ).
The Open University (mathematics) course books have a protocol of their own. Understanding this, and applying it well to your preferred study method, is a success factor for passing a course. I expect many different study methods though. Until now I have always added supplementary textbooks. It gives a different perspective on the topic.
And by comparing the two I have learned to appreciate the quality of the Open University course books. An Open University course is more than a set of course books and an exam though. I really like the overall course plan, on paper, and the different views of the plan on the Student Home Page. You are assigned a tutor, which you can reach via email or by phone, there are the student forums and there are various services on the website.
I read that 97% of the Open University students are satisfied with their course(s). I believe that ( although 3% counts for a lot of courses / students ).
Monday, January 17, 2011
URM Emulator  Continued
I completed the URM Emulator in Mathematica. Amazingly little code is required to emulate an URM, and thus a real computer. Example:
Input
{3,4,3,0,0}
Program
{{j,1,3,12},{j,2,3,11},{c,1,3},{s,5},{j,2,5,11},{z,4},{s,3},{s,4},{j,1,4,4},{j,1,1,7},{c,3,1}}
( performs multiplication of r1 and r2 )
Memorytrace
{3,4,3,0,0}
{3,4,3,0,1}
{3,4,3,0,1}
{3,4,4,0,1}
{3,4,4,1,1}
{3,4,5,1,1}
{3,4,5,2,1}
{3,4,6,2,1}
{3,4,6,3,1}
{3,4,6,3,2}
{3,4,6,0,2}
{3,4,7,0,2}
{3,4,7,1,2}
{3,4,8,1,2}
{3,4,8,2,2}
{3,4,9,2,2}
{3,4,9,3,2}
{3,4,9,3,3}
{3,4,9,0,3}
{3,4,10,0,3}
{3,4,10,1,3}
{3,4,11,1,3}
{3,4,11,2,3}
{3,4,12,2,3}
{3,4,12,3,3}
{3,4,12,3,4}
{12,4,12,3,4}
About time to look at the differences between a URM and a pure Turing Machine.
P.S.
Since 4 out of 9 questions of M381 TMA01 are about URMs I don't list the Mathematica code here, but that is logical anyway.
Input
{3,4,3,0,0}
Program
{{j,1,3,12},{j,2,3,11},{c,1,3},{s,5},{j,2,5,11},{z,4},{s,3},{s,4},{j,1,4,4},{j,1,1,7},{c,3,1}}
( performs multiplication of r1 and r2 )
Memorytrace
{3,4,3,0,0}
{3,4,3,0,1}
{3,4,3,0,1}
{3,4,4,0,1}
{3,4,4,1,1}
{3,4,5,1,1}
{3,4,5,2,1}
{3,4,6,2,1}
{3,4,6,3,1}
{3,4,6,3,2}
{3,4,6,0,2}
{3,4,7,0,2}
{3,4,7,1,2}
{3,4,8,1,2}
{3,4,8,2,2}
{3,4,9,2,2}
{3,4,9,3,2}
{3,4,9,3,3}
{3,4,9,0,3}
{3,4,10,0,3}
{3,4,10,1,3}
{3,4,11,1,3}
{3,4,11,2,3}
{3,4,12,2,3}
{3,4,12,3,3}
{3,4,12,3,4}
{12,4,12,3,4}
About time to look at the differences between a URM and a pure Turing Machine.
P.S.
Since 4 out of 9 questions of M381 TMA01 are about URMs I don't list the Mathematica code here, but that is logical anyway.
M381 site opens
TMA Cutoff dates are as follows:
 1: 31 Mar ( Thu )
 2: 09 Jun ( Thu )
 3: 04 Aug ( Thu )
 4: 15 Sep ( Thu ).
Carefully reading the wording in each TMA is important. The TMA booklets contain a special glossary. Two examples:
 "prove, show, explain" : Clear reasoning and explanation for all steps are called
for.
 "solve" : Working must be shown. A numerical answer alone is not sufficient.
TMA01 covers
Number Theory
unit 1 ( Foundations ) Q1, Q2, Q3
unit 2 ( Primes ) Q4, Q5
and Mathematical Logic
unit 1 ( URMs ) Q6, Q7, Q8 and Q9.
I think it's time for a replanning already. Go into sequential Number Theory > Logic  Number Theory mode. Not entirely as advised by the OU, have to think about it.
 1: 31 Mar ( Thu )
 2: 09 Jun ( Thu )
 3: 04 Aug ( Thu )
 4: 15 Sep ( Thu ).
Carefully reading the wording in each TMA is important. The TMA booklets contain a special glossary. Two examples:
 "prove, show, explain" : Clear reasoning and explanation for all steps are called
for.
 "solve" : Working must be shown. A numerical answer alone is not sufficient.
TMA01 covers
Number Theory
unit 1 ( Foundations ) Q1, Q2, Q3
unit 2 ( Primes ) Q4, Q5
and Mathematical Logic
unit 1 ( URMs ) Q6, Q7, Q8 and Q9.
I think it's time for a replanning already. Go into sequential Number Theory > Logic  Number Theory mode. Not entirely as advised by the OU, have to think about it.
Sunday, January 16, 2011
URM Emulator
I wanted to code the multiplication algorithm as described in URM code in Mathematica. I ended the day on my way to building a URM emulator in Mathematica. An example of not strictly necessary study time. In this case related to M381. But a task like this is what makes studying so much fun.
I am programming a Mathematica Function that takes a list like this
{
{{J}, {1,3,12}},
{{J}, {2,3,11}},
{{C}, {1,3}},
{{S}, {5}},
{{J}, {2,5,11}},
{{Z}, {4}},
{{S}, (3}},
{{S}, {4}},
{{J}, {1,4,4}},
{{J}, {1,1,7}},
{{C}, {3,1}},
}
as input and then processes it as of it was the following URM program.
1 J(1,3,12)
2 J(2,3,11)
3 C(1,3)
4 S(5)
5 J(2,5,11)
6 Z(4)
7 S(3)
8 S(4)
9 J(1,4,4)
10 J(1,1,7)
11 C(3,1)
I hope to complete it somewhere this week.
Alan Turing must have been quite a visionary I doubt it however if he ever imagined students, using advanced computers, trying to emulate his Turing Machine or similar URM.
I am programming a Mathematica Function that takes a list like this
{
{{J}, {1,3,12}},
{{J}, {2,3,11}},
{{C}, {1,3}},
{{S}, {5}},
{{J}, {2,5,11}},
{{Z}, {4}},
{{S}, (3}},
{{S}, {4}},
{{J}, {1,4,4}},
{{J}, {1,1,7}},
{{C}, {3,1}},
}
as input and then processes it as of it was the following URM program.
1 J(1,3,12)
2 J(2,3,11)
3 C(1,3)
4 S(5)
5 J(2,5,11)
6 Z(4)
7 S(3)
8 S(4)
9 J(1,4,4)
10 J(1,1,7)
11 C(3,1)
I hope to complete it somewhere this week.
Alan Turing must have been quite a visionary I doubt it however if he ever imagined students, using advanced computers, trying to emulate his Turing Machine or similar URM.
Saturday, January 15, 2011
[M381Logic]  #2: Unit 1
Already comfortable with the notion that Saturday is "M381Logicday". The purpose of the Logic half of M381 is finding answers to the following two questions.
Section 1 defines the URM.
The URM can store an infinite number of positive integers in registers R1 upto Rn and has the following instruction set:
( Name: Notation ; Effect )
Zero: Z(n) ; Replace the number in R(n) by 0.
Successor: S(n) ; Add 1 to the number in R(n).
Copy : C(m,n) ; Replace number in R(n) by number in R(m).
Jump : J(m,n,q) ; If R(m) = R(n) jump to q otherwise next.
From this limited instruction set a powerful programming language can be constructed.
Section 2 investigates which functions can be implemented on a URM.
The following URM program for example
1 J(1,3,12)
2 J(2,3,11)
3 C(1,3)
4 S(5)
5 J(2,5,11)
6 Z(4)
7 S(3)
8 S(4)
9 J(1,4,4)
10 J(1,1,7)
11 C(3,1)
implements multiplication on a URM.
So far. To be continued next week.
P.S.
Trivia:
 Russell's Paradox. A is the set of sets which do not contain A.
 Russell and Whitehead proved that 1+1=2: more on this page.
Is there an algorithm for deciding which statements of number theory are true?  Leibniz’s Question
Can the consistency of number theory be proved using only nondubious principles of finitary reasoning?  Hilbert’s Question
Section 1 defines the URM.
The URM can store an infinite number of positive integers in registers R1 upto Rn and has the following instruction set:
( Name: Notation ; Effect )
Zero: Z(n) ; Replace the number in R(n) by 0.
Successor: S(n) ; Add 1 to the number in R(n).
Copy : C(m,n) ; Replace number in R(n) by number in R(m).
Jump : J(m,n,q) ; If R(m) = R(n) jump to q otherwise next.
From this limited instruction set a powerful programming language can be constructed.
Section 2 investigates which functions can be implemented on a URM.
The following URM program for example
1 J(1,3,12)
2 J(2,3,11)
3 C(1,3)
4 S(5)
5 J(2,5,11)
6 Z(4)
7 S(3)
8 S(4)
9 J(1,4,4)
10 J(1,1,7)
11 C(3,1)
implements multiplication on a URM.
So far. To be continued next week.
P.S.
Trivia:
 Russell's Paradox. A is the set of sets which do not contain A.
 Russell and Whitehead proved that 1+1=2: more on this page.
Black Mathematics
Most sciences have a dark side. Psychiatrists prescribe addictive 'moodbalancing' drugs to improve the 'wellbeing' of their patients, and every day hundreds ( 2,730 every day according to W'pedia ) of people get electroshocked, to cure them from 'mental illness'. Physicists and engineers work on weapons of mass destruction to 'ensure peace'. BAE systems UK, the biggest arms manufacturer in the world, generates over 95% of its revenue from arms sales.
What do mathematicians contribute to the art of black science?  Charles Seife ( science writer and teacher of journalism in New York City) wrote the book Proofiness: the dark arts of mathematical deception.
Links:
 Proofiness  (Review in the New York Times).
(
 Psychiatry: An industry of Death.
 BAE Systems.
)
What do mathematicians contribute to the art of black science?  Charles Seife ( science writer and teacher of journalism in New York City) wrote the book Proofiness: the dark arts of mathematical deception.
Links:
 Proofiness  (Review in the New York Times).
(
 Psychiatry: An industry of Death.
 BAE Systems.
)
Two days and counting
I wonder why they open the M381 site on Monday the 17th and not, for example today Saturday, the 15th. Why would that be? Perhaps the Course Leader receives an automated email on Monday with instructions to manually open the site ( by updating some status field from off/closed/0 to on/open/1, whatever! ) If the site opens on Monday before office hours then I think it is a waste of a good study weekend for many students. As you may have noticed...
I can't wait for the sites to open!
I can't wait for the sites to open!
Friday, January 14, 2011
Despatch status
Since the materials were planned to be despatched today I checked the despatch status for both my courses:
M373:
BlocksUnits/Handbook/CDR  Mailed on 10 January 2011 for use 5 February 2011
M381:
Units/Course Gde/handbook  Mailed on 12 January 2011 for use 5 February 2011
Units/Bookmark/SEP  Scheduled to be mailed 20 May 2011 for use 11 June 2011
Receiving the course packs is one of those moments. Almost as good as receiving a pass letter. I haven't received anything yet though. But I live in another universe.
Continental Europe.
M373:
BlocksUnits/Handbook/CDR  Mailed on 10 January 2011 for use 5 February 2011
M381:
Units/Course Gde/handbook  Mailed on 12 January 2011 for use 5 February 2011
Units/Bookmark/SEP  Scheduled to be mailed 20 May 2011 for use 11 June 2011
Receiving the course packs is one of those moments. Almost as good as receiving a pass letter. I haven't received anything yet though. But I live in another universe.
Continental Europe.
Thursday, January 13, 2011
POLL: OU Library Services
We are a privileged generation because we live in a time where vast areas of the Internet are (still?) free. If you are able and willing to pay ( in one way or the other for example through course fees ) for Internet services, the value of the Internet increases several orders of magnitude. Registered ( Open University ) students have access to the Library Services of their university. If you like Google News you will be surprised by the ( historical ) news databases the Open University is subscribed to. You will only be using Google News to get a feeling for the current events, not more. ( Even more so if you are aware of, have woken up to, the fact that the mainstream media are controlled by a limited number of corporations and are merely distributing propaganda and disinformation. )
The truth is out there.
Just above the posts area you'll find a POLL about the usage of the Open University Library Services. Do you use them? If so, how often?
The truth is out there.
Just above the posts area you'll find a POLL about the usage of the Open University Library Services. Do you use them? If so, how often?
Wednesday, January 12, 2011
Mathematical Masterpieces
I would like to share the following book with you. If you are a math lover, and love beautiful books, then you will probably like this one.
A.Knoebel, R.Laubenbacher, J.Lodder, D. Pengelley
Mathematical Masterpieces
Further Chronicles by the Explorers
Springer 2007
Mathematical Masterpieces has 345 pages for four independent chapters, each a story anchored around a masterpiece of mathematical achievement.
The chapters are ( with between brackets the OU course they relate to )
1. The Bridge Between Continuous and Discrete ( MS221 )  This chapter is about the Discrete Calculus and explains calculating sums using the Pascal Triangle.
2. Solving Equations Numerically ( M373 )
3. Curvature and the Notion of Space ( None, don't know )
4. Patterns in Prime Numbers: The Quadratic Reciprocity Law ( M381N )
The book is intended for advanced undergraduates who know at least a year of calculus and have some maturity with mathematics at the upperdivision level.
I am currently reading ( studying ) chapter 4. ( The chapters can be read in any order ).
A.Knoebel, R.Laubenbacher, J.Lodder, D. Pengelley
Mathematical Masterpieces
Further Chronicles by the Explorers
Springer 2007
Mathematical Masterpieces has 345 pages for four independent chapters, each a story anchored around a masterpiece of mathematical achievement.
The chapters are ( with between brackets the OU course they relate to )
1. The Bridge Between Continuous and Discrete ( MS221 )  This chapter is about the Discrete Calculus and explains calculating sums using the Pascal Triangle.
2. Solving Equations Numerically ( M373 )
3. Curvature and the Notion of Space ( None, don't know )
4. Patterns in Prime Numbers: The Quadratic Reciprocity Law ( M381N )
The book is intended for advanced undergraduates who know at least a year of calculus and have some maturity with mathematics at the upperdivision level.
I am currently reading ( studying ) chapter 4. ( The chapters can be read in any order ).
Tuesday, January 11, 2011
[M3731] status: started
Started ( unofficially ) on M373 today using the unit1 course sample.
Unit 1 is called 'Introduction to Iterative Methods.' and is a repeat and continuation of MS221 B1 Iteration. The Newton Raphson method is also discussed which was part of MST121/MS221.
Proofs of theorems are given ( of course ) but questions on the exam will not be about proofs. The emphasis is on 'handson' calculation using computers where possible. Again, the use of MathCad is encouraged.
Haven't finished the unit yet, have to do more exercises.
Unit 1 is called 'Introduction to Iterative Methods.' and is a repeat and continuation of MS221 B1 Iteration. The Newton Raphson method is also discussed which was part of MST121/MS221.
Proofs of theorems are given ( of course ) but questions on the exam will not be about proofs. The emphasis is on 'handson' calculation using computers where possible. Again, the use of MathCad is encouraged.
Haven't finished the unit yet, have to do more exercises.
Monday, January 10, 2011
[M381N1] status: started
Started ( unofficially ) on M381N today using the unitn1 course sample. This course is all about proofs. From this intext problem early on in unit1:
"Show that: n is triangular IFF 8n+1 is square."
it is clear that MS221 + M208 are prerequisites for M381.
Regular study habits are a key success factor, so I have read ( and learned ). Last year my free Tuesday was scheduled for TMA's. This year I intend to study M381N on Monday evening, M373 on Tuesday and M381L on Saturday. If I have no commitments on a Sunday that day is scheduled for 'other' projects.
"Show that: n is triangular IFF 8n+1 is square."
it is clear that MS221 + M208 are prerequisites for M381.
Regular study habits are a key success factor, so I have read ( and learned ). Last year my free Tuesday was scheduled for TMA's. This year I intend to study M381N on Monday evening, M373 on Tuesday and M381L on Saturday. If I have no commitments on a Sunday that day is scheduled for 'other' projects.
Time to start debunking Euler ?
Some names are only to be spoken of with respect.
Apple has been around a lot longer than Google. Apple created the iTunes software and related store where you can buy songs for only $0.99 and they created the iPhone gadget. If you are a fan of that gadget I am sure you bought at least four of them in the last three years although the first one is still working it doesn't look that good compared to the iPhone 4. Besides that Apple created billions of cash for Steve Jobs who has no intention to share it with the poor like Bill Gates from Microsoft does.  Google on the other hand created free software like gmail, google docs and google maps, a free feature we take for granted. Oh, and they created Google Search of course. Since a while, also thanks to Google, we have digital access to the books of Leonhard Euler ( and all the other great mathematicians of course ).
Have a look at the page where Euler defines the number Pi, here at Google Books. ( If you don't read Latin, why not use Google Translate? ) I checked the first 120 decimals of Euler's Pi with those from Mathematica 8 (2010) . They are exactly the same.
When I looked at Pi in 120 decimals I knew immediately that Euler did not make these calculations by himself. He just could not. OK, he was a genius but he did not have more time than a mere humanoid like us. How many hours ( if not days, i.e. check doublecheck ) would it have cost to do that calculation? It's not just that calculation. It's about the sheer size and depth of his legacy. Although Euler was still an active mathematician when he died at age 76, it is beyond the capabilities of one human being to create that in one lifetime.
Euler must have used students and computing staff to write his books. We know that he was blind during the last years of his life but even in that period he continued to publish. Perhaps 'Euler' was merely a brand. I don't know. Who was 'Euler'?
Perhaps the comparison of Euler with Ramanujan would be somewhat like a comparison of Rembrandt with van Gogh. We know for sure that everything we read in Ramanujan's notebooks is written by Ramanujan himself. I am not sure that everything in the books of Euler is written by Euler himself, the same is true for Rembrandt and are the cause of the fact that so many Rembrandt's turned out to be work of his students.
Sunday, January 9, 2011
Mathematics and ballet
The movie Black Swan ( from the director of Pi ) convinced me that ballet is mathematics, even more so than physics. I surfed the Internet on both topics and found an interesting short video on the topic. What would Catherine Asaro say about it?
Find the transcript here. Is she saying in between the lines that girls are better suited for mathematics but that the (male) mathematics teachers don't understand how to teach mathematics to girls? An interesting point.
Find the transcript here. Is she saying in between the lines that girls are better suited for mathematics but that the (male) mathematics teachers don't understand how to teach mathematics to girls? An interesting point.
Friday, January 7, 2011
Looking back on 2010  2
I selfstudied mathematics years before I enrolled on my first Open University course MST121. Years went by in which I learned a lot of new topics but at the end of the year I had nothing to show for it. "What have I learned last year?" ( Perhaps you recognize the question? ) I passed M208 but that is not enough. Although I became more fluent in LaTeX and more skilled in using Mathematica it does not really show. (*) It could have been much worse though.
Somewhere in July or so I was spending twice as much time on the 30 point MT365 than on the 60 point M208. The real problem was that I procrastinated most of that time because MT365 was not what I expected from it at all. Part 1 Graph Theory was okish, but all in all I dare to say MT365 does not deserve the title mathematics course. (I'll deal with this issue in a separate post.) The situation deteriorated fast. I gave up entirely. I stopped submitting MT365 and M208 TMA's. I had given up. End of study.
After a month or so I realized that I had collected more than enough TMA points for a seat at the M208 exam. Not everything was lost, 60 out of 90 isn't too bad. Besides that, I never thought that getting a degree in mathematics would be easy. "Expect problems. But handle them, immediately."  I passed M208 grade 2 with sending in 6 out of 8 TMA's. I am content with that.
If you are not a fulltime student and if you want to do more than just your course work in your spare time, don't go for 90, let alone 120. I practice what I preach. I registered for 60 points in 2011. Considering what I went through last year my course(s) had to be challenging and fascinating. I looked very carefully at MST209 but it did not qualify for either of the requirements. Since MST209 is compulsory I postponed it to 2012 next to M336 Groups and Geometry. I'll deal with the MST209 issue later.
Now that leaves me with planning my free '30' points. An Open University study ends at 540 points. At that point the young and bright land in a Ph.D. position somewhere. They start teaching ( and deepening their knowledge of ) Calculus I. For them their study of mathematics begins, not ends. They are ( among ) the professionals. 'We' are what they call in mathematical circles 'amateurs', ( when there is a flaw in your work you are demoted to 'nutcase' ) disparaging maybe, but they know they can be beaten because all a mathematician need is his/her imagination and a pencil.
At 540, you must
 have chosen your field of expertise;
 be able to read articles of the journals in that field;
 understand the important open problems in that field;
 ( technically ) have the skills to write a paper and get it published.
The extra work I do is with this in mind. More on that another time.
(*) Did you know that you can copy a Mathematica formula and paste it as LaTeX in your favorite LaTeX IDE?  My fascination for Mathematica fuels my motivation to learn more mathematics and vice versa.
Somewhere in July or so I was spending twice as much time on the 30 point MT365 than on the 60 point M208. The real problem was that I procrastinated most of that time because MT365 was not what I expected from it at all. Part 1 Graph Theory was okish, but all in all I dare to say MT365 does not deserve the title mathematics course. (I'll deal with this issue in a separate post.) The situation deteriorated fast. I gave up entirely. I stopped submitting MT365 and M208 TMA's. I had given up. End of study.
After a month or so I realized that I had collected more than enough TMA points for a seat at the M208 exam. Not everything was lost, 60 out of 90 isn't too bad. Besides that, I never thought that getting a degree in mathematics would be easy. "Expect problems. But handle them, immediately."  I passed M208 grade 2 with sending in 6 out of 8 TMA's. I am content with that.
If you are not a fulltime student and if you want to do more than just your course work in your spare time, don't go for 90, let alone 120. I practice what I preach. I registered for 60 points in 2011. Considering what I went through last year my course(s) had to be challenging and fascinating. I looked very carefully at MST209 but it did not qualify for either of the requirements. Since MST209 is compulsory I postponed it to 2012 next to M336 Groups and Geometry. I'll deal with the MST209 issue later.
Now that leaves me with planning my free '30' points. An Open University study ends at 540 points. At that point the young and bright land in a Ph.D. position somewhere. They start teaching ( and deepening their knowledge of ) Calculus I. For them their study of mathematics begins, not ends. They are ( among ) the professionals. 'We' are what they call in mathematical circles 'amateurs', ( when there is a flaw in your work you are demoted to 'nutcase' ) disparaging maybe, but they know they can be beaten because all a mathematician need is his/her imagination and a pencil.
At 540, you must
 have chosen your field of expertise;
 be able to read articles of the journals in that field;
 understand the important open problems in that field;
 ( technically ) have the skills to write a paper and get it published.
The extra work I do is with this in mind. More on that another time.
(*) Did you know that you can copy a Mathematica formula and paste it as LaTeX in your favorite LaTeX IDE?  My fascination for Mathematica fuels my motivation to learn more mathematics and vice versa.
Thursday, January 6, 2011
Registration 2011 confirmed on studenthome
The M381 website opens on the 17th, while M373 opens on the 24th of January. Materials for both courses are scheduled for despatch on Friday next week ( the 14th ).
The assessment strategy for M381 is 4 TMA's with a weighting factor of 25% each while substitution is applied for one TMA and one (1) exam. Final grading as usual.
The assessment strategy for M373 is 4 TMA's with weighting factors of 10%, 30%, 30% and 30% while substitution is applied for one TMA excluding the first and one (1) exam. Final grading as usual.
Tuesday, January 4, 2011
[M381Logic]  #1: Objectives unit 1.
( Kickoff study M381Logic )
After working through Logic Unit1 ( Computability ) you should be able to:
(a) recall the definitions of the basic URM instructions and of a URM program;
(b) write down a trace table for the computation of a given URM program with a given input and state what is the output;
(c) draw a flow diagram for a given URM program;
(d) determine, in simple cases, which function of a given number of variables is computed by a given URM program;
(e) construct a URM program to compute a given function;
(f) write down the concatenation of given URM programs;
(g) calculate the values of a function defined by primitive recursion;
(h) write down a URM program to compute a function obtained by substitution or primitive recursion from given URMcomputable functions.
(... From the Student Hat.  Probably the most valuable LRH ever said was: "When reading a book, be very certain that you never go past a word you do not fully understand. The only reason a person gives up a study or becomes confused or unable to learn is because he or she has gone past a word that was not understood." Studying then becomes 'handling misunderstoods', of which finding a definition is only the first step. With the definition one should be able to give the concept sufficient 'mass'. But mass is not sufficient either, studying the topic must have purpose.  Take URM for example. Where do these letters stand for in the context of M381L? Can I describe the topic so that someone else would understand it? ( I.e. what does it look like, and so forth. ) Where and when will I need this topic? How does it relate to my long and shortterm goals? ...)
After working through Logic Unit1 ( Computability ) you should be able to:
(a) recall the definitions of the basic URM instructions and of a URM program;
(b) write down a trace table for the computation of a given URM program with a given input and state what is the output;
(c) draw a flow diagram for a given URM program;
(d) determine, in simple cases, which function of a given number of variables is computed by a given URM program;
(e) construct a URM program to compute a given function;
(f) write down the concatenation of given URM programs;
(g) calculate the values of a function defined by primitive recursion;
(h) write down a URM program to compute a function obtained by substitution or primitive recursion from given URMcomputable functions.
(... From the Student Hat.  Probably the most valuable LRH ever said was: "When reading a book, be very certain that you never go past a word you do not fully understand. The only reason a person gives up a study or becomes confused or unable to learn is because he or she has gone past a word that was not understood." Studying then becomes 'handling misunderstoods', of which finding a definition is only the first step. With the definition one should be able to give the concept sufficient 'mass'. But mass is not sufficient either, studying the topic must have purpose.  Take URM for example. Where do these letters stand for in the context of M381L? Can I describe the topic so that someone else would understand it? ( I.e. what does it look like, and so forth. ) Where and when will I need this topic? How does it relate to my long and shortterm goals? ...)
URM stands for Unlimited Register Machine its a development of the TURING Machine and is used to investigate the theoretical limitations of computers. Whereas the Turing machine uses an infinite paper tape. The URM machine is based on infinite shift registers. It's meant to model the process of computing in a manner closer to current digital computers.
The whole theory of computability or rather its limitations plays a crucial part in developing Godel's theorem probably one of the most significant theorems showing the limits of formal systems. Essentially it showed that the hope of reducing all of mathematics to logic as Russell, Frege and Hilbert hoped was no longer tenable.
As to how it relates to your long and short term goals only you can decide. I would have thought some understanding of Godel's theorem was an essential part of any one who aspires to be a mathematician.
( By Chris from Chris's Maths Physics Philosophy and Music Blog )
Infinite series videos
Two videos by Mika Seppala about infinite series:
On the Harmonic Series :
(
In Mathematica I would demonstrate the divergence as follows:
f[n_]:=HarmonicNumber[n,1]
s[n_]:=1+Sum[2^k/2^(k+1),{k,2,Floor[Log[2,n]]}]
t[m_]:=Table[{f[k],s[k]},{k,1,m}]//TableForm//N
)
and about Zeno's Paradox :
On the Harmonic Series :
(
In Mathematica I would demonstrate the divergence as follows:
f[n_]:=HarmonicNumber[n,1]
s[n_]:=1+Sum[2^k/2^(k+1),{k,2,Floor[Log[2,n]]}]
t[m_]:=Table[{f[k],s[k]},{k,1,m}]//TableForm//N
)
and about Zeno's Paradox :
Registration
Today I completed the final step in the registration process. Although it was rather difficult to contact the Open University by phone today ( already at my second try an excuse message was included in the answering system ) when I finally got through to the callcenter in Manchester the registration was handled in less than five minutes. Tomorrow, or Thursday the confirmed reg ( of M373 and M381 ) will be visible on my Student Home page and the materials will be shipped by the warehouse somewhere next week.
Monday, January 3, 2011
Infinite series ( and Euler's identity revisited )
The importance of infinite series and sequences can not be underestimated in my opinion, they literally pop up everywhere. To my surprise however, there isn't a single course at the Open University ( or any other university to my knowledge ) that deals exclusively with the topic of 'Infinite Series'. Instead the theory is stuffed away somewhere else, as if it is not really important. Like in M208 for example.  I have been searching for books on the subject and there is a ( recent ) book on the subject. It is called 'Real Infinite Series' by D. Bonar / M. Khoury published my TMAA. Tests not in M208 but in this book are for example Raabe's Test, Rummer's Test. Cauchy's Condensation Test, Abel's Test, and Dirichlet's Test as well as Bertrand's Test. It includes an entire chapter on the harmonic series with different divergent proofs. In the appendix there is an overview of the literature on infinite series.
P.S.
This video shows the proof of $e^{i\pi}+1=0$ using infinite series.
P.S.
This video shows the proof of $e^{i\pi}+1=0$ using infinite series.
Sunday, January 2, 2011
Prime numbers and the search for E.T.
I am glad that I decided to do M381 this year. ( Plans are M381 / M373 this year and MST209 / M336 next year. ) I am looking forward to Book 2 of M373: Linear Programming using the Simplex Method. And about the Logic part of M381 I can only say that it would be nice to finally understand Godel, Escher and Bach ( reread perhaps ) as it leads up to understanding Godel's incompleteness theorems.
Number Theory is what truly fascinates me about maths. Because it is deep, I suppose. Numbers are everywhere. Numbers are universal. As in universeal perhaps ( ... ). Imagine an intelligent civilization on a distant planet in the universe. If they are intelligent than they must have numbers and thus prime numbers. The movie 'Contact' based on the book of Carl Sagan was about how we would establish contact with extraterrestrial life. Almost inevitably prime numbers are involved. There is a short paper written by a number theorist that explains how this communication might work, 'Prime numbers and the search for E.T.' ( pdf )
Number Theory is what truly fascinates me about maths. Because it is deep, I suppose. Numbers are everywhere. Numbers are universal. As in universeal perhaps ( ... ). Imagine an intelligent civilization on a distant planet in the universe. If they are intelligent than they must have numbers and thus prime numbers. The movie 'Contact' based on the book of Carl Sagan was about how we would establish contact with extraterrestrial life. Almost inevitably prime numbers are involved. There is a short paper written by a number theorist that explains how this communication might work, 'Prime numbers and the search for E.T.' ( pdf )
Saturday, January 1, 2011
[Sign of the Times]  Journal of Number Theory YouTube videos
The Journal of Number Theory ( access via Open University Library Services ) created a channel on YouTube with interviews and various short presentations about papers to be published in the journal.
A conjecture about perfect numbers (  continued )
Conjecture
Show that: if $p$ is odd then$$ 2^{p1}(2^p1) = \sum_{k=1}^{\frac{p+1}{2}1} (2k1)^3$$
( Notice that $2^{p1}(2^p1)$ yields a perfect number if $(2^p1)$ is prime. )
Plan
The plan of the proof is as follows. Find expressions for $1^3 + 2^3 + 3^3 + ... + k^3$
 and $2^3 + 4^3 + 6^3 + ... + (2k)^3$
 Subtract both expressions
 Create $f(p)$ by injecting $2^p1$ into the upperindex
Proof
We use the Pascal Triangle to determine $\sum_{k=1}^{n} k^3$.$\underline{n}$
0: 1
1: 1  1
2: 1  2  1
3: 1  3  3  1
4: 1  4  6  4  1
5: 1  5  1010  5  1
Since $n^3={n \choose 1} + 6{n \choose 2} + 6{n \choose 3}$ we seek the second column, one row down or $\sum_{k=1}^{n} k^3 = {n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 3}$.
Clearly, $\sum_{k=1}^{n} 2k^3 = 8({n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 4})$.
If
$$f(n)=\sum_{k=1}^{n} k^3 = {n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 4}$$
and
$$g(n)=8 \sum_{k=1}^{n} k^3 =8({n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 4})$$
then the function we require is $$s(n) = f(n)  g( \lfloor \frac{n}{2} \rfloor ) \text{ for } n=1,3, \cdots $$.
$$\begin{array}{ll}
\underline{n} & \underline{s(n)}\\
1 & 1 \\
3 & 28 \\
5 & 153 \\
7 & 496 \\
9 & 1225 \\
11 & 2556 \\
13 & 4753 \\
15 & 8128
\end{array}$$
Since $n=1,3, \cdots $ anyway, we can remove the difficult to handle floor function by $\frac{n1}{2}$ giving $$s(n) = f(n)  g(\frac{n1}{2}) \text{ for } n=1,3, \cdots .$$
Finally we rework
$$s(p)=f(2^{\frac{p+1}{2}}1)  g(\frac{(2^{\frac{p+1}{2}}1)1}{2})$$
to $$s(p)=2^{p1} \left(2^p1\right)$$
yielding for $p=1 \cdots 7$
$$\begin{array}{ll}
\underline{p} & \underline{s(p)}\\
1. & 1. \\
2. & 6. \\
3. & 28. \\
4. & 120. \\
5. & 496. \\
6. & 2016. \\
7. & 8128.
\end{array}$$
( Since $2^31$, $2^51$ and $2^71$ are prime $28$ and $496$, $8128$ are perfect. )
Subscribe to:
Posts (Atom)
Popular Posts

Among lectures on Calculus I,II and III, ( Introduction to ) Linear Algebra and ( Introduction to ) Differential Equations from the UCCS ( ...

Problem: We want to calculate the sum of the elements of a list of numbers. Suppose this list is named l and has been assigned the value {1,...

Today I started to read the Ramanujan biography ( The ebook version, of course. ) The book looks promising. What was it like to communicate...

I found a set of video lectures on Abstract Algebra. MATH E222 Abstract Algebra  http://www.extension.harvard.edu/openlearning/math222/ E...

Ramanujan's genius (r) was discovered by Hardy (l) At a very young age Ramanujan designed the following formula for a 3 by 3 magic sq...
Welcome to The Bridge
Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before
(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)