Rahul
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 8h comment What is the sum of all the Fibonacci numbers from 1 to infinity. You seem to use the words "clearly" and "rigid" in... interesting ways. 8h comment Two decreasing, convex functions agreeing on a closed set For continuity one can show that $|f(x)-f(y)|\le|x-y|$ and that the construction of $g$ preserves this property. 10h comment Given a Matrix A, prove that 1/9A is an orthogonal matrix. An orthogonal matrix is one whose columns are orthonormal. 10h comment Two decreasing, convex functions agreeing on a closed set The complement of $B$ is a countable union of disjoint intervals. Let $f(x)=e^{-x}$. This fixes $g$ on $B$. Extend $g$ to the complement of $B$ via linear interpolation on the intervals. 20h comment Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it? There are several perfectly good formulations of "algorithm"... It sounds like you're essentially trying to define problem-solving as the task of finding an algorithm, is that right? 20h comment Angles Subtended by a Tetrahedron's Vertices and an Interior Point The natural generalization of the fact that the three edges of the triangle subtend a total angle of $2\pi$ radians is the fact that the four faces of the tetrahedron subtend a total solid angle of $4\pi$ steradians. 1d comment Please Explain Kuratowski Definition of Ordered Pairs What's important to understand is that the Kuratowski definition is merely one of many possible encodings of ordered pairs into the language of set theory. All we need from an encoding is to be able to decode it, that is, to recover $x$ and $y$ unambiguously from it. Can you see how to do that from the set $\{\{x\},\{x,y\}\}$? Can you see why you couldn't do that from just the set $\{x,y\}$? 1d comment Is isotropy preserved under uniaxial compression? Depends on the distribution. If the points were drawn from a Poisson process (i.e. each point independently of the others) then yes. In general no. For example, if the points were drawn from a blue-noise process (each point at least a certain distance away from all others) then this property will no longer hold afterwards. 2d comment Distribution of $aXa^T$ for normal distributed vector $a$ "If $X$ is symmetric matrix, then the above is a Wishart distribution." Are you sure about that? It's the distribution of the linear combination of $n$ chi-squared random variables, but that's not the same thing. Apr 30 revised Graphing/visualizing a complex parametric plot without using mathematica added 15 characters in body Apr 30 awarded Enlightened Apr 30 awarded Nice Answer Apr 29 comment Is there a simpler function with this shape? What do you mean, "out of syllabus"? Apr 25 comment What are some pairs of mathematically-important functions that differ only at a few points? Then let's get this out of the way: $\lceil x\rceil$ and $\lfloor x\rfloor+1$ Apr 25 comment How to prove a regular pentagon is formed by knotting a rectangular strip of paper? Here's a purely geometrical formulation of the problem: Let the vertices of the pentagon be labeled $PQRST$ (with $P$ being the one where edges $a$ and $e$ meet). Because $QT$ and $RS$ are opposite sides of a rectangular paper strip, they are parallel and a unit distance apart. The same is true for the pairs of line segments $PQ$ and $RT$, $QS$ and $TP$, and $QR$ and $PS$. Given this information we have to prove that the pentagon $PQRST$ is regular. Apr 25 comment Maximization of quadratic form on a sphere From $x^TAx+b^Tx\le \lambda_{\max}c+b^Tx$ how do you conclude that the optimum is $x=\sqrt cv_{\max}$? Apr 23 awarded Nice Answer Apr 21 comment Proof of orthogonality in the gradient descend algorithm. It depends on how you choose $\eta$. If it is a specified constant, then the orthogonality property is not true. If instead you choose $\eta$ to minimize $E(\mathbf w_{t+1})$ at each update, then orthogonality follows from the optimality condition for $E(\mathbf w_{t+1})$. Apr 19 comment Showing that a matrix is symmetric positive definite Please stop adding and removing a tag just to bump the question to the front page. If you want to bring more attention to your question, read What should I do if no one answers my question? Apr 17 comment why is the geometric mean less than the logarithmic mean? I think you mean $\frac{b-a}{\log b-\log a}$.