Saturday, 31 August 2013

Linear Equations in Two Variables

Linear equations in two variables

A common form of a linear equation in the two variables x and y is
y = mx + b,\,
where m and b designate constants (parameters). The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x2, y1/3, and sin(x) are nonlinear.

Forms for 2D linear equations

Linear equations can be rewritten using the laws of elementary algebra into several different forms. These equations are often referred to as the "equations of the straight line." In what follows, x, y, t, and θ are variables; other letters represent constants (fixed numbers).

General (or standard) form

In the general (or standard) form the linear equation is written as:
Ax + By = C, \,
where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis (where, y is zero), is C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is C/B, and the slope of the line is −A/B. The general form is sometimes written as:
ax + by + c = 0, \,
where a and b are not both equal to zero. The two versions can be converted from one to the other by moving the constant term to the other side of the equal sign.

Slope–intercept form

y = mx + b,\,
where m is the slope of the line and b is the y-intercept, which is the y-coordinate of the location where line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. It may be helpful to think about this in terms of y = b + mx; where the line passes through the point (0, b) and extends to the left and right at a slope of m. Vertical lines, having undefined slope, cannot be represented by this form.

Point–slope form

y - y_1 = m( x - x_1 ),\,
where m is the slope of the line and (x1,y1) is any point on the line.
The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, y − y1) is proportional to the difference in the x coordinate (that is, x − x1). The proportionality constant is m (the slope of the line).

Two-point form

y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1),\,
where (x1y1) and (x2y2) are two points on the line with x2x1. This is equivalent to the point-slope form above, where the slope is explicitly given as (y2 − y1)/(x2 − x1).
Multiplying both sides of this equation by (x2 − x1) yields a form of the line generally referred to as the symmetric form:
(x_2 - x_1)(y - y_1)=(y_2 - y_1)(x - x_1).\,

Intercept form

\frac{x}{a} + \frac{y}{b} = 1,\,
where a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form is in standard form with A/C = 1/a and B/C = 1/b. Lines that pass through the origin or which are horizontal or vertical violate the nonzero condition on a or b and cannot be represented in this form.

Matrix form

Using the order of the standard form
Ax + By = C,\,
one can rewrite the equation in matrix form:
\begin{pmatrix} A&B \end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}C\end{pmatrix}.
Further, this representation extends to systems of linear equations.
A_1x + B_1y = C_1,\,
A_2x + B_2y = C_2,\,

A_2 & B_2
\end{pmatrix} = 
Since this extends easily to higher dimensions, it is a common representation in linear algebra, and in computer programming. There are named methods for solving system of linear equations, like Gauss-Jordan which can be expressed as matrix elementary row operations.

Parametric form

x = T t + U\,
y = V t + W.\,
Two simultaneous equations in terms of a variable parameter t, with slope m = V / T, x-intercept (VUWT) / V and y-intercept (WTVU) / T.
This can also be related to the two-point form, where T = ph, U = h, V = qk, and W = k:
x = (p - h) t + h\,
y = (q - k)t + k.\,
In this case t varies from 0 at point (h,k) to 1 at point (p,q), with values of t between 0 and 1 providing interpolation and other values of t providing extrapolation.

2D vector determinant form

The equation of a line can also be written as the determinant of two vectors. If P_1 and P_2 are unique points on the line, then P will also be a point on the line if the following is true:
\det( \overrightarrow{P_1 P} , \overrightarrow{P_1 P_2} ) = 0.
One way to understand this formula is to use the fact that the determinant of two vectors on the plane will give the area of the parallelogram they form. Therefore, if the determinant equals zero then the parallelogram has no area, and that will happen when two vectors are on the same line.
To expand on this we can say that P_1 = (x_1 ,\, y_1), P_2 = (x_2 ,\, y_2) and P = (x ,\, y). Thus \overrightarrow{P_1 P} = (x-x_1 ,\, y-y_1) and \overrightarrow{P_1 P_2} = (x_2-x_1 ,\, y_2-y_1), then the above equation becomes:
\det \begin{pmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{pmatrix} = 0.
( x - x_1 )( y_2 - y_1 ) - ( y - y_1 )( x_2 - x_1 )=0.
( x - x_1 )( y_2 - y_1 ) = ( y - y_1 )( x_2 - x_1 ).
Then dividing both side by ( x_2 - x_1 ) would result in the “Two-point form” shown above, but leaving it here allows the equation to still be valid when x_1 = x_2.

Special cases

y = b\,
Horizontal Line y = b
This is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope m = 0. The graph is a horizontal line with y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept.
x = a\,
Vertical Line x = a
This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to a. The slope is undefined. There is no y-intercept, unless a = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept.

Connection with linear functions

A linear equation, written in the form y = f(x) whose graph crosses the origin (x,y) = (0,0), that is, whose y-intercept is 0, has the following properties:
 f ( x_1 + x_2 ) = f ( x_1) + f ( x_2 )\
 f ( a x ) = a f ( x ),\,
where a is any scalar. A function which satisfies these properties is called a linear function (or linear operator, or more generally a linear map). However, linear equations that have non-zero y-intercepts, when written in this manner, produce functions which will have neither property above and hence are not linear functions in this sense. They are known as affine functions.

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Thursday, 22 August 2013

"Business Logistics" by Ajit Mishra's Online Classroom

Business logistics

A forklift stacking a logistics provider's warehouse of goods on pallets.
One definition of business logistics speaks of "having the right item in the right quantity at the right time at the right place for the right price in the right condition to the right customer". As the science of process, business logistics incorporates all industry sectors. Logistics work aims to manage the fruition of project life cycles, supply chains, and resultant efficiencies.
Logistics as a business concept evolved in the 1950s due to the increasing complexity of supplying businesses with materials and shipping out products in an increasingly globalized supply chain, leading to a call for experts called "supply chain logisticians".
In business, logistics may have either an internal focus (inbound logistics) or an external focus (outbound logistics), covering the flow and storage of materials from point of origin to point of consumption . The main functions of a qualified logistician include inventory management, purchasing, transportation, warehousing, consultation, and the organizing and planning of these activities. Logisticians combine a professional knowledge of each of these functions to coordinate resources in an organization.
There are two fundamentally different forms of logistics: one optimizes a steady flow of material through a network of transport links and storage nodes, while the other coordinates a sequence of resources to carry out some project.

Production logistics

The term production logistics describes logistic processes within an industry. Production logistics aims to ensure that each machine and workstation receives the right product in the right quantity and quality at the right time. The concern is not the transportation itself, but to streamline and control the flow through value-adding processes and to eliminate non–value-adding processes. Production logistics can operate in existing as well as new plants. Manufacturing in an existing plant is a constantly changing process. Machines are exchanged and new ones added, which gives the opportunity to improve the production logistics system accordingly. Production logistics provides the means to achieve customer response and capital efficiency.
Production logistics becomes more important with decreasing batch sizes. In many industries (e.g., mobile phones), the short-term goal is a batch size of one, allowing even a single customer's demand to be fulfilled efficiently. Track and tracing, which is an essential part of production logistics due to product safety and reliability issues, is also gaining importance, especially in the automotive and medical industries.

Warehouse management systems and warehouse control systems

Although there is some overlap in functionality, warehouse management systems (WMS) can differ significantly from warehouse control systems (WCS). Simply put, a WMS plans a weekly activity forecast based on such factors as statistics and trends, whereas a WCS acts like a floor supervisor, working in real time to get the job done by the most effective means. For instance, a WMS can tell the system that it is going to need five of stock-keeping unit (SKU) A and five of SKU B hours in advance, but by the time it acts, other considerations may have come into play or there could be a logjam on a conveyor. A WCS can prevent that problem by working in real time and adapting to the situation by making a last-minute decision based on current activity and operational status. Working synergistically, WMS and WCS can resolve these issues and maximize efficiency for companies that rely on the effective operation of their warehouse or distribution center.

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"Differential Calculus" by Ajit Mishra's Online Classroom

Differential calculus

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. In mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. (The function it produces turns out to be the doubling function.) The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f′, pronounced "f prime." For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative, the doubling function.
If the input of the function represents time, then the derivative represents change with respect to time. For example, if f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball.
If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as y = mx + b, where x is the independent variable, y is the dependent variable, b is the y-intercept, and:
m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.
This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f. (a, f(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore (a + h, f(a + h)) is close to (a, f(a)). The slope between these two points is
m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.
This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The secant line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:
\lim_{h \to 0}{f(a+h) - f(a)\over{h}}.
Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f.
Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x2 be the squaring function.
\begin{align}f'(3) &=\lim_{h \to 0}{(3+h)^2 - 3^2\over{h}} \\
&=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\
&=\lim_{h \to 0}{6h + h^2\over{h}} \\
&=\lim_{h \to 0} (6 + h) \\
&= 6.
The slope of tangent line to the squaring function at the point (3,9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function, or just the derivative of the squaring function for short. A similar computation to the one above shows that the derivative of the squaring function is the doubling function.

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Monday, 19 August 2013

"Reverse Logistics" by Ajit Mishra's Online classroom .

Reverse logistics

Reverse logistics stands for all operations related to the reuse of products and materials. It is "the process of planning, implementing, and controlling the efficient, cost effective flow of raw materials, in-process inventory, finished goods and related information from the point of consumption to the point of origin for the purpose of recapturing value or proper disposal. More precisely, reverse logistics is the process of moving goods from their typical final destination for the purpose of capturing value, or proper disposal. Remanufacturing and refurbishing activities also may be included in the definition of reverse logistics." The reverse logistics process includes the management and the sale of surplus as well as returned equipment and machines from the hardware leasing business. Normally, logistics deal with events that bring the product towards the customer. In the case of reverse logistics, the resource goes at least one step back in the supply chain. For instance, goods move from the customer to the distributor or to the manufacturer.
When a manufacturer's product normally moves through the supply chain network, it is to reach the distributor or customer. Any process or management after the sale of the product involves reverse logistics. If the product is defective, the customer would return the product. The manufacturing firm would then have to organise shipping of the defective product, testing the product, dismantling, repairing, recycling or disposing the product. The product would travel in reverse through the supply chain network in order to retain any use from the defective product. The logistics for such matters is reverse logistics.
 See this diagram to understand

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"Relation between Exponential Function & Complex Number" by Ajit Mishra's Online Classroom

Relationship to exponential function and complex numbers

Euler's formula illustrated with the three dimensional helix, starting with the 2-D orthogonal components of the unit circle, sine and cosine (using θ = t ).
It can be shown from the series definitions that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:
 e^{i\theta} = \cos\theta + i\sin\theta. \,
This identity is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the complex plane, parametrized by e ix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent.
Euler's formula can also be used to derive some trigonometric identities, by writing sine and cosine as:
 \sin\theta = \frac{e^{i \theta} - e^{-i \theta}}{2i} \;
 \cos\theta = \frac{e^{i \theta} + e^{-i \theta}}{2} \;
Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:
\sin z = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} = \frac{e^{i z} - e^{-i z}}{2i}\, = \frac{\sinh \left( i z\right) }{i}
\cos z = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n)!}z^{2n} = \frac{e^{i z} + e^{-i z}}{2}\, = \cosh \left(i z\right)
where i 2 = −1. The sine and cosine defined by this are entire functions. Also, for purely real x,
\cos x = \operatorname{Re}(e^{i x}) \,
\sin x = \operatorname{Im}(e^{i x}) \,
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments.
\sin (x + iy) = \sin x \cosh y + i \cos x \sinh y,\,
\cos (x + iy) = \cos x \cosh y - i \sin x \sinh y.\,
This exhibits a deep relationship between the complex sine and cosine functions and their real (sin, cos) and hyperbolic real (sinh, cosh) counterparts.

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