Polski projekt

Skoro polskie literki w WordPress dobrze się sprawują, to zdecydowałem się właśnie tutaj umieścić, krok po kroku, dłuższy tekst po polsku, złożony z szeregu stron. Przekonam się jak to działa od strony organizacji takich tekstów. Jako minimum, będę prowadził własnoręcznie sporządzony spis rzeczy. Byle linki działały. Spróbuę:

Ufffffffffff… – link działa. Może jakoś to będzie. Mam nadzieję, że na stronach WordPress wszystko będzie działało równie dobrze jak w postach WordPress.

Polish diacritics?

ą  ć  ę  ł  ń  ó  ś  ź  ż
Ą  Ć  Ę  Ł  Ń  Ó  Ś  Ź  Ż

Any luck?

Wow, the Polish diacritics + Polish programmer font work in the context of WordPress! Congratulations, WordPress! 🙂

Snow

This year, which is this January month, we have quite a bit of snow (we had some also in December last year).

January is almost gone, just one week left. I’ll meet neuro-science biologists, B.Y. and another from Illinois, on January 29 (if I make it for a morning seminar :-)) and January 30.  B.J. And I met twice at Glencoe Crossing Starbucks last December.

There were a number of Introduction to Mathematics which meant to introduce readers to some topics not covered at high school, and to give an idea of the general mathematical style, away from the high school mathematics. They were mostly limited affairs, not terribly ambitious, and often rather shallow. Thus when I recently started to write a text which would teach the entire mathematics (up to a level–I need only four thousand years to get my text advanced) I called it Learn Mathematics. I wrote the initial few KBytes and… stopped. I hope to continue once I finish this blog entry. So far the text resides on my Eee (PC) only, which is not a good idea. I need to upload it to wlod.net.

Once again (again and again) I have too many plans. For instance, today I participated in a meeting of writers from my senior citizen building (these meetings are conducted by a woman who is one way or another a professional writer; she does it well). Thus I already feel like starting another Tangia site, this time under ipage. This time I feel like doing it totally alone (if at all; it would me more rational to give up on this idea).

In addition to Learn Mathematics I also want to start another project, which is with me for years: defining mathematics in my own way. I want to treat mathematics like a computer, I want to define it’s architecture (like a computer architecture). There is nothing wrong with this because ordinarily one would not use the assembler language, one would use advanced languages. I mean that I am not restricting mathematics in any way by limiting its architecture. Mathematics is finite at any moment, meaning that only finitely many characters are devoted to mathematics. Thus the countable ($$\aleph_0)$$ size of the architecture is not a true limitation.

In my architecture there are functions but they don’t have any explicit parameters. There is just $$f$$, and never $$f(x)$$ or $$f(x\ y)$$ or similar. Actually, implicitly, everything is a function of everything or at least of a countable set of variables. The mathematics/computer is always in a state. There is a countable set of variables, a countable set of constants, and the theorems make assumptions about the state of the computer (values of variables), and derive conclusions from these assumptions. That’s what happens literally. The literal description is limited. But there is also (full of imagination :-)) interpretation. Most of the true or classical mathematics happens via interpretations.

A few years ago J.M. mentioned to me Hilbert’s quantifier free logic (or logical notation, which in a sense is the same). This one statement made a great impression on me. It told me how mechanical is mathematics. Any philosophy about it should be forbidden :-). Logic is but a consequence of the mechanical nature of the Nature (of extremely simplified Nature). On one hand there is the hopeless chaos of Physics, and on the other hand there is a mechanical fragment of the World (the mechanical laws which govern the strings of characters), of some of its aspects, which allow us to study the Nature to our best. But that mechanics leads to chaos too, in more than one way. Even when we limit ourselves to the classical, so-called deterministic Newtonian mechanics (there is nothing really deterministic about it except in a fictional, formal sense), and even if we limit ourselves still much more to the mechanism of strings, we still run into Chaos very fast. Fortunately, we still have time to enjoy mathematics–and that fluctuation is a part of the Chaos as well 🙂

Enough of this disgusting rumble.

Leometry and groupls

As a youngster I have discovered an un-obvious symmetry for a family of certain (generalized) tic-tac-toe games. This immediately has induced me into defining an attractive topic (its formulation is so natural that it had to exist forever–I am not claiming any credit, this topic belongs naturally to the public domain): given a family of sets  $$F$$,  study the group of all permutations of  $$\bigcup F$$  onto itself, which preserve  $$F$$.  I mentioned this topic many times but I never studied it, which is a pity. Of course this topic generalizes finite geometries or some infinite too (e.g. affine geometry and projective geometry, with their groups–affine and projective–which preserve the family of straight lines).

Now I will formulate a more special topic in the same spirit. (First let me publish what I have already written here–I’d like to make sure that mathjax, or is it jaxmath, works, or more specifically the inline $$\LaTeX$$).

(It does not. Let me check and eventually try to extend the respective java script).

(My fault!!! MathJax does not support the in-line code between the single dollar signs, but it does so between slash-parenthesis 2-term sequences. Now the $$\LaTeX$$ code should show up. It is also possible to enable the single dollar option. BTW, I wanted this and the previous comment written in a small font–how do I do it? the small-tag pair doesn’t work.)

Let  $$(L\ G)$$  be an ordered pair consisting of a set  $$L$$,  called line, and a group  $$G$$  of permutations (i.e. bijections onto itself) of  $$L$$.  Then let  $$G_\bullet$$  be  $$G$$  together with all constant functions  $$c:L\rightarrow L$$.

The geometries (or rather leometries) described below are interesting (nontrivial) only when  $$|L|\ge 2$$.

A leometry (i.e. line geometry) over  $$(L\ G)$$  is an ordered triple  $$\Gamma\ :=\ (X\ P\ \Lambda)$$,  where:

• $$X$$  is a set called  space,  its elements are called points;
• $$P$$  is a set of functions  $$\pi : X\rightarrow L$$,  called functionals or projections;
• $$\Lambda$$  as a the set of all injections  $$\lambda: L\rightarrow X$$,  called parametrizations or orbits,  and such that

$\forall_{\lambda\in\Lambda}\forall_{\pi\in P}\quad \pi\circ\lambda\in G_\bullet$

Now we define a $$\Gamma$$-line as an image  $$L’ := \lambda(L)$$  for arbitrary  $$\lambda\in\Lambda$$;  and let  $$\mathcal L$$  be the set of all $$\Gamma$$-lines.

Given a leometry  $$\Gamma$$,  the goal is to describe the groupl  $$G_\Gamma$$  of  $$\Gamma$$,  which is defined as the group of all bijections  $$T:X\rightarrow X$$  such that:

$\forall_{L’\in\mathcal L}\quad T(L’) \in \mathcal L$

I am back (am I?)

It’s been several days since my last post, and even more so since anything meaningful.

I have moved to a new place; for three days (Jan 4-6) I rented two places at the same time. I live on the 3′ floor, have a wi-fi connection on the 8′ floor, in the large dining room, common to the whole building. There are two races: pro’s and artists. Unfortunately I belong to the latter, which implies that my wi-fi arrangement is not adequate for me. (Something funny happens to the cursor; suddenly it sneaks into a random position, creating havoc with my test, forcing me to some extra copy/cut and paste operations. It’s frustrating.)

Wikispaces entered my email again. They claim to be into education (in their way they are), thus I had started two wikis there:

I had to stop soon because their style and attitude is horrible, it must be routinely coming from the world of official (so to speak) education and educators, which is the lowest of the worlds. They didn’t allow any usage of multiple white spaces for the sake of format; and they left me with no doubts whatsoever that they will ever care about doing anything about it, or about any other useful (virtually necessary) editing features. Thus it was a good bye. I just wasted my time there, and my emotional energy (enthusiasm). World is but Chaos–no wonder that it’s mostly ugly.

A quick dollar_latex test of the back-slash LaTeX command

Will it compile? — $\LaTeX$

Exploring mathjax latex etc. (I hope for etc too, the sooner the better)

I am looking at web page:

Wait, is this the way to insert links here? What about my own wlod.net? let me try:

Well?–Nothing! WordPress is pathetic, ridiculous!

Let me try again, by hand:

Hm, well?–this time I got working links!!! And the first one is correct, while the second one resulted not in the wlod.net home page but in my blastog. Still not too bad :-).

Fine, back to mathjax $\LaTeX$ (this in-line TeX-code perhaps doesn’t work here?)–let me check the slash-bracket tags:

$\forall_{a\ b\ >\ 0}\ \ a^{\log(b)}\ =\ b^{\log(a)}$

How is it?–Works like a charm!

Hm, on that page they propose a different way to install mathjax on wordpress.org blog. I’ll stay with what I have already. They show how to install it under different conditions, also on an html page, when one has an access to the header. Thus on ipages I can use both the method which I have already tried in wiggles, and their. Theirs has actually two script-tagged commands, not one, i.e. the pairs of script tags occur twice.

Great moment! :-) MathJax $$\LaTeX$$ etc.

Finally somehow I’ve found how to install it for html files on ipage, and also for the whole wordpress.org blog like this one! (but no such luck for wordpress.com blogs, too bad). Let me try mathjax latex:

$$3^2 – 2^3 = 1$$

OK, let me have a look at it. ………. YES!!!!! It works!!!!!

Let’s compare:

• wordpress latex:   $\qquad\frac 53-\frac 85 = \frac 1{15}$
• mathjax   $$\LaTeX:\qquad\frac 53-\frac 85 = \frac 1{15}$$

I need to learn more of MathJax. Double\$ tags display the text inside on a separate line, or rather not in-line. It’s strange that wordpress latex doesn’t understand the command which displays sign LaTeX. Let me try differently: … — well? No, it doesn’t.

Euclidean ball and sphere

Let

•   $B^n\ :=\ \,\{x\in R^n : |x| \le 1\}$
•   $S^{n-1} := \{x\in R^n : |x| = 1\}$
•   $\Delta^n := \{(x\ x) : x \in B^n\}$

I’d like to compute explicitly the linearly determined function  $F : B^n\times B^n \setminus \Delta^n \rightarrow S^{n-1}$ such that  $F(a\ b) = b$  whenever  $b\in S^{n-1}$. I want to show its well known continuity.

Let  $a\ b\ \in\ B^n$,  and  $a \ne b$.  Let

$c_t\ :=\ (1-t)\cdot a + t\cdot b$

I’d like to find  $t := t_a \le 0$  such that  $|c_t|=1$,  i.e.  $c_t^2=1$.  This last equation is equivalent to the following ones:

•   $(1-t)^2\cdot a^2 + 2\cdot(1-t)\cdot t\cdot a\cdot b + t^2\cdot b^2\ \ =\ \ 1$
•   $(a - b)^2\cdot t^2 + 2\cdot a\cdot (b-a)\cdot t + a^2-1\ \ =\ \ 0$
•   $t^2 + 2\cdot\frac{a\cdot (b-a)}{(a-b)^2}\cdot t + \frac{a^2-1}{(a-b)^2}\ \ =\ \ 0$

Define   $\delta\ :=\ \left(\frac{a\cdot (b-a)}{(a-b)^2}\right)^2 + \frac{1 - a^2}{(a-b)^2}$.  Both summands are non-negative (since  $|a| \le 1$).  Thus  $\delta$  is a non-negative real number. The two solutions  $t_a\ t_b$  of the above quadratic equation are given by

•   $t_a := -\frac{a\cdot (b-a)}{(a-b)^2} - \delta^{\frac 12}$
•   $t_b := -\frac{a\cdot (b-a)}{(a-b)^2} + \delta^{\frac 12}$

Thus  $t_a \le 0$  (of course), and  $t_a = 0$  ⇔  $|a| = 1$   —   indeed, when   $|a| = 1 \ge |b|$   then

$a\cdot(b-a) = a\cdot b - a^2 = a\cdot b - 1 \le |a|\cdot|b| - 1 \le 0$

and it’s clear that  $t_a = 0$. The inverse implication is simple.

Acceleration of convergence of slow series

It’s fun (to me too) to accelarate classical slow series like

$\log(2)\ =\ \frac 11 - \frac 12 + \frac 13 - \ldots$

or

$\frac{\pi}4\ =\ \frac 11 - \frac 13 + \frac 15 - \ldots$

The latter is called Gregory-Leibniz series. What about the former? Is it Gregory-Leibniz too? Anyway, I played with them in the past and have reproduced, after Euler, versions which converge almost as fast as linear combinations of Taylor series. Possibly my web pages are still somewhere buried deep quietly.

Let  $a_n\ (n=0\ 1\ \ldots)$  be an arithmetic progression of positive real numbers. Let  $d := a_1-a_2$  be the difference of this progression. Finally, let  $k$  be a positive integer. Then

$\sum_{n=0}^\infty\ \frac 1{\prod_{j=0}^k a_{n+j}}\ \ =\ \ \frac1{k\cdot d}\cdot\frac1{\prod_{j=0}^{k-1} a_{n+j}}$

This allows to accelerate similar series by approximating them by the ones based on arithmetic series as above. A given series gets split into one of the above and the difference, which is a new series. But the new series converges faster. Euler was able to iterate this simple acceleration operation infinitely many times in a way which resulted in fast converging series.

This time I’d like to look at this method a bit more systematically than in the past. Instead of selecting the above easily summable auxiliary series artistically I’d like to do it perhaps optimally (or to make a compromise but a conscientious compromise).

Let me give an idea of what I am talking about on the initial examples. Thus first let’s transform the above G-L series::

$\log(2)\ \ =\ \ \sum_{n=0}^\infty\ \frac 1{(2\cdot n+1)\cdot(2\cdot n+2)}$

$\frac{\pi}4\ \ =\ \ \sum_{n=0}^\infty\ \frac 1{(4\cdot n+1)\cdot(4\cdot n+3)}\ \ =\ \ 2\cdot\sum_{n=1}^\infty\ \frac 1{(4\cdot n-1)\cdot(4\cdot n+1)}$

Now

$\frac 1{(2\cdot n+1)\cdot(2\cdot n+2)}\ \ =\ \ \frac 1 {(2\cdot n+\frac 12)\cdot(2\cdot n+\frac 52)}\,\ -\,\ \frac 34\cdot\frac 1{(2\cdot n+\frac 12)\cdot(2\cdot n+1)\cdot(2\cdot n+2)\cdot(2\cdot n+\frac 52)}$

hence

$\log(2)\ \ =\ \ 1\ -\ 3\cdot\sum_{n=0}^\infty \frac 1{(2\cdot n+1)\cdot(2\cdot n+2)\cdot(4\cdot n+1)\cdot(4\cdot n+5)}$