Part 3 of a six-part series of summaries and adaptations from Terence Tao's book "Structure and Randomness". [1.9] Amplification, arbitrage, and the tensor power trickGiven an inequality $f(x) \leq g(x)$ with imbalances in symmetry between the left-hand side (LHS) and right-hand side (RHS), amplification is a mathematical trick that can exploit that imbalance to derive a stronger inequality (i.e. the LHS and RHS are closer). As for why mathematicians might need such a technique, see "Why do we need strong inequalities?" below.
Consider some transformations $T$ that change $x$ such that $g$, but not $f$, is "symmetric" relative to $T$. That is, $f(T(x)) \leq g(T(x)) = g(x)$. Then we can choose $T$ to maximize the LHS $f(T(x))$ and "tighten" the inequality. Let's illustrate this trick by applying it to prove the Cauchy-Schwarz Inequality (actually, the special case of the familiar $n$-dimensional space $\mathbb{R}^n$): $$\lvert\mathbf{v} \cdot \mathbf{w}\rvert \leq \lVert\mathbf{v}\rVert\lVert\mathbf{w}\rVert \text{ for all } \mathbf{v}, \mathbf{w} \in \mathbb{R}^n \tag{3.1}$$
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[1.2] Compressed sensing and single-pixel camerasThis'll be quick; the only part of the original blog post that I understood comfortably was the brief (and slightly inaccurate) explanation of traditional image compression. It's how image formats like JPEG can reduce the memory space needed by an image file drastically while losing only a bit of quality. [1.1] Quantum mechanics and Tomb RaiderTao uses the analogy of the game Tomb Raider as a model to give some intuition for the reasons behind the "weird" consequences of quantum mechanics (QM), in particular the so-called "many worlds interpretation". The game consists of two worlds:
(At this point, Tao apologizes for the violent analogy. Well, I've added a little drama... and fixed a loophole.) Each save point before a lethal puzzle causes Lara's world (from her point of view) to split into many possible "developments", some of which involve her failure to solve the puzzle and thus death, while others have her survive. Figure 1 illustrates a sample puzzle: A tomb whose only exit a wooden trapdoor on the floor that leads to an underground passage. The tomb will collapse in, say, five minutes, and Lara ("L") must escape through the passageway, but the trapdoor is locked. The Player (smiley face) helps by creating a save point.
To any sentient being, I think the most important thing is to doubt yourself. No matter how sure you are of yourself, you must always bear in mind that you can be wrong. In fact, you have been wrong on many (most?) occasions, and you will be wrong for many times to come. This is a serious and crucial realization, because most of our actions arise from our conscious and unconscious beliefs. Our horizons are bounded. The thoughts and opinions we can have are limited by our cultural experience (e.g. socially accepted norms). Our confidence in our beliefs is bolstered by the fact that we can only operate inside our tiny little box, which makes everything in the box so familiar and comfy and, you know, "obviously true". However, to make good decisions in life, we must accumulate much "experience" to guide us. That "data" comes in the form of a diverse array of opinions and beliefs, which may not agree with each other. We must expand our shoebox of a mind to a warehouse of different perspectives which allow us to appreciate things and happenings more completely. This will aid our decision-making process by allowing us to evaluate the consequences of choices more accurately. Hence, to grow as a person, we should:
Is it possible to do mathematics WITHOUT background knowledge?... Is it humanly possible to do mathematics on own without research or is the information content too much to discover identities, or methods of proofs on one's own?... But, how to get started then? How to discover mathematics on my own? I wouldn't say that I've been "discovering mathematics", but I definitely have been playing around with many of my original math research ideas, some of which I am happy with. So I introspected a little about my amateur math explorations and how my experience can help people who ask this question. Here is my slightly edited response.
I'm completely unfamiliar with art, but I've had this interesting thought lately. First of all, Art is more than the sum of its parts. (Statement 1) There are many ways how an art piece can be more than its constituents. The visual arts give a few easy examples, where emotion can be injected into a medium (the "sum of its parts"). The emotion, message, or whatever special qualities that qualify something as art, was not present in the medium itself, yet it present in its final form.
I was proud of this idea, as it was probably the best early research idea I've come up with.
I became very interested in matrices and linear algebra after reading a paper on modelling origami using rotation and translation matrices (Belcastro and Hull, 2012). I began to play with expressing all kinds of things as matrices and seeing what "meanings" matrix operations had in those contexts. Somehow complex numbers cropped up, and I decided that they were a good candidate for this "interpretation" because multiplying by a complex number meant a rotation and dilation of the complex plane - or an "amplitwist" (Needham, 1996). So I represented a complex number as a rotation matrix together with a scaling factor. Given a complex number $z = r\mathrm{e}^{\mathrm{i}\theta}$, let $\mathbf{Z} = r\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$ Some interesting properties follow immediately from this correspondence: Another old idea. Given a $2 \times 2$ matrix $\mathbf{A}$,
$$\det(\mathbf{A}) = \begin{bmatrix}1&0\end{bmatrix}\mathbf{A}^T\begin{bmatrix}0&1\\-1&0\end{bmatrix}\mathbf{A}\begin{bmatrix}0\\1\end{bmatrix}.$$ Prove this and extend it to general $n \times n$ matrices. Can you go further than that? |
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