## ¡ This Is the Test Area !

Continuously Under Construction

### Different Formatting Elements

•  This is some pre-formatted text:

   Column1     Column2     Column3
-------------------------------
123.45       23.11        -3.1
===============================


•  A little bit of pop art (using Google's monospaced webfont 'Cousine'):

         airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer
airkreuzer         airkreuzer


•  Here's a very nice example using pre-formatted text (found at Christian Hamann's page on the History of Slide Rules ):

The Principle of the Slide Rule:
----------------------------------------------------------------------

Multiply = Add Logarithmic Scales, e.g.       2×2=4 or 2×3=6 or 2×4=8
Divide = Subtract Logarithmic Scales, e.g.  4÷2=2 or 6÷3=2 or 8÷4=2

1                2         3      4     5   6   7  8 9 1 ...
|                |         |      |     |   |   |  | | |
____________________________________________________________

|                |         |      |   |  |
1                2         3      4   5  6 ...

----------------------------------------------------------------------


•  Here are some lines of Pascal code:

program First;
begin
WriteLn ('Hello again, Martin.')
end.

•  This is what the J interpreter has to say:
   'Hello again, Martin'
Hello again, Martin

[More examples from various programming languages may be found on the ACM "Hello, world!" project page.]

Using the MathJax Content Delivery Network (CDN) by linking MathJax (JavaScript based) into the web pages that are to include mathematics; this solution calls the combined in-line configuration "TeX-MML-AM_CHTML" on the server to make simultaneous use of ASCIIMath and TeX/LaTeX coding possible:

⋄  First, some ASCIIMath input:

The two possible solutions to a quadratic equation  ax^2 + bx + c = 0\ ,\ a!=0  are given by  x_(1;2) = (-b +- sqrt(b^2-4ac))/(2a) .

⋄  Second, using TeX/LaTeX input:

Given the quadratic equation  $a x^2 + b x + c = 0 \,, \,\, a \ne 0$   then possible solutions are  $x_{1;2} = \cfrac{-b\pm\sqrt{b^2-4ac}}{2a}\,.$

⋄  Third, more TeX/LaTeX input (mixed at some stages with ASCIIMath):

•  Porting PYTHAGORAS' theorem to the unit circle:  $sin^2(x)+cos^2(x)=1$  .

•  Probability for continued BERNOULLI trial  $P(E) = {n \choose k}\, p^k\, (1-p)^{ n-k}$ where  $P(E)$  is the probability to find exactly  $k$  successes in  $n$  trials when the probability for success at each stage of the experiment is  $p$  and that of failure is  $1 - p$  .

Imagine e.g. tossing a fair coin repeatedly; let  $E_{k<3}$  be "There will be less than three heads in five trials".
The probability is then given by $P(E_{k<3}) = \binom{5}{0} \left(\frac{1}{2}\right)^{0} \left(\frac{1}{2}\right)^{5} + \binom{5}{1} \left(\frac{1}{2}\right)^{1} \left(\frac{1}{2}\right)^{4} + \binom{5}{2} \left(\frac{1}{2}\right)^{2} \left(\frac{1}{2}\right)^{3} = \left(\binom{5}{0} + \binom{5}{1} + \binom{5}{2}\right) \left(\frac{1}{2}\right)^{5} = \left(1 + 5 +\frac{5\cdot4}{1\cdot2}\right) \left(\frac{1}{32}\right) = \frac{16}{32} = 0.5 = 50\%$  (do not expect that result every time you try).

•  The Golden Ratio  phi=(1+sqrt(5))//2  resp.  varphi=(-1+sqrt(5))//2

   ] Phi=. -: >: %: 5
1.61803398875
] phi=. -: <: %: 5
0.61803398875
Phi - phi          NB. difference
1
Phi * phi          NB. product
1
% Phi              NB. reciprocal 1/Phi
0.61803398875
phi = % Phi        NB. does phi equal 1/Phi ? (boolean)
1

is the only number having a continued fraction representation this simple $\phi=\frac{1+\sqrt{5}}{2}=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}}\approx 1.61803$ consisting of 1s only.
   10 # 1             NB. a list (vector) of 10 1s
1 1 1 1 1 1 1 1 1 1
(+ %)/ 1 #~ 10     NB. building the continued fraction (feeding 10)
1.61818181818
(+ %)/ 1 #~ 15     NB. cf (feeding 15)
1.61803278689
(+ %)/ 1 #~ 25     NB. cf (feeding 25)
1.61803398867
(+ %)/ 1 #~ 50     NB. cf (feeding 50)
1.61803398875


•  EULER's Product formula for calculating the Totient (number of co-prime integers below and including $n$) $\varphi(n) = n\cdot\prod\limits_{i=1}^{i=k} (1-\frac{1}{p_{i}}) = \prod\limits_{i=1}^{i=k} (p_{i}-1)\cdot p_{i}^{(e_{i}-1)}=\bigg\lbrace $$\begin{split} & n-1, \: \text{if}\: n\: \text{prime} \\ & \!\! < n-1, \: \text{if}\: n\: \text{composite} \end{split}$$$

   5 p: 11            NB. 11 is prime; Totient of 11 is 10
10
5 p: 12            NB. 12 is composite (non-prime); Totient of 12 is 4
4
5 p: 5039          NB. 5039 is prime
5038
5 p: !7            NB. 5040 is composite (non-prime)
1152
(- ~:) &. q: !7    NB. using Andrew Nikitin's translation of Product formula
1152


•  EULER's Prime-Generating Polynomial  n^2 + n + 41  is known to generate (distinct) primes for  n = 0..39  and has less success for  n > 39  .

   1 p: 41 1 1 p. i. 40
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
v=. 41 1 1 p. 39 + i. 12
(] ,:1&p:) v
1601 1681 1763 1847 1933 2021 2111 2203 2297 2393 2491 2591
1    0    0    1    1    0    1    1    1    1    0    1


•  This is one of my favourites  $\int_0^\infty e^{-x}\,\mathrm{d}x = 1$ besides the famous  $e^{i \pi} + 1 = 0$

•  This definite integral is used to calculate the area of one of the graph's spikes [the indefinite integral I looked up at the Wolfram|Alpha site]: $\int_{0}^{\pi} \frac{sin^2(x)}{1 + cos^2(x)} dx = (\sqrt{2} - 1) \,\pi \approx 1.3$

•  One of S. RAMANUJAN's formulae for  $\pi$  reads $\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{{k=0}}^{\infty}\frac{(4k)!\;(1103+26390k)}{(k!)^{4}\;396^{{4k}}}$ and converges really fast, as for  $k=0$  it already yields 7 (correct) leading digits,  $\pi\approx \frac{9801}{1103 \cdot 2 \cdot \sqrt{2}}=\color{green}{3}.\color{green}{141592}\color{gray}{73001...}$ and will compute a further 8 decimal places of π with each term of the series as can be seen comparing the results of these lines of code

   9801 % 1103 * (* %:) 2   NB. Ramanujan's approximation (k=0; sum=1103)
3.141592730013306
pir 0                    NB. piR approx (k=0)
3.141592730013306
pir 1                    NB. piR approx (k=1)
3.141592653589794
1p1                      NB. value of π (to 16 leading digits)
3.141592653589793


•  And then there's  $1\cdot1=1$  at the very base …

### Working with Forms

For proper alignment, this form is actually a borderless table.
To use the "Submit" feature, your browser needs to be set up for EMail services.
 I find this Web page interesting. plain dull. I am a first time visitor to this page and will never come back. Remark Name EMail Address

### Server Side Includes

The SSI feature depends on being provided by the server platform. These single lines embedded in HTML code may, among other things, be used to call and execute a script file. Example:

!--#exec cgi="/cgi/envvar00.cgi" --

Enclosing the expression with "< >" (so that it actually looks like a comment) will make the thing "live". So that innocent enough looking line above will generate the output given below.

[an error occurred while processing this directive]

Server environment variables may also be called by PHP of course; here is the result of a reverse DNS lookup for your machine (probably revealing some information on your ISP), managed by a SSI call to a PHP script on the same server:
[an error occurred while processing this directive]

The last line on this page is also the result of a SSI call.

### Server Time

My first provider is maintaining MET/MEST (CET/CEST) since the server farm used resides within Germany and it serves mainly German customers.

Another site is running on a different provider's server, where they are maintaining GMT.
[Sidetrack: On 2009-03-07 we became aware that this particular site (among others) had been hacked in early 2009 to send out malware (a trojan); the situation was brought back to normal with the provider's help a day later on 2009-03-08.]

### Local Computer Time

 This code snippet (applet) uses your computer clock and requires Java to be enabled. [The annoying flicker is caused by Java's repaint() procedure; double buffering not implemented yet …]

### Time Servers

 ienvenue au ureau International des Poids et Mesures This is the home of UTC and SI, the Bureau International des Poids et Mesures ( BIPM ). Have a view at their UTC/TAI Time Server applet. It is presented here as a link only, to avoid excessive loading times. Note the estimate given for the transmission delay.

### Gopher

Just for the pleasure of it there is my Gopher page, a remeniscence to earlier times of the internet. Hosted by a (presently) non-commercial site, it is not available 24/7 (as one might exspect).

Hint: If you are an Opera user you will want to configure a Gopher Proxy Server at e.g.
File > Preferences > Network > Proxy servers to be able to view it.

In case you want to send a message, please use this link.

[an error occurred while processing this directive]
[an error occurred while processing this directive]