Messages

1

Basic Maths / Square Root of Matrices

« on: May 31, 2013, 11:27:58 AM »

Like numbers, definition of square root can be extended for matrices. A matrix B is said to be a square root of A if  B²=A. For example square roots of the matrix are , and their additive inverses.

There are a number of methods for calculating square roots of matrices. One of the frequently used method is diagonalization method. An n×n matrix A is diagonalizable if there is a matrix V and a diagonal matrix D such that A = VD^-1V . This happens if and only if A has n eigenvectors which constitute a basis for Cⁿ. In this case, V can be chosen to be the matrix with the n eigenvectors as columns, and a square root of A is VSV^-1, where S is any square root of D. Indeed, (VD^1/2V^-1)² = VD^1/2(V^-1V)D^1/2V^-1 = VDV^-1 = A .

As in the example, the matrix can be diagonalized as , where and . D has principal square root , giving the square root .



Ref: Wikipedia

2

Applied Maths / Re: Conservation of Mass

« on: May 20, 2013, 10:13:52 AM »

Some times we misuse mass and weight. But these are totally different things. Mass, the total amount of elements, of a particle becomes constant at it's every states as is described in the article. On the other hand, weight, a force that equals (mass)x( gravitation), varies from position to position of the particle.   

3

Applied Maths / Re: Mathematics and the Modern World Changing Words, Sound, and Pictures into Number

« on: May 20, 2013, 09:12:18 AM »

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have specific properties that make them useful for signal processing. Wavelets can be combined, using a "reverse, shift, multiply and sum" technique called convolution, with portions of a known signal to extract information from the unknown signal.

For more detailed visit http://en.wikipedia.org/wiki/Wavelet.

4

Applied Maths / Re: Mathematics and the Modern World 2. Computers

« on: May 20, 2013, 09:02:25 AM »

Mathematicians are always the real bosses. But they are not rewarded perfectly, aren't they?

5

Math Helps / Importance of Complex Numbers

« on: May 20, 2013, 08:44:25 AM »

Complex number is a number that combines real and imaginary numbers. For example 2+3i, 1-5i, -2i etc are complex numbers where i is the imaginary unit defined as i²=-1.

Real numbers are commonly used  where as complex numbers are mainly dealt by scientists, engineers and researchers. Every real life problem can be described by a special kind of mathematical model consisting group of simultaneous equations. The solution of a mathematical model provides a set of information (data consisting real and imaginary numbers) regarding the related problem. For these reasons development of complex numbers has a significant role in the areas of scientific research.

Here are some examples of complex numbers.

In electronics, the state of a circuit element is described by two real numbers (the voltage V across it and the current I flowing through it). A circuit element also may possess a capacitance C and an inductance L that (in simplistic terms) describe its tendency to resist changes in voltage and current respectively.

These are much better described by complex numbers. Rather than the circuit element's state having to be described by two different real numbers V and I, it can be described by a single complex number z = V + i I. Similarly, inductance and capacitance can be thought of as the real and imaginary parts of another single complex number w = C + i L. The laws of electricity can be expressed using complex addition and multiplication.

Another example is electromagnetism. Rather than trying to describe an electromagnetic field by two real quantities (electric field strength and magnetic field strength), it is best described as a single complex number, of which the electric and magnetic components are simply the real and imaginary parts.

6

Applied Maths / Importance of Complex Numbers

« on: May 19, 2013, 12:53:02 PM »

Complex number is a number that combines real and imaginary numbers. Real numbers are commonly used  where as complex numbers are mainly dealt by scientists, engineers and researchers. Every real life problem can be described by a special kind of mathematical model consisting group of simultaneous equations. The solution of a mathematical model provides a set of information (data consisting real and imaginary numbers) regarding the related problem. For these reasons development of complex numbers has a significant role in the areas of scientific research.

Here are some examples of complex numbers.

In electronics, the state of a circuit element is described by two real numbers (the voltage V across it and the current I flowing through it). A circuit element also may possess a capacitance C and an inductance L that (in simplistic terms) describe its tendency to resist changes in voltage and current respectively.

These are much better described by complex numbers. Rather than the circuit element's state having to be described by two different real numbers V and I, it can be described by a single complex number z = V + i I. Similarly, inductance and capacitance can be thought of as the real and imaginary parts of another single complex number w = C + i L. The laws of electricity can be expressed using complex addition and multiplication.

Another example is electromagnetism. Rather than trying to describe an electromagnetic field by two real quantities (electric field strength and magnetic field strength), it is best described as a single complex number, of which the electric and magnetic components are simply the real and imaginary parts.

7

Mathematics / Re: Algebra

« on: May 15, 2013, 09:08:32 AM »

Now-a-days a group of mathematicians are highly interested in studying algebra using computers. This branch of algebra in known as computational algebra. The central target of computational algebra is developing algorithms for different mathematical and engineering problems that can be executed by using computers. A number of computational algebra systems (CAS) like Matlab, Mathematica, Maxima, Axiom, Reduced etc are already developed by some renowned research groups. These CASes are have been being used for last few years by both mathematicians and engineers.     

8

Web based Developer Forum / Re: Basic Operatins with Fractions (PHP code)

« on: May 13, 2013, 01:23:41 PM »

Sample Return Values:

1.  FracSum("2/3", "1/4")  returns  "11/12"

2. FracDifference("2/3", "1/4")  returns  "5/12"

3. FracProduct("2/3", "1/4")  returns  "1/6" 

4. FracQuotient("2/3", "1/4")  returns  "8/3"

9

Faculty Forum / Re: Finding LCM of Several Integers, PHP Code

« on: February 03, 2013, 10:25:59 AM »

Sample codes:

<?php
$A[0] = 5; $A[1] = 3; $A[2] = 2; $A[3] = 7;
$lcm = NumNLCM($A);
echo "LCM of 2, 3, 5 and 7 is ".$lcm;
?>

Output: LCM of 2, 3, 5 and 7 is 210

10

Basic Maths / Re: Shortcut Methods for Partial Fractions

« on: December 10, 2012, 11:02:55 AM »

Important Example:

      4               1        4        1         1               4        1             1                 4          4      1          1             
------------  =  ---- . ------- ( ----  -  ------ )   =   ---- ( ------  -  ----------- )  =  ------  -  ---- ( ----  -  ------- )
 x³(x²+3)        x      3-0       x²      x²+3           3       x³        x(x²+3)          3x³        3      x²      x²+3

                        4           4                4     
                 =  ------  -  ------  +  ------------
                      3x³        3x²         3(x²+3)

11

Faculty Forum / Re: Finding LCM of Several Integers, PHP Code

« on: December 05, 2012, 06:51:52 PM »

Sample codes:

<?php
$A[0] = 4; $A[1] = 6; $A[2] = 12; $A[3] = 16;
$lcm = NumNLCM($A);
echo "LCM of 4, 6, 12 and 16 is ".$lcm;
?>

Output: LCM of 4, 6, 12 and 16 is 48

12

IT Forum / Re: Finding GCD of Several Integers, PHP Code

« on: December 05, 2012, 06:45:18 PM »

Sample codes:

<?php
$A[0] = 4; $A[1] = 6; $A[2] = 12; $A[3] = 16;
$gcd = NumNGCD($A);
echo "GCD of 4, 6, 12 and 16 is ".$gcd;
?>

Output: GCD of 4, 6, 12 and 16 is 2

13

Faculty Forum / Finding LCM of Several Integers, PHP Code

« on: November 26, 2012, 09:57:36 AM »

The  function NumLCM($A, $B) returns the least common multiple (LCM) of the integers contained in $A and $B. The function NumNLCM($A) calls NumLCM() repeatedly and returns LCM of several integers contained in the array $A.

function NumLCM($A, $B)
 {
      if($A<0) { $A = -$A; }
      if($B<0) { $B = -$B; }
      if($A>=$B) { $bigger=$A; }
      else  { $bigger=$B; }
      for($n=$bigger;$n<$A*$B;$n+=$bigger)
          {
               if($n%$A==0 && $n%$B==0) { $lcm=$n.""; return $lcm; }
          }
      $lcm=$A*$B.""; 
      return $lcm;   
 }

function NumNLCM($A)
  {
       $nlcm=$A[1]; $i =1;
       while($A[$i] != NULL)
            {
                  $nlcm = NumLCM($nlcm, $A[$i]);
                  $i++;
            }
       return $nlcm;
  }

14

IT Forum / Finding GCD of Several Integers, PHP Code

« on: November 22, 2012, 01:14:31 PM »

The  function NumGCD($A, $B) returns the greatest common divisor (GCD) of the integers contained in $A and $B. The function NumNGCD($A) calls NumGCD() repeatedly and returns GCD of several integers contained in the array $A.

function NumGCD($A, $B)
 {
   if($A==0 || $B==0) { return NULL;}
   if($A<0)  $A=-$A;
   if($B<0)  $B=-$B;
   if($A>=$B) { $bigger=$A; $smaller=$B; }
   else { $bigger=$B; $smaller=$A; }
   while(1)
      {
         $C = $bigger%$smaller; if($C==0) { $gcd=$smaller.""; return $gcd; }
         $bigger = $smaller; $smaller = $C;
     }
 }

function NumNGCD($A)
  {
        $ngcd=$A[1]; $i=1;
        while($A[$i]!=NULL)
             {
                   $ngcd=NumGCD($ngcd, $A[$i]);
                   $i++;
             }
        return $ngcd;
  }

15

Prophet's(SW) Guideline / Re: Allah's Pharmacy

« on: November 22, 2012, 12:42:16 PM »

Thanks for very nice collection. I just whisper, " O' Allah, its quite impossible for human beings to predict how much mysteries you kept in your creations! "