Happy numbers in python

I'll show you a method in python to check if a number is happy.
When a number is happy? Well, you have to follow this algorithm:

1. Take a number n.
         n = 23
2. Dissect it into digits.
         2 and 3
3. Square them all and add them up
         2^ 2 + 3 ^ 2 = 4 + 9 = 13
4. You get a new number m.
         m = 13
5. If m = 1, n is happy; otherwise set n = m and repeat at 1.
         1^2 + 3^2 = 1 + 9 = 10
         n = 10
         1 ^ 2 + 0 ^ 2 = 1
         23 is happy!
6. If you run into a loop, n is not a happy number (is sad).

I've used recursion because it is fun (talking about happy numbers).

def happy(n, past = set()):
    m = sum(int(i)**2 for i in str(n))
    if m == 1:
  return True
    if m in past:
  return False
    past.add(m)
    return happy(m,past)
 
print [ x for x in range(1,100) if happy(x, set())]

Here 'past' is a set, needed to check if we are in a loop.
The output shows the set of happy numbers below 100.

[1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97]

logistic map in python

Simple logistic map using python and matplotlib.

import math
import matplotlib.pyplot as plt

def logistic(xa =2.9 , xb=4.0 , imgx = 240 , imgy = 500, 
    maxit = 200, f=lambda x,r: r * x * (1 - x) ):
 xs = []
 ys = []
 for i in range(imgx):
  r = xa + (xb - xa) * float(i)/(imgx - 1)
  x = 0.5
  for j in range(maxit):   
   x = f(x,r)   
   if j > maxit / 2:
    xs.append( i ) 
    ys.append(int(x * imgy))
 return [xs,ys]
 
 
myfunction = lambda x,r: r * (math.sin(x)**2) 
points = logistic( xa = 2.1 ,xb = 3.1, imgy=200 , f=myfunction)
ax = plt.subplot(121)
ax.scatter(points[0], points[1], s= 1)

points = logistic()
bx = plt.subplot(122)
bx.scatter(points[0], points[1], s= 1)
plt.show()

Similarity Matrix in Text Mining

( in this post i'm testing http://www.mathjax.org for latex math formulas )
What is a similarity matrix ( in text mining ) and why is important?

CORPUS

We have to start from a corpus composed by k documents:

$$ \left\{ D_i \right\}_{i=1}^k $$

 ( A corpus is merely collection of documents )

SIMILARITY MATRIX

A way to find semantic structures in the corpus is to study the occurrence and the
co-occurrence for every pair of words in the corpus.
A good tool to find something interesting is a similarity matrix.

DEFINITION

To define a similarity matrix we must define the similarity between two objects ( words )

$$ s(w_i, w_j ) $$

a similarity matrix becomes simply a matrix that contains the ratio of similarity between
the objects of index i and j for the generic position {i,j}

a good similarity matrix can follow from this definition: $$ s(w_i,w_j) = \dfrac {c(w_i,w_j) } { f(w_i) \cdot f(w_j) }$$ where:  $$ c(w_i,w_j) $$ is the co-occurrence between two words ( the number of documents containing both
words )

and:  $$f(w_i) $$

is the occurrence of the word.

MATRIX REPRESENTATIONS


- GRAPH

The created matrix is symmetric and could be visualized using a undirected weighted graph.
The nodes represents the words and the similarity between the two words is given by the
weight between two nodes.



this visualization is nearly useless (easily more than 10000 nodes!!!) .

- METRIC SPACE

A way to handle this info is to position k points in an n-dimensional space so that the mutual
distance between a couple of elements previously defined could reflect the weight between
the related pair of words.$$w_i \mapsto p_i | s(w_i,w_j) = \dfrac{1}{||p_i - p_j||}  \forall i,j \leq k$$
( higher weight - closer distance )

a problem related with this approach is that is not always operable (matrix could not be
compatible with metrics constraints ) and the requested dimension of the target space
is a $$O(k^2).$$
so we need to use a technique to reduce the dimension preserving the significant information
( reducing the dimension brings a certain loss of information).

SEMANTIC STRUCTURES

Using the representation in an n-dimensional space is important to analyze clusters of points.
A cluster could be defined as a subset of points whose mutual distances are much smaller
than the average distance of the complete set.

A cluster is a reflection of some kind of statistical structure of the corpus.

Structures able to create a cluster can either be:

1. language related rules ( eg: syntactic structures ) or
2. semantic meanings ( eg: topics )

Verlet integration in Haskell

Verlet integration algorithm written in 3 minutes

-- Verlet integration

projOneTwo ( a, b, c , d ,e ) = ( a, b)

nextStepInt (_ ,_ , 0 ,_ ,_ ) = []

nextStepInt ( xt ,vt , n , acc , h ) =

let

atph = acc . ( xt + ) $ h

vtph = vt + atph * h * 0.5

xtph = xt + h * vt + 0.5 * acc xt * h^2

in

( xtph, vtph , n-1 , acc , h ) : nextStepInt ( xtph, vtph , n-1 , acc , h )

timeInt ( xt ,vt , n , acc , h ) = map ( projOneTwo ) $ nextStepInt ( xt ,vt , n , acc , h )

example ( spring ) :

timeInt (0 , 1 , 10000 , (\x -> 0.1 * ( - x ) ) , 0.2 )

Cairo-Chaos Haskell

Haskell chaos and lib-cairo.

Is only a fixed-point iteration over this function:

lgs x r = r * x * exp(- x )

Dollar $ operator in Haskell

Do you know what is $ operator in Haskell?
$ means simply , 'apply the left function at the right value'.

f $ x := f x

it seems really trivial, isn't it ?
But, for example in this kind of situation, is really usefull:

zipWith ( $ ) ( cycle [ \x -> div (x + 1) 2 , \x -> div x 2 ] ) [1..]

here you have a infinite list of function:

a = cycle [ \x -> div (x + 1) 2 , \x -> div x 2 ]
( ie: [\x -> div (x + 1) 2 , \x -> div x 2 , \x -> div (x + 1) 2 , \x -> div x 2, ... ] )

and you want to apply every element of the list at the element
at the same index in the second list:

b = [1..]

zipWith, for every index i takes the element a(i) of the left list
and b(i) of the right list and execute what is requested inside the parentheses.
In this situation is specified $ so:

a(i) $ b(i ) := a(i) ( b(i) )

the result must be the following:

[ 1 ,1 , 2, 2 , 3 ,3 ... and so on.

Prime Numbers in haskell

well, do you want to know how to find 'prime numbers' in a quick and dirty
way using Haskell ?
try this!

import Data.List
nubBy ( \x y -> mod y x == 0 ) [2..]

Haskell is so easy and charming...

( ps: if you want to speed up a little bit:
nubBy ( \x y -> ( x*x-1 <= y ) && ( mod y x == 0 ) ) [2..]
)

re: Mandelbrot Set in Haskell

import Data.Complex

conv p=(\n->".,:;|!([$O0*%#@?"!!(n-1))(ceiling (p*8)::Int)

gr=map(\y->[(x:+y)|x<-[-2,-1.97..0.7]])[-1.2,-1.13..1.2]

--px w=magnitude(foldl1(\z c ->z^2 +c)(replicate 10 w))

px w=(magnitude.last.take 10.takeWhile(\r->(magnitude r )<6).iterate(\z->z^2+w))w

image=map((map(\el->case(px el<2)of{true->conv(px el);_->' '})))gr

main=mapM_ putStrLn image

better...

Mandelbrot set in Haskell

import Data.Complex
gr = map(\y-> [( x:+y )|x<-[-3,-2.95..1]])[-2,-1.9..2]

px (a:+b) = magnitude(foldl (\z c ->z^2+c) 0 (take 10([((a*x):+b)|x<-[1,1..]])))

image = map((map(\el->case(px el<2)of{true->'*';_ ->'-'})))gr
main = mapM_ putStrLn ( (map(\el->show el) )image )

A first interpretation ( a little bit unsatisfactory ) of the Mandelbrot set in Haskell

Numbers in Haskell

The core of numbers post in Haskell:

idens p q x = case ( rem ( x ^ 2 ) p ) of { 1 -> 1 ; _ -> 0 }
numList p x = map ( idens p x ) ( map ( x * ) [ 1..( p-1 ) ] )
matIde p = map ( numList p ) [ 1..(p-1) ]

Mandelbrot in Rebol

And this is my interpretation of Mandelbrot fractal using Rebol:

mandelb.r

This language allow to write a gui using few rows of code.

There is still a problem with zoom functionality ( I hope to fix it soon ).

You can see also my ruby mandelbrot .

My first time in Rebol and numbers

Rebol is a fantastic language!
You can try this code in the Rebol/view Console:
numbers.r

REBOL [Title: "Numbers Test"]
ena: 121
positions: []
dimPal: 3

unitar: func [ ele enne ][ either (( (ele // enne) == 1 ) or (( ele // enne) == (enne - 1)))[ 1  ] ["-"] ]

evaluateUni: func [ en ] [
for j 1 en 1[ st: copy ""
for i 1 en 1[
append st unitar i * j en
sr: square-root en
if ( unitar i * j en ) == 1  [ append positions ( as-pair i j  ) * 350 / en + 23x23 ]
]
]
positions
]
evaluateUni ena

refre: does[
positions: copy []
evaluateUni ena
pis: copy []
posi: copy positions
foreach p posi[ insert tail pis  reduce[ 'fill-pen 160.209.215.100 'circle p dimPal ]  ]
out: form ( length? positions ) / 2 == ( ena - 1 )
insert tail pis reduce [  'text 230x5 join "numero primo: "   out ]
scrn/effect/draw: copy []
append scrn/effect/draw pis
show scrn
]

refreNoEval: does[
pis: copy []
posi: copy positions
foreach p posi[ insert tail pis  reduce[ 'fill-pen 160.209.215.100 'circle p dimPal ]  ]
out: form ( length? positions ) / 2 == ( ena - 1 )
insert tail pis reduce [  'text 230x5 join "numero primo: "   out ]
scrn/effect/draw: copy []
append scrn/effect/draw pis
show scrn
]
lay: layout [

scrn: box 400x400 black effect [
draw [ text "unit" ]
rotate 50
gradmul 180.180.210 180.60.255
]
slider 200x16 ena / 300 [
ena: to-integer value * 300
inpNum/data: form ena
inpNum/text: form ena
show inpNum
refre
]
inpNum: field form ena [
print to-integer inpNum/text
ena: to-integer inpNum/text 
]
at 330x460
btn "Change"  [
refre
]
slider 100x15 dimPal / 10 [
dimPal: to-integer  value    * 10
refreNoEval
]
]

view lay 

Mandelbrot in Scala

I'm also working on a more complex mandelbrot set program in Scala.
Soon i'll post the source code.
This is the output:

Esoteric ruby Mandelbrot

Inspired from this article i've decided to
play with code in order to create my version of Madelbrot set program in ruby.
This is the code:

( not true, the code is here because the blogger editor didn't work fine when a code contain <
or > )


The code is intentionally obscure.

Is possible to decide: position, zoom factor, number of iterations, and number of lines.
The output should be like this:

Vedic Math first lesson

the Sutra says: "By one more than the previous one".
With this sutra you can calculate mentally this fraction:





easy, isn't it?
In the next few days i'll give the explanation of this method.

Missing number

the result has 8 digits and the missing number is ... 8
soon i'll post some of Vedic maths

Fast Fourier transform

Fast Fourier transform in Scala

http://sites.google.com/site/snakelemma/snippets/fftShort.scala?attredirects=0