Eikonal Blog

2016.08.24

Approximate numerics

Filed under: Uncategorized — sandokan65 @ 21:22

~1700BCE Babylon: \sqrt{2} \approx 1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} \approx 1.414,213

pi

Filed under: Uncategorized — sandokan65 @ 21:19

Seen somewhere recently:

  • \frac\pi4 = \frac11 - \frac13 + \frac15 - \frac17 + \cdots
  • \frac{\pi^2}6 = \frac1{1^1} + \frac1{2^2} + \frac1{3^2} + \cdots
  • \frac2\pi = \left(1-\frac13\right)\left(1+\frac15\right)\left(1-\frac17\right)\left(1-\frac1{11}\right)\left(1+\frac1{13}\right)\left(1+\frac1{17}\right)\cdots where fractional terms \frac1p are over all primes, and the sign in front of them is minus if p=4n-1 and plus otherwise.

2013.01.03

Brain games

Filed under: Uncategorized — Tags: , , , , , — sandokan65 @ 14:02

Sites

Dual N-back

Other


Elsewhere in this blog: Intelligence (IQ) – https://eikonal.wordpress.com/2010/10/27/intelligence/

2012.05.05

Shell-In-A-Box

Filed under: Uncategorized — Tags: , , , — sandokan65 @ 22:19

2011.02.10

File systems over anything

Filed under: Uncategorized, web tools — Tags: , , — sandokan65 @ 16:25

2010.07.20

Conservative motion of a 3D particle in potential-like force fields

Filed under: Uncategorized — Tags: , — sandokan65 @ 09:20

Let’s consider the right-hand oriented orthonormal triad {\cal B} :\equiv \{\underline{n},\underline{m},\underline{k}\} (\underline{n}\cdot\underline{m}= \underline{n}\cdot\underline{k} = \underline{m}\cdot\underline{k}= 0 and \underline{n}^2 = \underline{m}^2 = \underline{k}^2 = 1) assigned to a curve c in 3-dimensional Euclidean space {\Bbb R}^3. Let this curve be parametrized by the “time” parameter t associated to the “current” position \underline{r}(t) of a particle moving along that curve. Instead of defining the triad {\cal T} as a tangential vectors (i.e. the tangential, normal and bi-normal vectors) to the c at \underline{r}, define it based on the radius vector \underline{r} by \underline{r} \equiv: \underline{n} r.

The triad vectors satisfy following evolution equations:

  • \underline{n}' \equiv: \varphi' \underline{m},
  • \underline{m}' \equiv: - \varphi' \underline{n} + \theta' \underline{k},
  • \underline{k}' = -\theta' \underline{m},

where \phi(t) and \theta(t) are two angles defining the position of the particle.

In following we will look at the several choices for the force field \underline{F} that guides the motion of the particle via Newton’s equation \underline{r}'' = \underline{F}(\underline{r}, \underline{r}').

The simplest case

For \underline{r}'' = f(r) \underline{r} one finds:

  • r'' = f(r) r + \frac{l^2}{r^3} (radial equation of motion),
  • r^2\phi' = l (= const|_{t}) (preservation of angular momentum),
  • \theta' \equiv 0 (i.e. the trajectory is planar).

A simple generalization

In force field \underline{F} = f(r) \underline{r} + g(r) \underline{r}' one finds:

  • r'' = f(r) r + g(r) r' + r \varphi'^2 (radial equation of motion),
  • l(t) :\equiv r^2\phi' (definition of non-constant angular momentum),
  • l' = g(r) l (equation guiding the evolution of the angular momentum),
  • \theta' \equiv 0, i.e. the trajectory is (still) planar.

This system can be somehow simplified by use of \varphi instead of t, and by introduction of the Binet variable u(\varphi) :\equiv \frac1{r(t)}:

  • - \frac{d^2 u}{d\varphi^2} = u +  \frac{f[u]}{l^2 u^3}
  • \frac{d l}{d\varphi} = \frac{g[u]}{u^2}.

with f[u] :\equiv f(r), g[u] :\equiv g(r), \frac{d}{dt} = l u^2 \frac{d}{d\varphi}, r'(t) = - l \frac{d u}{d\varphi} and r''(t) = - l u^2 \frac{d}{d\varphi}\left(l \frac{d u}{d\varphi}\right).

Further generalization

For \underline{r}'' = f(r) \underline{r} + g(r) \underline{r}' + h(r) \underline{r}\times\underline{r}' we get:

  • r'' = f r + g r' + r \varphi'^2,
  • l :\equiv r^2 \varphi',
  • l' = g(r) l,
  • \theta' = h(r) r – the trajectory is not planar any more.

which, in Binet coordinates has following form

  • - \frac{d^2 u}{d\varphi^2} = u +  \frac{f[u]}{l^2 u^3}
  • \frac{d l}{d\varphi} = \frac{g[u]}{u^2}.
  • \frac{d \theta}{d\varphi} = \frac{h[u]}{l u^3}.

2010.05.13

PCI DSS (Payment Card Industry Data Security Standard)

Filed under: Uncategorized — Tags: — sandokan65 @ 15:39

2010.02.03

Threats of cloud computing

Filed under: Uncategorized — Tags: , , , , — sandokan65 @ 12:28

The Edge (edge.org) has an article by Charles Leadbeater (“CLOUD CULTURE: THE PROMISE AND THE THREAT” – http://www.edge.org/3rd_culture/leadbeater10/leadbeater10_index.html) on the dangers of the budding “cloud culture”.

The second replicators (memes) are getting organized better and better.


Related here:

2010.01.28

BCS (Baker–Campbell–Hausdorff) formula

Filed under: Uncategorized — Tags: , — sandokan65 @ 17:12

In this posting the matrix C(t) is defined by e^{C(t)}:\equiv e^{t(A+B)} where A and B are constant matrices.

The Baker–Campbell–Hausdorff theorem claims that:

C(t) = B + \int_0^1  dt g(e^{t a} e^b ) A,

where g(z):\equiv \frac{\ln(z)}{z-1} = \sum_{m=0}^\infty \frac{(1-z)^m}{m+1}, and the lower-case letters a and b represent the adjoint actions of the corresponding matrices A and B (e.g. a X:\equiv ad(A) X :\equiv [A,X]).

One is frequently seeing the following series expression:

C(t) = t (A+B) + \frac{t^2}2 [A,B] + \frac{t^3}{12} ([[A.B],B]-[[A,B],A]) + \cdots = t (A+B) + \frac{t^2}2 a B + \frac{t^3}{12} (b^2 A + a^2 B) + \cdots


Source: T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

Astronomic tables and calculators

Filed under: Uncategorized — Tags: , , , — sandokan65 @ 15:11

Related here: Astronomy & Astrophysics – https://eikonal.wordpress.com/2011/05/09/astronomy-astrophysics/ | Astronomic tables and calculators – https://eikonal.wordpress.com/2010/01/28/astronomic-tables-and-calculators/ | Solar planetary system – https://eikonal.wordpress.com/2011/02/17/solar-planetary-system/ | Inside black holes – https://eikonal.wordpress.com/2011/04/12/inside-black-holes/ | Physics Sites – https://eikonal.wordpress.com/2010/02/12/physics-sites/

2010.01.22

Guenin on commutators

Filed under: Uncategorized — Tags: , — sandokan65 @ 15:46

A B^n = \sum_{j=0}^n \binom{n}{j} (-)^j B^{n-j} \ ad(B)^j A.

B^n A = \sum_{j=0}^n \binom{n}{j} (-)^j  \ (ad(B)^j A) B^{n-j}.

\partial_s f(F(s)) = \sum_{j=0}^\infty \frac{(-)^j}{(j+1)!} f^{(j+1)}(F(s)) \ ad(F(s)^j \partial_s F(s) = \sum_{j=0}^\infty \frac{1}{(j+1)!}  \ (ad(F(s))^j \partial_s F(s) ) f^{(j+1)}(F(s)).

[H, f(A)] = \sum_{j=0}^\infty \frac{(-)^j}{j!} f^{(j)}(A) \ ad(A)^j H = -  \sum_{j=0}^\infty \frac1{j!}\ (ad(A)^j H) f^{(j)}(A).
—-
Sources:

  • T1266: M. Guenin: “On the Derivation and Commutation of Operator Functionals”; Helv. Phys. Act. 41, 75-76, 1968.

2010.01.09

E8 as a crystal symmetry

Filed under: Uncategorized — Tags: — sandokan65 @ 14:24

2010.01.08

Iva Zanicchi – La riva bianca, la riva nera

Filed under: Uncategorized — Tags: — sandokan65 @ 22:33

At YouTube: http://www.youtube.com/watch?v=NWjZD37rkhM

Lyrics ((Testa- Sciorilli) ed. Mascotte):

Signor capitano si fermi qui, 
sono tanto stanco, mi fermo si, 
attento sparano, si butti giù... 
sto attento, ma riparati anche tu. 

dimmi un pò soldato, di dove sei... 
sono di un paese vicino a lei... 
però sul fiume passa la frontiera. 
la riva bianca, la riva nera, 
e sopra il ponte vedo una bandiera, 
ma non è quella che c'è dentro il mio cuor. 

tu soldato, allora, non sei dei miei... 
ho un'altra divisa lo sa anche lei... 
non lo so perché non vedo più, 
mi han colpito e forse sei stato tu... 

signor capitano, che ci vuol far... 
questa qui è la guerra, non può cambiar. 
sulla collina canta la mitraglia... 
e l'erba verde diventa paglia... 
e lungo il fiume continua la battaglia, 
ma per noi due è già finita ormai. 

signor capitano io devo andar... 
vengo anch'io che te non mi puoi lasciar... 
non non ti lascerò, io lo so già, 
starò vicino a te per l'eternità.

Tutto è finito, tace la frontiera, 
la riva bianca la riva nera, 
mentre una donna piange nella sera 
e chiama un nome che non risponderà.

Signor capitano si fermi qui, 
sono tanto stanco, mi fermo si...

3D Fractals

Filed under: Uncategorized — Tags: — sandokan65 @ 16:36

MandelBulb:

A 3D iterative mapping similar to mandelbrot mapping.

Harmonograph

Filed under: Uncategorized — Tags: — sandokan65 @ 16:30

Source:

2010.01.07

Book: Enterprise Security For the Executive

Filed under: Uncategorized — Tags: , — sandokan65 @ 16:20

One more infosec book for management types. I am not sure yet if it is worth reading – it got favorable SlashDot review by Ben Rothke, whose opinion I usually trust.

Info:

2010.01.04

Schrodinger equation

Filed under: Uncategorized — Tags: — sandokan65 @ 20:26

\Delta \psi + \frac{2m}{\hbar^2}(E-U)\psi = 0

2010.01.03

Test

Filed under: Uncategorized — sandokan65 @ 22:48

Math samples:

  • a=\int_0^\infty dz \frac{f(z e^{w})}{z-w}

Hello world!

Filed under: Uncategorized — sandokan65 @ 22:39

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