This blog will be more alive for 2012!

Comments: 0January 8th, 2012 by

Whew, after a lapse of nearly one year, we will be back to enliven our WordPress Blogs hosted at optimal-learning-systems.org, namely
Life, Research, Education http://optimal-learning-systems.org/wp and
Engineering Problem Solvers http://optimal-learning-systems.org/eps

Posted in Optimal-Learning-Systems

Apologies! Our latex addin has been changed.

Comments: 0September 25th, 2011 by

Our main blogging site, Digital Explorations, has no problems presenting Latex formulas, but this alternate site encounters Latex problems from time to time. So today, we changed our Latex addin and we are sorry that former good looking formulas are not presented properly.

We apologize for any inconvenience, and we promise to look into this matter seriously. Not only our readers but also the main author of this bog now suffers.
Be patient.

Posted in Uncategorized

Part 7: Conics, equation of tangent lines to conics

Comments: 0September 25th, 2011 by

Given the general conic equation Ax^2 + Bxy + C y^2 + Dx + Ey + F=0, the equation of the tangent
line passing through the point (x_1, y_1) is given by

    \[(Ax_1 +\frac{By_1}{4} + \frac{D}{2})x + (Cy_1 \frac{Bx_1}{4}+ \frac{E}{2}) y +\]

    \[\frac{Bx_1y_1}{4} + \frac{Dx_1}{2}+\frac{Ey_1}{2}+F = 0.\]

The slope m is equal to

    \[m = \frac{Ax_1+By_14 + D/2}{y_1+Bx_1/4 + E/2}\]

    \[\begin{array}{lll} \hline \mbox{Conic} & \mbox{Equation} & \mbox{Equation of Tangents} \\ \hline \mbox{circle} & x^2 + y^2 = r^2 & y = mx + \pm r \sqrt{1 + m^2} \\ \mbox{ellipse} & (x/a)^2 + (y/b)^2 = 1 & y = mx \pm \sqrt{a^2m^2+ b^2}\\ \mbox{parabola} & y^2 = 4ax &ny = mx + a/m\\ \mbox{hyperbola} & (x/a)^2 - (y/b)^2 = 1 & y = mx \pm \sqrt{a^2m^2- b^2} \\ \mbox{hyperbols} & 2xy = a^2 & y = mx + a \sqrt{-2m}\\ \hline \end{array}\]

Posted in Geometry

Part 6: Conics, the Hyperbola

Comments: 0September 25th, 2011 by

The hyperbola is the locus of a set of points such that the difference of its
distances from to fixed points is a constant.

Case 1. Horizontal axis, y = k.

    \[\begin{array}{ll} \mbox{Conditions} & AC < 0, C < A\\ \mbox{Equation} & \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1 \\ & a^2 = G/A, b^2 = -G/C, G = \frac{E^2}{4C}+ \frac{D^2}{4A} -F\\ \mbox{Assymp. Intersection} & (h, k)=(\frac{-D}{2A}, \frac{-E}{2C}) \\ \mbox{Foci} & (h\pm \sqrt{a^2+b^2}. k)\\ \mbox{Latus Rectum} & \frac{2b^2}{a} \\ \mbox{Dist. (h,k) to a focus} & d = \sqrt{a^2 + b^2} \\ \mbox{Vertices} &(h+a, k), (h-a, k)\\ \mbox{x-Intercepts} &h \pm \frac{a}{b}\sqrt{b^2 + k^2}\\ \mbox{Assymptotes} & y = \pm\frac{b}{a}x +k\\ \end{array}\]

Case 2. Vertical axis, x = h

    \[\begin{array}{ll} \mbox{Conditions} & AC < 0, C < A\\ \mbox{Equation} & \frac{(y-h)^2}{a^2} - \frac{(x-h)^2}{b^2} =1 \\ & a^2 = G/C, b^2 = -G/A, G = \frac{E^2}{4C}+ \frac{D^2}{4A} -F\\ \mbox{Assymp. Intersection} & (h, k)=(\frac{-D}{2A}, \frac{-E}{2C}) \\ \mbox{Y-Intercepts} &y= k \pm \frac{a}{b}\sqrt{a^2 + h^2}\\ \mbox{Assymptotes} & y = \pm\frac{b}{a}x +k\\ \end{array}\]

Case 3. Special Rectangular hyperbola

    \[\begin{array}{ll} \mbox{Conic Equation} & Ax +By + xy + C = 0\\ \mbox{Equation} & (x-h) (y-k) = 0\\ & k = -A, h = -B, c = AB -Ca^2 \\ \mbox{Dist. (h, k) to vertex} & \sqrt{|2c|}\\ \mbox{Dist. (h, k) to foci} & 2\sqrt{|c|}\\ \mbox{Assymptotes} & x = h, y = k\\ \end{array}\]

Posted in Geometry, Uncategorized

Part 5: Conics, the Ellipse

Comments: 0September 25th, 2011 by

The ellipse is the locus of a set of points such that the sum of its distances from two fixed points,
called the foci, is a constant.

\noindent Case 1. Major axis parallel to X-axis, y = k

    \[\begin{array}{ll} \mbox{Conditions} & AC > 0, A\ne C, C > A\\ \mbox{Equation} & \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} =1 \\ & a^2 = G/A, b^2 = G/C, G = \frac{E^2}{4C}+ \frac{D^2}{4A} -F\\ \mbox{Center} & (h, k)=(\frac{-D}{2A}, \frac{-E}{2C}) \\ \mbox{Focus}_1 & (h+\sqrt{a^2-b^2},k)\\ \mbox{Focus}_2 & (h-\sqrt{a^2-b^2},k)\\ \mbox{Major Axis} &y = k\\ \mbox{Minor Axis} &x = h\\ \mbox{Latus Rectum} &2b^2/a\\ \mbox{Eccentricity} & e = \frac{\sqrt{a^2 - b^2}}{a}\\ \mbox{Distance, center to foci} & d = \sqrt{a^2 - b^2}\\ \mbox{Tangent line at }(x_1, y_1) &y = \frac{-b^2}{a^2}\frac{(x_1 - h)(x-x_1)}{y_1-k} + y_1\\ \end{array}\]

\noindent Case 2. Major axis parallel to Y-axis, x = h

    \[\begin{array}{ll} \mbox{Conditions} & AC > 0, A\ne C, C < A\\ \mbox{Equation} & \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} =1 \\ & a^2 = G/C, b^2 = G/A, G = \frac{E^2}{4C}+ \frac{D^2}{4A} -F\\ \mbox{Center} & (h, k)=(\frac{-D}{2A}, \frac{-E}{2C}) \\ \mbox{Focus}_1 & (h, k+\sqrt{a^2-b^2})\\ \mbox{Focus}_2 & (h, k-\sqrt{a^2-b^2})\\ \mbox{Major Axis} &x = h\\ \mbox{Minor Axis} &y = k\\ \end{array}\]

Posted in Geometry, Uncategorized

Part 4: Conics, the Parabola

Comments: 0September 25th, 2011 by

The parabola is the locus of a set of points such that the diatace measured perpindicularly from a given line,
called the directrix is equal to the distance to a given pont called the focus.

Case 1. Vertical axis, x = h

    \[\begin{array}{ll} \mbox{Conditions} & A\ne 0, C = 0 \\ \mbox{Equation} & (x -h)^2 = 4p(y-k) \\ \mbox{Vertex } & (h, k)=(\frac{-D}{2A}, \frac{D^2-4AF}{4AE}) \\ \mbox{Vertex radius} & 2 p\\ \mbox{Latus rectum} & 4p\\ & \mbox{whre } p = \frac{-E}{4A}\\ \mbox{Focus} & (h, p+k)\\ \mbox{Directrix} & y = -p + k\\ & \mbox{if }p > 0, \mbox{curve opens upward}\\ & \mbox{if }p < 0, \mbox{curve opens downward}\\ \end{array}\]

Case 2. Horizontal axis,y = k

    \[\begin{array}{ll} \mbox{Conditions} & A\ne 0, C = 0 \\ \mbox{Equation} & (y-k)^2 = 4p(x-h) \\ \mbox{Vertex } & (h, k)=(\frac{E^2 -4CF}{4CD}, \frac{-E}{2C}) \\ \mbox{Vertex radius} & 2 p\\ \mbox{Latus rectum} & 4p\\ &\mbox{where } p = \frac{-D}{4C}\\ \mbox{Focus} & (p+h, k)\\ \mbox{Directrix} & x = -p + h\\ & \mbox{if }p > 0, \mbox{curve opens to the right}\\ & \mbox{if }p < 0, \mbox{curve opens to the left}\\ \end{array}\]

Posted in Geometry

Part 3: Conics, the Circle

Comments: 0September 25th, 2011 by

A circle is the locus of a set of points equidistant from a given point called the center.

Let T = D^2 - E^2 - 4 AF. Then,

    \[T:\begin{cases} >0 ,\mbox{a circle}\\ = 0 ,\mbox{a point}\\ < 0 ,\mbox{no locus}\\ \end{cases}\]

Equations for a circle:

    \[\begin{array}{ll} \mbox{Condition} & A = C \\ \mbox{Simplified eqn} & (x-h)^2 + (y -k)^2 = r^2\\ \mbox{Center} & (h, k)= (-\frac{D}{2A}, -\frac{E}{2A}) \\ \mbox{Radius} & r =\frac{1}{2A}\sqrt{T}\\ \mbox{tangent line at $(x_1, y_1)$} & y = \frac{r^2 - (x-h)(x_1 -h)}{y_1 - k}+ k\\ \end{array}\]

Posted in Geometry

Part 2: Conics and Rotation of Axes

Comments: 0September 25th, 2011 by

Let X’, Y’ be new axes obtained by rotating the X and Y axes by \alpha degrees from its original position.
Trasnformations between (x, y) in X, Y coordinates and (x’, y`) in X’,Y’ coordinates are given by
the matrix equations:

    \[ \left(\begin{array}{l} x \\ y\\ \end{array}\right)= \left(\begin{array}{ll} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \\ \end{array}\right) \left(\begin{array}{l} x' \\ y'\\ \end{array}\right) \]

    \[\left(\begin{array}{l} x' \\ y'\\ \end{array}\right)= \left(\begin{array}{ll} \cos(\alpha) & \sin(\alpha) \\ \cos(\alpha) & -\sin(\alpha) \\ \end{array}\right) \left(\begin{array}{l} x \\ y\\ \end{array}\right)\]

The transformed equation in X’,Y’ coordinates is given by

A'x'^2 + B'x'y' + C'y'^2 + D' x' + E' y' + F' = 0

where

    \[\begin{array}{ll} F' &= F\\ E' &= -D \sin(\alpha) + E \cos(\alpha) \\ D' &= D \cos(\alpha) + E \sin(\alpha) \\ C' &= A\sin^2(\alpha) -(B/2) \sin(2\alpha) + C \cos^2(\alpha) \\ B' &= (-A + C) sin(2\alpha) + B cos (2\alpha) \\ A' &= A cos^2(\alpha) + (B/2) sin(2\alpha) + C\sin^2 (\alpha) \\ \end{array}\]

If the angle of rotation of axes , \alpha has the value \alpha = (1/2) tan^{-1}\frac{B}{A-C}, then
B’ = 0 and the term containing x’y’ drops out!

We shall deal from hereon with the reduced second order equation

    \[Ax^2 + Cy^2 + Dx + Ey + F = 0\]

Posted in Uncategorized

Part 1: Formulas for Conics

Comments: 0September 25th, 2011 by

\section{Geometry of the General Second Order Polynomial Equation in x and y.}

Consider Ax^2 + Bxy + Cy^2 + D x + E y + F = 0, which we shall call the conic equation. This equation represents a conic or lines or points
depending on the values of the following computed parameters:

Let \Delta = \left| \begin{array}{lll} 2A &B & D \\ B & 2C & E\\ D & E & 2F\\ \end{array} \right|,

Q = \sqrt{B^2 - 4AC},

and T = D^2 + E^2 -4 (A+C) F.

The following table shows what type of conic exists when \Delta \ne 0, and any degeneracies when
\Delta = 0.

    \[\begin{tabular}{lll}\hline Conditions & Conic & Degeneracies\\ \hline $Q < 0, A \cdot \Delta <0, A\ne C $ &Ellipse & point \\ $Q < 0, A \cdot \Delta <0, A= C$ &Circle & point\\ $Q = 0$ & Parabola & 2 lines if ($T > 0$)\\ & & 1 line if $T=0$\\ $Q> 0$ & Hyperbola & 2 intersecting lines\\ \hline \end{tabular} \]

No conic is formed when either Q=0, \Delta = 0, T <0 or when
Q< 0, \Delta \ne 0, A > C.

Posted in Geometry

Can you believe Firefox will not survive?

Comments: 0March 26th, 2011 by

Just read Why Internet Explorer will survive and Firefox won’t and it made me unhappy. Seems that Ed Bott is claiming that the pace of version releases of the various browsers is showing that Microsoft IE and Chrome are healthy and will survive in the future whereas Firefox will not! This is a mistaken conclusion.

First, anyone betting on using a single browser is subject to bugs which may be focused on that browser. It is the reason I keep three browsers in my Ubuntu box: Chrome, Firefox, and Opera.

Second the claim that Google Apps and soon to be released web versions of MS Office is better than Firefox’s extensions missed the simple fact that Google Apps can also run in Firefox! More, some of the best developers of Chrome were formerly working in Firefox. Am pretty sure that they still have their hearts beating for Firefox NOT to GO AWAY.

Third and most importantly, IE with supporting OS is an insecure ecosystem. Most hackers target IE and still do. It is the reason why some companies shifts to Firefox and LInux to avoid paying for the required antivirus software which in my opinion is senseless expense. Antivirus software require upgrades and the maintenance costs may in fact increase percentage-wise of business operating expenses.

It is up to the reader who may be in position to recommend software purchase in a company. Firefox is free and so is Chrome. The best advise is not to put ones eggs in one basket. People were saying Opera will die, it is still alive and in fact introduced most of the innovations! Now we have people saying that Firefox will not survive, and for how long? five years? ten years? a hundred years? The ZDnet author does not say for how long. So consider it as a strike against open source software, and a possible FUD feeder.

Be vocal, blog about your stand, send in your comments. Lets show that Firefox deserves to survive.

Posted in Browsers

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