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Copy file name to clipboardExpand all lines: _posts/2018-01-01-geometry-lagrange-multiplier.md
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The Lagrangian Multipliers is an important strategy in mathematical optimization to find the local maxima or minima of a function subject to equality constraints.
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A lot of interesting stuff happens here at the interection point. First, you can easily verify that both the curves have a common tangent equations. Nice. Next, the point has two normals acting on it, in the opposite directions. This is what we care about.
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<fontcolor="blue">Normals.</font>
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<fontcolor="red">Normals.</font>
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Let's focus on normals. As we know the normal is given by the gradient of the curve, (since the gradient of a function is perpendicular to the contour lines)
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$$ \nabla_{x,y} f = \Bigl( \frac{\partial{f}}{\partial{x}}, \frac{\partial{f}}{\partial{y}} \Bigr) $$
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So, we never know whether these normal are balanced. It depends on the curve and elipse equations. Thus we introduce a scalar, non-zero constant, <fontcolor='blue'> $\lambda$ </font> (to balance the point of interection, where the maxima or minima lies).
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So, we never know whether these normal are balanced. It depends on the curve and elipse equations. Thus we introduce a scalar, non-zero constant, <fontcolor='red'> $\lambda$ </font> (to balance the point of interection, where the maxima or minima lies).
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<center>
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<divclass="boxx">
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$$ \nabla_{x,y} f = \lambda \nabla_{x,y} g $$
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</div>
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</center>
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This constant is called the <fontcolor='blue'>Lagrange multiplier.</font>
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This constant is called the <fontcolor='red'>Lagrange multiplier.</font>
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We continue solving this equation,
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Note, setting $\lambda = 0$ is a solution if $f(x,y)$ is at the level (horizontal) regardless of $g(x,y)$.
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Let us define another function $\mathscr{L}(x,y,\lambda) \equiv f(x,y) + \lambda g(x,y)$ to include all the conditions into one equation and solve for $\nabla_{x,y,\lambda}\mathscr{L}(x,y,\lambda) = 0.$ [ this function is also called the <fontcolor="blue">Lagrangian Function</font>, also note that $\lambda$ can be positive or negative ]
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Let us define another function $\mathscr{L}(x,y,\lambda) \equiv f(x,y) + \lambda g(x,y)$ to include all the conditions into one equation and solve for $\nabla_{x,y,\lambda}\mathscr{L}(x,y,\lambda) = 0.$ [ this function is also called the <fontcolor="red">Lagrangian Function</font>, also note that $\lambda$ can be positive or negative ]
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As there are three unknowns, $x,y$ and $\lambda$, would need to solve three equations.
References include <fontcolor="blue">my father's lecture notes on numerical methods and FEM at NITC (riverbed) </font>, <ahref="https://en.wikipedia.org/wiki/Lagrange_multiplier">Wikipedia</a>, <ahref="https://math.stackexchange.com">math.stackexchange</a>, <ahref="https://www.microsoft.com/en-us/research/people/cmbishop/">Pattern Recognition and Machine Learning, Bishop</a> and <ahref="https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-13-lagrange-multipliers/">MIT OWC Multivariable Calculus. (necklace intuition)</a>
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References include my father's lecture notes on numerical methods and FEM at NITC (riverbed), <ahref="https://en.wikipedia.org/wiki/Lagrange_multiplier">Wikipedia</a>, <ahref="https://math.stackexchange.com">math.stackexchange</a>, <ahref="https://www.microsoft.com/en-us/research/people/cmbishop/">Pattern Recognition and Machine Learning, Bishop</a> and <ahref="https://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-13-lagrange-multipliers/">MIT OWC Multivariable Calculus. (necklace intuition)</a>
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All images are mine and are licensed under CC-BY-4.0. Softwares used include MATLAB, OSX Grapher, Preview.app and Adobe illustrator.
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All images are mine and are licensed under CC-BY-4.0. Softwares used include MATLAB, OSX Grapherand Adobe Illustrator.
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<hr>
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I hope you had as much fun reading than I had writing this up!
Copy file name to clipboardExpand all lines: _posts/2018-03-09-school-resolution.md
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My CRISPR slide.</div>
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</div>
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**Observations:** Visualizing protein structure using AR would be pretty cool. <sup><ahref="#first">1</a></sup>
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**Observations:** Visualizing protein structure using AR would be pretty cool.
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---
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Explaining this kids a simple contraint problem! Copyrights: (NAE Bridge/Lars Blackmore/SpaceX) </div>
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</div>
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### Sorcery.
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---
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<imgsrc="/assets/school/me.png"width="50%">
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<divclass="thecap"style="text-align:center">
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When you allow kids to draw on your photo. Sorcery!</div>
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</div>
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haha, someone started writing the formular for angular velocity!!?, someone drew horns, someone a hat, someone gave me muscles!, I have no clue what that red thing is on the chair!
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The best feeling I got is when someone wrote "come again!".
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I will!
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<hr>
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#### EDITS + UPDATES
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<divid="first">
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<b>1:</b>
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I tried to visualize a DNA strand using Vuforia and Unity and results are great! The only bad review from kids was the 3D model which I had freely downloaded. :(
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I hope I'll try to make my own blender, perhaps model and open source this!
Copy file name to clipboardExpand all lines: _posts/2019-09-10-Visions.md
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**Note:** This is [a comment I wrote on Reddit](https://www.reddit.com/r/Indian_Academia/comments/cqttnh/best_skillset_to_invest_in_now/ewzmozd/). As I'm right now going extremely slow on an original, quality post -- I hope this will suffice the need reading something new in my blog.
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**Note:** This is [a comment I wrote on Reddit](https://www.reddit.com/r/Indian_Academia/comments/cqttnh/best_skillset_to_invest_in_now/ewzmozd/) as right now I'm going extremely slow on an original, quality post -- I hope this will suffice the need to read something new in my blog. I expect the new post to be out around December last week.
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