• 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 ... add ===>> <<=== subtract ----------------------------------------------------------------------

• 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) 1is 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 \(n\)) \[\varphi(n) = n\cdot\prod\limits_{i=1}^{i=k} (1-\frac{1}{p_{i}}) = \prod\limits_{i=1}^{i=k} (p_{i}-1)\cdot p_{i}^{(e_{i}-1)}=\bigg\lbrace \begin{equation}\begin{split} & n-1, \: \text{if}\: n\: \text{prime} \\ & \!\! < n-1, \: \text{if}\: n\: \text{composite} \end{split}\end{equation} \]

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 …

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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

[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.

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.]

This code snippet (applet) uses your computer clock and requires Java to be enabled. |
||||

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. |

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]

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