31
Faculty Sections / Re: Mathematics help in Savings and credits
« on: November 15, 2012, 03:46:16 PM »
Its of course true. Note that in every cases related to benefits there are hidden losses where math works rarely.
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32
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($B));
else { echo "Undefined Operation !"; return NULL; }
}
function FracOper($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($B));
else { echo "Undefined Operation !"; return NULL; }
}
33
Web based Developer Forum / Re: Learn PHP
« on: November 15, 2012, 11:12:07 AM »
PHP is really easy to learn and understand because one of the main positive sides of PHP is it can proceed any kind of variables as strings. But, it should note that after any kind of numerical operations the return value must be a string.
34
Web based Developer Forum / Re: PHP codes to simplify a fraction
« on: November 15, 2012, 10:35:25 AM »
NumLCM() function returns least common multiple(LCM) of two numbers only and the function NumNLCM() returns LCM of several numbers.
function NumLCM($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 $nlcm;
}
function NumLCM($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 $nlcm;
}
35
Web based Developer Forum / Re: PHP codes to simplify a fraction
« on: November 15, 2012, 10:33:49 AM »
The NumGCD() function returns the greatest common divisor(GCD) of two numbers only and the function NumNGCD() returns GCD of several numbers.
function NumGCD($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 $ngcd;
}
function NumGCD($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 $ngcd;
}
36
Web based Developer Forum / PHP codes to simplify fractions
« on: November 15, 2012, 10:30:38 AM »
The FracSymp($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 $ret;
}
function FracSymp($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;
}
37
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)
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)
38
Pure Maths / Re: Algebraic Geometry
« on: November 14, 2012, 06:44:57 PM »
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.
39
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.
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.
40
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.
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.
41
Applied Maths / Re: Finite Element Method
« on: November 14, 2012, 06:31:17 PM »
Example: An example of application of FEM would be in numerical weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas.
42
Applied Maths / Re: Finite Element Method
« on: November 14, 2012, 06:27:36 PM »
Application: FEM is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation).
43
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
- 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
44
Applied Maths / Re: Approximation Theory
« on: November 14, 2012, 06:16:49 PM »
The objective of approximation theory is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floating point arithmetic. This is accomplished by using a polynomial of high degree, and/or narrowing the domain over which the polynomial has to approximate the function. Narrowing the domain can often be done through the use of various addition or scaling formulas for the function being approximated. Modern mathematical libraries often reduce the domain into many tiny segments and use a low-degree polynomial for each segment.
45
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
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