Topics

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

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.

3

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.

4

Programming Competition / Calculating Square Root of Integer, PHP Code

« on: November 21, 2012, 06:37:27 PM »

The function SquareRoot( $number ) returns square root of integer up to 4 decimal places.

function SquareRoot( $number )
{
$number = $number."00000000";
$number_length = strlen($number);
if($number_length % 2 == 0)
    {
       $dividant = $number[0].$number[1];
       for($i=2;$i<$number_length;$i++) 
           {
                $number[$i-2] = $number[$i];
           }
       $number[$number_length-1] = NULL;
       $number[$number_length-2] = NULL;
       $number_length = $number_length-2;   
    }
else
    {
       $dividant = $number[0]; 
       for($i=1;$i<$number_length;$i++) 
         {         
              $number[$i-1] = $number[$i];
         }
       $number[$number_length-1] = NULL;
       $number_length = $number_length-1;   
    }
$root = 0;
while($dividant != 0)
   {
        $multiplier = 0;
        $product = 0;
        while($product <= $dividant)
              {
                   $divisor = $root*2;       
                   $multiplier = $multiplier+1;   
                   $divisor = $divisor*10+$multiplier;
                   $product = $divisor*$multiplier; 
              }
        $multiplier = $multiplier-1;
        $divisor = $root*20+$multiplier;
        $root = $root*10+$multiplier;
        $square_root = $square_root.$multiplier;
        if($number_length==8)
              {
                   $square_root = $square_root.".";
              }
        $product = $divisor*$multiplier;
        $dividant = $dividant-$product;
        if($number_length <= 0)
            {
                 break;
            }
        $dividant = $dividant.$number[0].$number[1];
        for($i=2;$i<$number_length;$i++) 
             {         
                   $number[$i-2] = $number[$i];
             }
        $number[$number_length-1] = NULL;
        $number[$number_length-2] = NULL;
        $number_length = $number_length-2;
   }
return $square_root;
}

5

Basic Maths / Shortcut Methods for Partial Fractions

« on: November 21, 2012, 10:55:57 AM »

In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function. The main motivation to decompose a rational function into a sum of simpler fractions is that it makes it simpler to perform linear operations on it.

The problem of computing derivatives, anti-derivatives, integrals, inverse Laplace transforms, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition.


Standard Form 1:

         p                   p           1            1
----------------  =  ------- ( --------  -  ------- )
 (x+a)(x+b)          b-a       x+a         x+b   
 

Example:

         5                   5             1             1               5       1             1
----------------  =  ---------  ( --------  -  ------ )   =   ---- ( -------  -  -------- )
 (x-1)(x+3)          3-(-1)        x-1         x+3             4      x-1         x+3


Standard Form 2:

           p                      p           1               1
--------------------  =  ------- ( ---------  -  --------- )
  (xⁿ+a)(xⁿ+b)          b-a       xⁿ+a         xⁿ+b   
 

Example:

         3                       3             1              1                 3        1              1
------------------  =  ---------  ( ---------  -  --------- )   =   ---- ( --------  -  --------- )
 (x²-2)(x²+3)         3-(-2)        x²-2         x²+3              5      x²-2         x²+3


Standard Form 3:

         p                     p            1             x-a
----------------  =  ---------- ( --------  -  ---------- )
 (x+a)(x²+b)         b+a²       x+a          x²+b   
 

Example:

        7                        7              1            x+2               1           x+2
-----------------  =  ------------  ( -------  -  --------- )   =  -------  -  ---------
 (x-2)(x²+3)          3+(-2)²        x-2         x²+3             x-2         x²+3

6

Math Helps / Completing Perfect Squares

« on: November 21, 2012, 10:52:19 AM »

Problem: For what integer value n is n² + 6x + 10 also a perfect square?

Solution: Let us first express the given polynomial n² + 6x + 10 in the form "a perfect square + constant". We have, n² + 6x + 10 = (n+3)² +1. If n² + 6x + 10 = m² for some integer m, then 1 = m² - (n+3)². The left hand side gives the difference of two perfect squares which is 1 as in the right side. The only perfect squares that differ by 1 are 0 and 1. Hence, (n+3)² = 0 , having the solution n = -3 which gives the required value of n.

7

Math Helps / Finding Maximum Value of Quadratic Polynomials

« on: November 21, 2012, 10:48:45 AM »

Problem: What is the maximum value of -x² + 4x + 10?

Solution: At first we rewrite the given polynomial in the form "a perfect square + constant". Here, -x² + 4x + 10 = - (x-2)² + 14 . Since squares are non-negative, - (x-2)² + 14 <= 14. Thus, the maximum value of the quadratic is 14, which is achieved for the minimum value of (x-2)² i.e. for (x-2)²=0 or at x=2.

8

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

« on: November 15, 2012, 11:22:48 AM »

The following function FracOper($A, $Oper, $B) returns the sum $A+$B, difference $A-$B, product $A*$B and quotient $A/$B of the fractions $A & $B according the operations ($Oper) "+", "-", "*" and "/" respectively. 

function FracOper($A, $Oper, $B)

 {
   $C=NumLCM(FracDenom($A), FracDenom($B));

   if($Oper=="+")
          return FracSymp(($C/FracDenom($A))*FracNumer($A)+($C/FracDenom($B))*FracNumer($B)."/".$C);

   elseif($Oper=="-")
          return FracSymp(($C/FracDenom($A))*FracNumer($A)-($C/FracDenom($B))*FracNumer($B)."/".$C);

   elseif($Oper=="*")
          return FracSymp(FracNumer($A)*FracNumer($B)."/".FracDenom($A)*FracDenom($B));

   elseif($Oper=="/")
          return FracSymp(FracNumer($A)*FracDenom($B)."/".FracDenom($A)*FracNumer($B));

   else { echo "Undefined Operation !";  return NULL; }
 }

9

Web based Developer Forum / PHP codes to simplify fractions

« on: November 15, 2012, 10:30:38 AM »

The FracSymp($A) function returns the simplest form of the fraction $A. 

function FracSymp($A)

  {
    if(FracCheck($A)==0 || FracDenom($A)==0) { return NULL; }   

    if(FracNumer($A)%FracDenom($A)==0) 

            $ret=FracNumer($A)/FracDenom($A)."";

    elseif(FracDenom($A)%FracNumer($A)==0)

           $ret="1"."/".FracDenom($A)/FracNumer($A);

    else
            {
                 $gcd = NumGCD(FracNumer($A), FracDenom($A));
                 $ret = FracNumer($A)/$gcd."/".FracDenom($A)/$gcd;
            }

    if(FracDenom($ret)<0) 

            { $ret = -FracNumer($ret)."/".-FracDenom($ret); }

    return $ret;
  }

10

Math Helps / Need help in Linear Algebra?

« on: November 15, 2012, 08:45:22 AM »

OASE (Online Algebra Solution Engine) is ready to help you in all access. Practicing a variety of examples is an easy way for you to learn the basic operations and quick applications of linear algebra. At present OASE is giving the following opportunities:
1. Find rank of matrix (by row reduction)
2. Evaluate determinant (by cofactor expansion & by row reduction)
3. Find inverse matrix (by cofactor method & by row reduction)
4. Solve system of linear equations (by using Cramer’s Rule, by matrix inverse method & by row reduction)

Online Algebra Solution Engine (OASE)

11

Pure Maths / Algebraic Geometry

« on: November 14, 2012, 06:43:47 PM »

Algebraic geometry is a branch of mathematics, classically studying properties of the sets of zeros of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

12

Pure Maths / Differential Geometry

« on: November 14, 2012, 06:39:45 PM »

Differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus, as well as linear algebra and nonlinear algebra, to study problems in geometry.

The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology, and to the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field.

13

Applied Maths / Finite Element Method

« on: November 14, 2012, 06:25:28 PM »

The finite element method (FEM)  or finite element analysis (FEA) is a numerical technique for finding approximate solutions to partial differential equations (PDE) and their systems, as well as (less often) integral equations. In simple terms, FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other. FEM is a special case of the more general Galerkin method with polynomial approximation functions. The solution approach is based on eliminating the spatial derivatives from the PDE. This approximates the PDE with

    - a system of algebraic equations for steady state problems,
    - a system of ordinary differential equations for transient problems.

These equation systems are linear if the underlying PDE is linear, and vice versa. Algebraic equation systems are solved using numerical linear algebra methods. Ordinary differential equations that arise in transient problems are then numerically integrated using standard techniques such as Euler's method or the Runge-Kutta method.


Ref: Online Resources

14

Applied Maths / Approximation Theory

« on: November 14, 2012, 06:15:35 PM »

In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by best and simpler will depend on the application.

A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials.

One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations.



Ref: Online Resources

15

Applied Maths / Areas of Applied Mathematics

« on: November 14, 2012, 06:07:15 PM »

Applied mathematics consisted principally of Applied Analysis, most notably differential equations; Approximation Theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and Applied Probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. In the early 20th century subjects such as Classical Mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and Fluid Mechanics may still be taught in applied mathematics departments. Engineering and computer science departments have traditionally made use of applied mathematics.


Ref: Online Resources