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An obsession?

Since I’d like to write about number theory, I can’t an urge to introduce the ring of integers from scratch. Thus first I am about to post about Abelian groups. Since I am about to post about Abelian groups, I jusrt have to introduce groups in general. This my unrealistic obsession with systematic writing is my weakness. I don’t know the secret of pragmatic writing from the middle, which is an art mixed with self-assurance or arrogance, depending on the author.

I’ve already wrote about axioms of groups, together with the first simple theorems, in the past at least twice. Once on Chimeryd site, a long time ago. That site unfortunately has dissappeared. Thus a couple years ago I wrote about groups again. That text must be still somewhere on the Internet, in one of my sites/pages, but I was unable to find it. Ok, I am not happy about it but I’ll still write about groups for the third time (the last? :)).

It’s amazing that three simple and natural axioms have at the same time so many interesting consequences (some of them deep), and they define a theory–a true profound theory, not just a loosely defined theme– which admits a whole world of models (examples), from finite to infinite, from discrete to analytic, from easy to understand to most complex.

The importance of the groups comes from their actions on other mathematical objects, thus throwing light on the structure of these objects, as well as on the group itself (we study groups by studying their actions). Thus it is good to remember about the group transformation spirit of groups. Even when studying a group alone by itself it’s good to view its elements as acting at least on itself.

A quick intro to groups (a sketch).

Let me write here a preliminary sketch, before I’ll post a more finished version on the main part of wlod.net.

Axioms

DEFINITION   A group is an ordered quadruple

Γ  :=  (G * Inv e)

where  G  is a set,  e ∈ G,  * and Inv  are a unary and binary operation in&nbsp G,  such that the following axioms hold:

1. (x*y)*z  =  x*(y*z)
2. e*x  =  x
3. Inv(x)*x  =  e

It’s good to interpret these axioms in terms of the action of the group on itself, meaning that with every element  a ∈ X  we associate a (left) map

L_a : G → G

given by:

∀ _{x ∈ G}   L_a(x)  := a*x

First goal

Axioms 2 3 above allow symmetric versions:

• x*e  =  x
• x*Inv(x)  =  e

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I will show below that these symmetric variations are theorems, that they follow from the axioms 1-3 of the theory group. It’s an elementary result (obviously :-)) but not trivial. (It goes without saying that this topic is classical; I don’t claim any of the results, of course not, and only the exposition is mine, at least to my limited knowledge).

Simple general properties

THEOREM 0   e*e = e

PROOF   substitute  x := e  in axiom 2 above.

THEOREM 1   If  a*x = a*y  then  x=y,  for arbitrary  a x y ∈ X. In other words, transformation  L_a  is an injection of  X  into itself.

PROOF   If  a*x = a*y  then:

x = Inv(a)*a*x = Inv(a)*a*y = y

END of PROOF

REMARK   More explicitly, equation  a*x = b  has at the most one solution, namely

x :=Inv(a)*b

But is this  x  actually a solution? At this moment this is still an open question, which will be answered in positive soon.

THEOREM 2   ∀_{x ∈ X}  x*e = x

PROOF

• Inv(x)*(x*e) = (Inv(x)*x)*e = e*e = e
• Inv(x)*x = e

Thus, by theorem 1,  x*e = x.  END of PROOF

THEOREM 3   ∀_{x ∈ X}  x*Inv(x) = e

PROOF By Theorem 2:

Inv(x)*e  =  Inv(x)  =  e*Inv(x)

=   (Inv(x)*x)*Inv(x) = Inv(x)*(x*Inv(x))

and by Theorem 1,  x*Inv(x) = e.   END of PROOF

This should be very easy on paper (those who play such games can do it most likely in their head). But how will it go in real time on-line? It feels to me awkward but let me try for awhile.

First try

Axioms

Let’s consider system  (X -),  where  –  is a binary operation in  X,  which satisfies the following axioms:

1. x – (y – z)  =  z – (y – x)
2. x-x = y-y
3. x – (y-y)  =  x

for every  x y z ∈ X.

Consequences

From axioms  1. and 3.  (let z:=y)  we get:

Let’s define:

• 0  :=  x-x     (the choice of  x  is irrelevant)
• Neg(x)   :=  0 – x
• x+y  :=  x – Neg(y)

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Wen can rewrite axioms 3 as:

THEOREM 0

Neg(Neg(x)) = x

PROOF

Neg(Neg(x))  =  (z-z) – ((y-y) – x)  =  x -((y-y) – (z-z)  =  x

END of PROOF

THEOREM 1

x+y = y+x

PROOF

x+y  =  x – (0 – y)  =  y – (0 – x)  =  y+x

END of PROOF

THEOREM 2

x+0 = x

PROOF

x+0  =  x – (0 – 0)  =  x – 0  =  x

END of PROOF

THEOREM 3

(x+y)+z  =  x+(y+z)

PROOF

(x+y)+z  =  (x – (0 – y)) – (0 – z)  =  z – (0 – (x – (0 – y))

z – ((0 – y) – (x – 0))  =  z – ((0 – y) – x)

By the same token (swap  x  and  z):

(z+y)+x  =  x – ((0 – y) – z)

But

x – ((0 – y) – z)  =  z – ((0 – y) – x)

hence

((x+y)+z  =  ((z+y)+x

and we already showed that operation + is commutative–therefore

((x+y)+z  =  (x+(y+z)

END of PROOF

THEOREM 4

x-y  =  x + Neg(y)

PROOF

x + Neg(y)  =  x – Neg(Neg(y))  =  x-y

END of PROOF

Ooooph

Oooph, I got lucky :-). It was hard for me to type, copy/paste, and manage html, and at the same time to think. Fortunately, I’ve selected the axioms well, and then there was very little left to think about, lucky me.

Now the road to integland is wide open–hurray!

Should I start another elementary mathematical topic (and which) or should I stop writing, and spend some time now on learning CSS (Cascading Style Sheets), and then JavaScript (in this order, because they are presented in my textbook in this order, and I’d like to be systematic, at least on this occasion)?

One of the topics is the beginning Number Theory. It’s rather clear that One should not start with Peano axioms, followed by the inductive definitions of the operations. It’d take too long and be too boring to get to the first juicy results. The start point can be a strong algebraic system, which includes the ordering of the natural or integer numbers. It’s a better option. But the list of axioms has to be quite extensive, which means back to boring. One would like to believe that elementary mathematics is simple but it’s not. And neither is the task of presenting elementary mathematics.

I could leave unclear what is assumed about natural or integer numbers. That’s what usually is done. But I would feel uncomfortable all the time.

Hey, I have a promising idea. In general, of the equivalent systems of axioms some seemingly weak simple ones and some seemingly very strong systems can be highly elegant, for different reasons of course. If you want to prove that a model satisfies the axioms then the weak system of axioms is useful. But when you want to derive new theorems then the strond system gives you a head start. In the theory of smooth manifolds John Milnor superbly selects axioms and definitions which are in between, neither weak nor very strong. It’s a true art. Beautiful and pragmatic at the same time. Let me attempt something similar for the beginning of the Number Theory. I’ll sketch my idea below (in the actual note the axioms will be spelled out in a more detailed way, and will be nicer–no Neg nor + will be mentioned). The axioms will describe only the group of rational integers. The rest–multiplication and ordering–will be defined afterward. Thus indeed this approach is stronger than Peano axioms, but avoids immediate definition of integers as an ordered ring.

DEFINITION   Integland  is an ordered triple   Z  :=  (Z  –  1), which satisfies the following axioms (this are not the ultimate axioms yet, but what the axioms should achieve):

1. Z  is a set;
2. 1 ∈ Z;
3. symbol  –  stands for a binary operation  – : Z² → Z;
4. 1-x ≠ x
5. if  1-x = x-1  then  x=1;
6. (Z + Neg 0)  is an abelian group, where  0 := 1-1,  and the unary operation  Neg,  and binary operation +:

   Neg : Z → Z   and   + : Z² → Z

   are defined as follows:

   Neg(x) := 0-x   and   x+y := x-Neg(y)

   for every  x y ∈ Z;

7. If  A ⊆ Z  is such that:
   • 1 ∈ A
   • ∀_{x y ∈ A}  x-y ∈ A

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   then  A = Z

END of DEFINITION

Now  *  is defined as the unique binary operation such that

• 1*x = x*1 = x
• (a-b)*x = a*x – b*x
• a*x = x*a

for every  a b x ∈ Z.

This is actually an inductive definition of *.

in order to get–withing a link–from here to the index page of wlod.net I need to go up twice. Otherwidse, until now, I was not able within this blog to create a working link to wlod.net. It was so frustrating!!!

You never have to leave your home and you will gain confidence, that helps in making your personal life better. price of viagra The biggest benefit of Penegra is, it successfully heals the problem by working tadalafil 20mg for women djpaulkom.tv on it without taking into account the tumor stage, the tumor grade, the man’s age along with the man’s overall wellness, some medical practitioners would advocate other treatment methods. Ingredients of Vital M-40 capsule: Asparagus Racemosus, Withania Somnifera, viagra ordination Cinnamomum Cassia, Onosma Bracteatum etc. However, the individuals who have had attended the purchase viagra in australia http://djpaulkom.tv/category/news/page/8/ sessions say this therapy can benefit the couples to a great extent. I still don’t feel comfortable about this issue. It seems to me that at wlod.net site, outside this wordpress blog, I need to go up one less time than within this blog–in the comparable situations. I must be wrong 🙂 Am I?

And I still am not able to write an absolute link from this blog to wlod.net. Am I hopelessly embarrassing myself? (Of course! What a stupid question :-)).

Tomorrow is a special day. Or taking into account time zones, it is already around.

I’ve checked once again on LaTeX on these pages. I am poor at such things. Nevertheless now I know, as expected, that one kind of latex works on the wordpress of this site. My knowledge and control over this important to me (to this site) issue is still almost non-existent.

At wlod.net I was so far using only html, even for mathematics. This was a step back. Should I get more ambitious? However I feel like developing the main part of this site outside the wordpress. I don’t feel free under wordpress. I’d like to limit wordpress on wlod.net just to my blog Blastog.

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I closed the topic of counting unicolor triangles. Getting the examples which showed that the bound of the main theorem (word main sounds so impossibly serious :-)) was rather trivial. The whole thing is very simple, just one simple idea, quite natural. Can it be new? Hard to believe. But then, why doesn’t it appear in presentations of bicoloring the complete 6-vertex graph  K₆?

Now it’s time for me to write about the probabilistic approach. I should also add a comment or to to what I’ve already written, especially about the relation of the 3 mod 4 case complication to the theorem 0 of combinatorics.

\[\LaTeX\] page:

$3^2-2^3=1$

Now a different mode:

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\[1+3+\ldots+(2\cdot n-1) = n^2\]

\[1^3+2^3+…+n^3 = (1+2+\ldots+n)^2\]

How was it?

Are adjectives unicolor and bicolor acceptable to native Anglo-Saxons? Should I use uni-colored and bi-colored instead? Can I use all four of them?

Anyway, in the past week or ten days I was writing about the count of unicolor triangles when the set of pairs (edges) is painted, each edge in one of the two colors, say blue and red. I don’t know if the result is new (perhaps not) or the method (I hope so). Most of this writing repeats my older notes. But I was happy that two days ago or so I got also something new–a very tiny something but it still makes me feel better. I have provided (simple!) extreme examples also in the

ξ ≡ 1 mod 4

case. Case  ξ ≡ 3 mod 4  is still open (to me).
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My counting idea is adequate for triangles of colored edges, but not for tetrahedra of colored triangles, not by itself–I need an additional fresh idea. One day I will get it.

I am in Starbucks. I forgot to bring my mouse, how dumb! and frustrating! Somehow today I am doing not bad without the mouse, using just the touch pad. I mean typing. Otherwise I am not doing much. Thus I better write some emails. Will I?

In less than 15m Starbucks gets closed but it’s almost full, everybody is comfortable, noboedy seems to be going home. It feels strange, surrealistic. A few minutes ago I looked for any employee, to ask about the closing time, just in case. I couldn’t find any. They happened to hide in the back in their room. TThey have cleared the shelves of the food, so it does look like the end of the day here. And still…

Somehow time goes by. Can I do anything about it? Do I have any ability to do anything about getting more intensive?

It is strange today. Ordinarily, they would already be cleaning the place for the night, sweeping the floor, but not yet, not tonight.

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Finally, a single guy a couple minutes ago, and a couple right now, have left. But then, there are only 7m left!

Yes, they are closing in a few minutes. It’s just a bit different today. What a pity! I’d love to stay for another hour, at least.