What is a lattice? - Or lattices in M336

#openuniversity #m336

The Open University course M336 contains two booklets which are dedicated to lattices. One booklet about two-dimensional lattices (GE3) and one about three-dimensional lattices ( and polyhedra ) (GE6). To a layman I would explain lattice as some regular grid of points ( connected by thin lines ).

Click to enlarge

In the example above the lattice is defined by two vectors and consists of all points $n \mathbf{a} + m \mathbf{b}$ where $n,m$ are integers.

Fields which use lattice theory are crystallography, finance, game ( maze ) programming, group theory and number theory. When I dug a little bit deeper I discovered that the field of lattices is -ginormous-. Gabriele Nebe and Neil Sloan ( yes him ) maintain a catalog of lattices which now contains over 160,000 lattices. Mathematicians like to generalize over n-dimensions so yes, that database contains lattices in dimensions higher than 3. Like lattices in 40 dimensions for example. Forty.

A catologue of lattices.
Junkyard article about lattices and geometry of numbers.

The mathematical universe is expanding with tremendous speed.

M336 - Group Theory - Fundamental Theorem of Abelian Groups

#openuniversity #m336 #video

One of the theorems that is discussed in the group theory track in the Open University Course 'M336 Groups and Geometry' is the Fundamental Theorem of Abelian Groups. Early on in Group Theory it becomes clear that there is a connection between group theory and number theory in Langrange's theorem and the Sylow Theorems ( also part of M336 ) but only after studying the Fundamental Theorem of Abelian Groups you'll get a notion of the depth of the connection between Group Theory and Number Theory.

MathDoctorBob ( his YouTube alias ) made a short video lecture on the topic. Precise as always.

Sets and multisets

A set is a collection of well defined and distinct objects. I remember it as I have learned the Set interface in Java, a Set has no duplicates and is not sorted: 'it models the mathematical set abstraction'.

But what if we want to study collections of well defined but not necessarily distinct objects? The easy way out is to simply define another base abstraction. The beauty of mathematics is that you don't have to. The body of mathematical knowledge is built from a minimal number of base abstractions. Then how should we define a multi-set?

Definition: Let S be a nonempty set. A multi-set M with underlying set S is a set of ordered pairs: $$M=\left\{ (s_i,n_i) | s_i \in S, n_i \in \mathbb{Z}^+ \right\},$$ where $n_i$ is the multiplicity of the element $s_i$.

A multi-set defined as, or using, a set.

An algebraic proof of Fermat's Little Theorem

Let $G$ be an abelian group. Define a scalar multiplication over $\mathbb{Z}$ as follows: $$n \cdot g = \underbrace{g+g+\cdots+g}_{n \ \text{times}}.$$ Note that in this case $|G| \ g=0$. ( We turned $G$ into a $\mathbb{Z}$-module. )

For primes, the multiplicative group $\mathbb{Z}_p$ is abelian, $| \mathbb{Z}_p | = p-1$ and the identity element is $1$. Let $a \in \mathbb{Z}_p$ and the multiplicative notation of $|G| \ g=0$ becomes $a^{p-1} \equiv 1 \bmod{p}$. But this is just Fermat's Little Theorem!

Fermat

Abstract Algebra E-222 video 26 Rings 2

#maths

Watched lecture 26 of the Harvard Abstract Algebra series. - What can you say about the complex number $z$ if $(2+i)z$ must be an integer?

Prof. Gross ... "ideals in the Gaussian integers $\mathbf{Z}\left[i\right]$ of type $\mathbf{Z}/p\mathbf{Z}$".

These lectures were recorded in 2003 and are basically saved for all generations to come. Imagine that the lectures of Gauss were recorded on video! Euler and Gauss will be remembered forever by their name and picture, but the great mathematicians of today and tomorrow will be remembered by their video lectures.

Abstract Algebra E-222 video 25 Rings 2

Every word counts in mathematics.

Every non zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. ( Fundamental theorem of algebra, Wikipedia )

The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.

$x^2-1$ has $4$ roots...

In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.

Abstract Algebra E-222 video 24 Rings 1

#M336

Just watched a video where Benedict Gross introduces Ring Theory. I don't think you can learn Ring Theory ( or any mathematics for that matter ) by just watching a video.

In the business of commercial education, in programming for example, teachers are often confronted with students ( sent by their employers ) who expect to leave as a qualified programmer just by hanging in their chairs during the course. Needless to say they leave as empty headed as they came in.

But if you watch prepared you can pick up a lot from this professor. In this first lecture he explains why there is such a field as Ring Theory in the first place. Where did it come from? And most of all: what are the important topics we have to watch in this field? ( I.e. Ideals and Unit Groups ). You may wonder why I gave this post the M336 ( Groups and Geometry ) hash-tag, it is because Rings and abelian Groups ( and Number Theory ) are intimately connected and one of the objectives of M336 is the classification of all abelian groups. - By the way, the word is abelian group and not Abelian group despite the fact that the word abelian comes from Niels Abel. Writing a name lowercase is the highest possible honor in mathematics. ( So I have been told... ).

$(\mathbf{Z/nZ})^{\times}$ has $\phi(n)$ elements

At the end of this lecture he mentions that Group Theory is a really hard subject and all that. The thing with Group Theory is that it has to sink in quite a while before it clicks and opens up to you.

Polynomials

I am studying some more about polynomials, the topic of symmetric polynomials for example, is an interesting one.

Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask...

...which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?

To be continued.

About the Open University

Last year's problems seem simple with what I am going through now. What is going on? I have to withdray money from my student account which I hold with the Open University to pay for this year's courses, the procedure is simple and fast, really. First you call the course registration line, then you simply ask the student advisor to arrange it for you. But...

... I can't get through.

Not that I haven't been trying. Last week I gave up and wrote an e-mail asking for advice on how to establish contact. They say that it can take 'up to three working days before you get a reply'. It's past three working days already. But...

... no reply.

What's going on? Does it have to do with budget cuts? Queries about the new fees? I never had a problem to get through. The Open University is not just any university. They are =huge=. On a piece on the OU site I read that the OU is the biggest university in the UK with
- 250,000 students
- 7000 tutors ( Associate Lecturers )
- 1200 academic staff, and
- 3500 support staff
It is also an -international- university, they have 3500 students in Ireland, 9000 in the EU and another 7500 outside the EU.

In the meantime I am still waiting. I'll have to figure out something because I can't wait to start studying again. When everything is OK I do the reflection post first.

I haven't dropped the Fearless Symmetry series, on the contrary: I am styding algebra again. I have posted a question about Galois Theory here. Basically a course in Galois Theory makes you understand -why- polynomials with rational coefficients and degree five or higher can't be solved by radicals. At least not in general, if the corresponding Galois group of the polynomial is soluble however then there is a solutiuon. - Which you won't find in the textbooks. And that's what I find disappointing, to say the least. Maybe this is why a study in mathematics never seems to stop. A course answers a few of your questions but you'll have more questions after the course than you had before.

If Galois Theory is a blank for you, then this article ( pdf) might fill it, a bit. ;-)

Example of a Galois Group of order 8 ( Introducing Math Doctor Bob )

Regular readers must have noticed my interest in Abstract Algebra, of which I am currently studying, in different ways, the topic of Galois Theory. If you have chosen a different route in mathematics ( computation, statistics, and so forth ) or if you are at the early undergraduate level you may have difficulty picturing what Galois Theory is -all about-. I am trying to communicate that idea by summarizing the popular introduction to the field 'Fearless Symmetry' which basically introduces Galois Theory to the general ( but educated ) public. ( Currently working on part 7 out of 23). But as they say, one picture says more than a thousand words. For those that want to get an idea, fast and easy, and *now*, I recommend the following video ( mini lecture ). Don't expect you can master the subject by watching a ten minute video but the ten minutes are well worth it. The video lecturer is 'Math Doctor Bob', who uploaded about 600 mini lectures on various mathematical topics.



See also:
- Fearless Symmetry

Peter Neumann speaks about Galois.

Found this video more or less by accident on YouTube.

Fearless Symmetry 6/23: Equations and varieties

I read the sixth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 6: Equations and varieties

Logic of Equality

An equation is a statement, or assertion, that one thing is identical to another. In mathematics we replace is by = and use symbols that stand for the terms.

History of equations

Long before algebra as we know it, ancient peoples were working with equations.
A triangle whose three sides have lengths 3, 4, and 5 is a right triangle which is an example of a Diophantic equation because the unknowns are restricted to integers. Around the late 1500s Descartes added the connection between algebra and geometry now known as analytic geometry. Descartes, as a philosopher believed that the physical universe was governed entirely by the laws of geometry. Newton ( and Leibniz ) discovered that this wasn't true, they had to invent calculus to solve their scientific problems mathematically.

Z-Equations

A rational number is any number that can be expressed as the ratio of two integers. Real numbers are rational iff it is a terminating decimal or a repeating decimal. The set of all rational numbers is usually denoted as Q. We will deal mostly with equations where all the constants are integers. Or equations "defined over the integers". A Z-Equation is an equality of polynomials with integer coefficients. One of the main problems in number theory is finding and understanding all solutions of Z-equations.

Varieties

Fix the attention on a particular Z-equation. Write S(Z) for the set of all integral solutions of that equation, S(Q) for the set of all rational solutions of it, and so on. We call S an "algebraic variety". The variety S defined by a Z-equation ( or a system of Z-equations ) is the function that assigns to any number system the set of solutions S(A) of the equation or system of the equations.

For example define the Variety S as x^2 + Y^2 = 1
Then
S(Z) = {{1,0),(0,1),(-1,0),(0,-1)}.
S(Q) = {t in Q | 1-t^2 / 1+t^2, 2t/1+t^2}.

We can reformulate Fermat's Last Theorem using varieties as follows.
For any positive integer n, let V_n be the variety defined by x^n + y_n = z^n. Then if n > 2, V_n(Z) contains only solutions where one or more of the variables is 0.

Systems of equations

The system
x^2+y^2=1
x>0
is valid and has solutions, but it does NOT define an algebraic variety because inequalities are not defined in C nor in any of the finite fields.

Take the system
x^2+y^2+z^2=w
w^4=1
x+y=z
then S(R) is the ellipse x^2+y^2+x y = 1/2.

Finding roots of polynomials

The easiest general class of varieties to look at would be those defined by a single Z-equation in a single variable, for instance, x^3 + x - 2 = 0. The study of this type of variety is dominated by the concept of the Galois group. ( More in 8 and 13 ). If f(x) is a polynomial the roots of f(x) are the numbers c such that f(c)=0.

Are There General Methods for Finding Solutions to Systems of Polynomial Equations?

On a purely number-theoretical level, leaving philosophy and logic behind, we also have the famous theorem of Abel and Ruffini: Unlike quadratic polynomials, for which we can use the quadratic formula, for polynomials f (x) of degree 5 or greater, there is no formula involving just addition, subtraction, multiplication, division, and nth roots (n = 2, 3, 4, . . .) that can solve f (x) = 0 in general.

Deeper understanding is desirable

The amazing discovery of Galois is that there is more structure to S(A). As we shall see, S(A) is not just a set; it is the basis for defining a representation of a certain group, called the Galois group. We will look at another series of very interesting and very important, though not so very simple, Z-varieties: elliptic curves. These two kinds of varieties will give us some of our main examples to help us understand Galois groups and their representations.

To be continued with 7. Quadratic reciprocity

Fearless Symmetry 5/23: Complex Numbers

I read the fifth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 5: Complex Numbers

Well, I think it is safe to assume that readers of this blog know the complex numbers. This chapter is in fact about a subset of the Complex Numbers called the Algebraic Numbers. In FS they use $\mathbf{Q}^{Alg}$ as notation, whereas I have seen mostly the notation $A$ for the Algebraic Numbers.

Every algebraic number can be expressed as the root of of a polynomial equation with integer coefficients. So $\pi$ is not a member of $\mathbf{Q}^{Alg}$, but $\sqrt{2}$ is because $\sqrt{2}$ is a solution of $x^2 - 2 = 0$.

Visualisation of the (countable) field of algebraic numbers in the complex plane.
( From Wikipedia )

To be continued with 6. Equations and varieties

Fearless Symmetry 4/24: Modular Arithmetic

I read the fourth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 4: Modular Arithmetic

Chapter 4 is all about modular arithmetic.

Considering the goal of the book somewhere fields have to be introduced and in this chapter we find the first definition of a field.

Definition: A field is a number system where we can divide by anything nonzero.

Anything more precise would scare off the laymen casual reader for who the book is intended. I kind of like the definition myself. '... where you can divide anything by nonzero'.

Modular arithmetic is introduced as clock arithmetic of course with examples like: "Today is Tue. What day is it in 25 days?" or "The analog clock shows 8. What time will it show in 33 hours?"

Also, the extremely important concept of an equivalence relation is defined. There is much more about modular arithmetic in the book, of course.

To be continued with 5. Complex Numbers

Fearless Symmetry 3/23: Permutations

I read the third chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 3: Permutations

In chapter 3 the concept of a permutation is explained and how they form groups.

Definition: A permutation is a one-to-one map from a set to itself.

Example 1:
Given the set {1,2} the possible one-to-one maps ( permutations ) are:
1->1, 2->2 and
1->2, 2->1.

Example 2:
For sets of three elements there are 6 = 3! possible permutations.

If we put all the permutations of a set in a set by itself and add the composition of permutations as the operation then this set becomes a group. Permutation groups are among the most important objects in group theory because every finite group is a subgroup of some permutation group.

There are two possible ways of notation when it comes to permutations. Let's consider the set {1,2,3,4} which has 24 possible permutations.

The permutation 1->1, 2->2, 3->4 and 4->3 can be written as [1 2 4 3] and also as (1)(2)(3 4) or short (3 4) this is the so called cycle notation. Thus (1 2 3) and [2 3 1] represent the same permutation. Clearly the cycle notation is more efficient, especially when considering permutations of large sets.

The composition of permutations means permuting one after the other. Unfortunately in some books it is done from left to right, in others from right to left.

Exercise 1:
Show that (ab)(cde)*(ae)(bc)(d)=(ac)(bde).

( I would solve it as follows: )
Right hand side:
A B C D E
_ D _ E B Apply (bde)
C D A E B Apply (ac)

Left hand side:
A B C D E
_ _ _ D _ Apply (d)
_ C B D _ Apply (bc)
E C B D A Apply (ae)

E C B D A
C D _ E _ Apply (cde)
C D A E B Apply (ab)

And it shows that LHS = RHS

Sets of permution form a group under composition because:
- composition leads to a new permutation ( closure )
- the neutral element is the do-nothing permutation, i.e. (1)(2)(3).
- every permutation has an inverse because it can be permuted back to the original positions.
- composition of permutations is associative.

Note that the composition of permutations is NOT ( always ) commutative.

To be continued with 4. Modular Arithmetic

Fearless Symmetry 2/23: Groups

I read the second chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 2: Groups

Definition: A group G is a set with a composition defined on pairs of elements, as long as three axioms hold true:
1. For any three elements x,y,z in G: x*(y*z) = x*(y*z)
2. G contains an element e such that for all x in G: x*e = e*x = x.
3. For any element x in G, there is an element y in G such that x*y=e.

Example 1:
The group of rotations of the sphere in R3: SO(3) or the Special Orthogonal Group in 3 dimensions. -The set G is the collection of all rotational symmetries of the sphere, i.e. if we rotate the sphere by any angle, the sphere doesn't noticeably change. The group property basically means that if we rotate the sphere over any angle A, after this over an angle B, it is the same if we would have rotated it in one go, but over some different angle. Also any rotation has an inverse: rotating it over the opposite angle. This makes the rotations a group. SO(3) is in fact a Lie group because these rotations can be done arbitrary small which is not the case when considering the symmetry group of for example a cube. Lie groups capture the concept of "continuous symmetries".

For me personally, this is the time to review chapters 1,2 and 3 of Naive Lie Theory by John Stillwell, Springer 2008. There you will find that the ( 4 dimensional ) quaternions are intimately related to the group SO(3) and that the quaternions can be expressed as 'complex 2-dimensional rotations' or complex 2 by 2 matrices. - This explains why quaternions are frequently used in 3D-(game)-programming.

To be continued with 3. Permutations

Fearless Symmetry 1/23: Representations

I read the first chapter of Fearless Symmetry. See: Reading ( conflicts time management ) for the history on this topic.

Part 1: Algebraic Preliminaries

Chapter 1: Representations

The goal of the book is 'Mod p linear representations of Galois groups' and how these representations help to clarify the general problem of solving systems of polynomial equations with integer coefficients.

It is very important to know that a mathematical definition can redefine a commonly word used elsewhere. ( A simple group is not 'simple' but complex. A tree is a graph, a 'tree' in the forest is not. ) Sometimes an object is defined by listing its properties and following that a proof is given of the existence of such an object.

Definition: A set is a collection of things which are the elements of the set.

Definition: A function f: A-> B from a set A to a set B is a rule that assigns to each element in A an element of B.

Definition: A morphism is a function from A to B that "captures at least part of the essential nature" of the set A in its image in B. ( Clearly "captures at least part of the essential nature" needs to be refined later. )

Definition: A representation is a morphism from a source object to a standard target object.

Example 1: Take A,B and the fact that B represents A. A may be a citizen, B her state rep and X the legal fact that B represents A by voting in the legislature on her behalf. A may be a(n abstract) group, B a group of matrices, and X a morphism from A to B. The ultimate in abstraction is representing A,B as dots and X as an arrow from A to B.

Example 2: In the context of counting, given any two finite sets A and B, a morphism is a one-to-one correspondence from A to B. A representation in this case is a morphism from a given finite set to one of the sets {1}, {1,2}, {1,2,3} and so on. So a flock of three sheep has the set {1,2,3} as its target.

To be continued with 2. Groups.

Exercise ( algebra )

Given that $$x^n-y^n = (x-y) (\sum_{k=1}^n x^{n-k}y^{k-1} )$$ with for example: $$x^4-y^4 = (x-y)(x^3 +x^2y +xy^2 + y^3)$$.
a) How would you factorize $x^5 + y^5$?
b) Generalize.
c) Prove the identity above for $x^n-y^n$ using mathematical induction.

Equivalence classes in Mathematica

An equivalence relation like 'has the same remainder after division by 5' partitions a set in equivalence classes.

The relevant Mathematica function is GatherBy[ set, equivalence relation] -> {equivalence classes}:

In[5]:= GatherBy[{1, 2, 3, 4, 5}, Mod[#, 5] &]

Out[5]= {{1}, {2}, {3}, {4}, {5}}

In[6]:= GatherBy[{49, 33, 11, 1, 2, 3, 4, 5}, Mod[#, 5] &]

Out[6]= {{49, 4}, {33, 3}, {11, 1}, {2}, {5}}

Or when using the equivalence relation 'Is Odd?' on the same set:

In[7]:= GatherBy[{49, 33, 11, 1, 2, 3, 4, 5},OddQ]

Out[7]= {{49, 33, 11, 1, 3, 5}, {2, 4}}

[Sign of the times] - Abacus still in use

The Corporation is optimistic about the potential of the abacus system of mental arithmetic, which it introduced in two of its primary schools on Friday. The system is aimed at improving the comfort of young students with numbers and the mathematical functions.

Source: express buzz

Link: Abacus ( see the comment )

Link: An introduction to the abacus

Link: Chinese Abacus + Manual