# Tag Info

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A good candidate would be elliptic curve cryptography. This is a direct practical application of finite fields, number theory, and other arithmetic geometry, that you would otherwise think have no purpose outside of pure mathematics.

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Perhaps a bit obscure -- finite topological spaces applied to digital analysis Perhaps a bit advanced -- spectral sequences applied to physics

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I would actually go with the quaternions $\mathbb{H}$. They form a 4 dimensional, associative division algebra. With the basis $i, j , k, 1$ which satisfies $$i^2 = j^2 = k^2 = ijk = -1$$ They first might seem not useful at all, until you notice they are easily created with matrices, what means they are easily computable. Quaternions are used to compute ...

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Tangentially on topic, I made video solutions to that practice test. They're organized here. My hope is that people will chime in with alternate (aka better) solutions in the comments. Hope it helps with the studies. Good luck!

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Here's one I like: almost all numbers fail to be computable. In particular, there are only countably many finite, terminating algorithms, and consequently there are only countably many real numbers that can be expressed (to arbitrary precision) via such algorithms. One example of a non-computable number is Chaitin's constant.

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Almost always still allows for an infinite yet uncountable amount to get through see Cantor Set for an example on how to build such sets. Another more trivial example would be the number is not an integer or the number is not of this "particular" countable set whatever that set may be. Other things that come to mind are things like Zigmondys theorem ...

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Solution 6 There are $n^2$ ordered pairs of computers. In $n$ of these pairs, both members are the same computer. All others come in pairs with reversed order. Thus there are $\frac{n^2-n}2=\frac{n(n-1)}2$ unordered pairs of different computers. Solution 7 Put the computers in a circle and count the number of connections that a computer has to computers ...

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by using Fourier series of $f(x)=1, x\in[0,1]$ $$1=\sum_{n=1}^\infty\frac{4}{(2n-1)\pi}\sin (2n-1)\pi x$$ integrate both sides when integration limits are $x=0 \rightarrow 1$ $$\int_{0}^{1}1.dx=\int_{0}^{1} \sum_{n=1}^\infty\frac{4}{(2n-1)\pi}\sin (2n-1)\pi x dx$$ $$1=\sum_{n=1}^\infty\frac{8}{(2n-1)^2\pi^2}$$ $$\sum_{n=1}^\infty\frac{1}{(2n-1)^2}=\frac{\pi^... 0 This sheet by Dave Renfro that I found online was beyond helpful! http://mathforum.org/kb/servlet/JiveServlet/download/206-1874348-6544585-538002/seq3.pdf 0 I tried to find the number of ways in which a number can be expressed in term of sum of two numbers and I ended up learning Partitions which showed me how everything can be expressed mathematically.... 0 You asked: What was the first bit of mathematics that made you realize that math is beautiful? For me, it was when I was 3 years old (possibly 4), contemplating my hands and fingers. I had the sudden epiphany that 5+5 absolutely had to equal 10 every time that you added them together -- not merely that they had done so repeatedly, mind you, but that ... 1 Following proof rely on this integral identity :$$\int_{a}^{1}\frac{\arccos x}{\sqrt{x^2-a^2}}\mathrm{d}x=-\frac{\pi}{2}\ln a\qquad ;\,a\in(0,1]$$We will prove it later on. Now, let's make a power series :$$\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\int_0^1\frac{1}{x}\sum_{n=1}^{\infty}\frac{x^n}{n}\,\mathrm{d}x=-\int_0^1\frac{\ln(1-x)}{x}\,\mathrm{d}x=...

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Let $X$ be an independent Laplace random variable with $X\sim L(0,1) = \frac12 \exp{(-|x|)}$, then its characteristic function : $$\varphi_X(t)=\mathbb{E}[e^{itX}]=\frac{1}{1+t^2} \newcommand{\var}[1]{\mathrm{var}\left[#1\right]}$$ By symmetry $\mathbb{E}[X]=0$ we write (generally) : $$\varphi_X(t)=\mathbb{E}[e^{itX}]=\mathbb{E}[1+itX-t^2X^2+\cdots\,]=1-\... 0 1). Suppose you want to build a 2-sphere (up to homeomorphism) using convex polygons cut out of paper. It is natural to want to glue two polygons only along their edges and the edges should be glued in pairs. Further, if two edges of two polygons are to be glued, they should be of the same length so we can overlap them. Knowing that the Euler ... 4 Feynman's trick, Frullani's theorem, symmetry tricks, Glasser's master theorem, the Laplace transform, Fourier (Legendre/Chebyshev) series expansions and the Euler beta function gave me at least the 80\% of my reputation points, but the day still has to come, to be prouder than this massacre through the residue theorem. I have just realized I forgot to ... 3 A structure of (unitary left) R-module over an abelian group G (written additively) is determined by a (unitary) ring homomorphism R\to\operatorname{End}(G), where \operatorname{End}(G) consists of the endomorphisms of G under the standard sum operation and map composition. To see why, suppose G is an R-module. For r\in R, define \lambda_r\... 1 In the particular example you gave, in fact you can make \mathbb{Z} / 5 \mathbb{Z} into a \mathbb{Z}[i]-algebra by "mapping i to 2" -- the specific formula for module multiplication would be:$$ (a+bi) \cdot c := (a+2b)c $$for a+bi \in \mathbb{Z}[i], c \in \mathbb{Z} / 5 \mathbb{Z}. (Here, the essential property of 2 which will make \alpha \cdot ... 7 The Bieberbach conjecture is an example of a lonely result in the sense that, while it generated much interest and almost competition, its ultimate solution by de Branges pretty much closed the field. It turned out that the result does not have many applications, and is a kind of a very high-level olympiad problem. 0 Here is a collection of the series with reciprocal binomial coefficients.$$\sum_{n=0}^\infty (-1)^n \left( \begin{matrix} 2n \\ n \end{matrix} \right)^{-1}=\frac{4}{5} \left(1-\frac{\sqrt{5}}{5} \ln \phi \right)\sum_{n=1}^\infty \frac{(-1)^n}{n} \left( \begin{matrix} 2n \\ n \end{matrix} \right)^{-1}=-\frac{2\sqrt{5}}{5} \ln \phi\sum_{n=1}^\...

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Although it is a bit messy, you may be interested in the following map: If anyone knows who made it, please let me know!

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Ramanujan's master theorem can be applied to a wide range of (sometimes extremely complicated) definite integrals, allowing them to be evaluated in less than a line of computations. The theorem states: If $f(x)$ has a series expansion of the form: $$f(x) = \sum_{k=0}^{\infty}(-1)^{k}\frac{\lambda(k)}{k!} x^k$$ then $$\int_0^{\infty}x^{s-1}f(x) dx = \... 8 My favourite example of this is @SangchulLee's solution to @VladimirReshetnikov's question, which asks to verify the correctness of the identity$$\int_0^{\infty} \frac{dx}{\sqrt[4]{7 + \cosh x}}= \frac{\sqrt[4]{6}}{3\sqrt{\pi}} \Gamma\left(\frac14\right)^2 .$$The other answers indicate the "toughness" of this integral, resorting to all sorts of special ... 5 There is a lot of 'tough looking' integrals which can be solved by various tricks, but usually it requires more than a few lines of proof. This is a really soft question, because 'tough looking' integral is a very subjective term (note that I use it instead of just 'tough' because I agree with Robert's comment). I suggest the book Inside Interesting ... 2 I believe that enumeration of finite groups of a given order is definitely among most wanted reproducible experiments. Here "enumeration" means providing complete and non-redundant list of groups, "complete" means that no groups are missing in this list, and "non-redundant" means that groups from this list are non-isomorphic pairwise. Guaranteeing these ... 0 Differential geometry is a basic tool in mathematical physics, in particular in mechanics. I strongly recommend the lecture of the book of Arnold "Mathematical Methods of the Classical mechanics", and the book of Penrose "The Road to Reality: A Complete Guide to the Laws of the Universe". Enstein theory of relativity (what is space-time) cannot even be ... 1 I think it's kinda fun to see how so many things elegantly follow from definitions or axioms: The axiom of regularity says that every nonempty set x has an element y that is disjoint from x:$$ \forall x : (x\neq \emptyset \rightarrow \exists y\in x : (y\cap x= \emptyset )) $$Conclude that: a \notin a a \notin b or b \notin a There is no ... 1 An example from Linguistics: Heller and Macris, in their book “Parametric Linguistics” arranged known sounds in a grid (matrix), in which a missing sound (blank cell) was obvious, and that is how this sound was discovered. (My memory is a little rusty on this, but it goes something like that.) 4 Wikipedia has a few here. Here is Graham himself explaining what Graham's number expresses. (It's an upper bound for an geometric problem). It has been constructed to make a proof work so it's not a arbitrary construct. However I'm not sure if you would still consider this as "used in real life". It's large enough that you can't express it in a regular ... 2 Suppose f(x,y,z) is a function that is: symmetric, i.e. f(x,y,z)=f(y,x,z)=f(x,z,y) and so on, positive, i.e. f(x,y,z)\ge0 for all x,y,z\ge0, and has power-law scaling with exponent p, that is, f(ax,ay,az)=a^pf(x,y,z) for all a\ge0. Suppose g(x,y,z) is a similar function but with exponent q\ne p. Then$$m(x,y,z)=\left(\frac{f(x,y,z)\,g(1,...

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Many results in the answers so far are examples from quantum physics. However there is also the example of quantum physics. As far as I remember, Planck wanted to explain the spectrum of black body emission by calculating a limit $\lim_{h\to 0}$ of a discretization. However, the results would only make sense if instead of letting $h\to 0$, he kept $h$ ...

1

Following on from @leonbloy's comment to the OP: Pythagoras's system of tuning is based on octaves, of frequency ratio 2:1, and perfect fifths, of frequency ratio 3:2. (Pythagoras observed that an interval whose frequency ratio is the ratio of small integers sounds pleasant.) Thus the frequency ratio of any interval in this system is the ratio of two 3-...

1

Let $\mathbf{x}$ denote a vector of positive real numbers (you can take it for your purposes to be 3 such numbers). Let $\mathbf{w}$ denote a vector of positive real numbers of the same length (i.e. 3-dimensional in this case), with $\sum_i w_i=1$; the simplest option is all $w_i$ being equal. Define $M_p:=\left(\sum_i w_i x_i^p\right)^{1/p}$ for $p\neq 0$. ...

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The Titius–Bode law (sometimes termed just Bode’s law) was an observation that the radius of the orbit of the $n\rm th$ planet in our solar system could be approximated by the formula $$r_n=(r_1+0.3\times 2^{(n-2)})\rm AU$$ for $n>1$, where “AU” represents an Astronomical Unit; i.e., the radius of Earth’s orbit (i.e., $r_3$). Setting $r_1$ to the ...

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There have been many attempts to prove Euclid's Parallel Postulate, which for around 2000 years was stumping many Mathematicians. Eventually it took Gauss to completely redevelop the notion of some of the equivalent properties in his development of hyperbolic geometry. Whilst it is true to say that Gauss' endeavours were not found by accident; it is true to ...

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Bell's theorem on the foundations of quantum mechanics showed that not all philosophical questions are impervious to experiment, to the extreme surprise of pretty much every physicist on Earth. (It also showed that Einstein was soundly wrong, which some people might also find surprising.)

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Arago's spot is a classical (and classic) example of a beautiful mathematical theory anticipating a beautiful physical fact. Briefly, the story goes like this: Back in the 1800's, scientists were debating whether light was a particle or a wave. Following some convincing experiments by Young showing wave-like properties of light, Fresnel developed a ...

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The memristor, the fourth passive electronic component (to accompany the resistor, capacitor, and inductor), was predicted by Leon Chua in 1971. An anomalous signal found by engineers in HP Labs in 2008 was, after much consternation, eventually attributed to the discovery of the memristor. The prediction follows from the relationship between voltage, charge,...

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Here's a rather different example which came up recently (see the Journal of Recreational Mathematics): A couple of mathematicians were studying juggling. They came up with a way to encode the 'ball catch' patterns as simple numeric sequences. Then they derived the sequences for all known juggling patterns, and inferred from them a set of rules governing ...

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I'm not sure if this is the sort of thing that you're looking for, but colimit (and limit) arguments prove really useful when dealing with $R$-modules. In particular we can think of direct limits (i.e. colimits over a directed poset) as a union (in some sense that can be made precise). As an example, working in $\mathsf{Ab}=R\hbox{-}\mathsf{mod}$, we have ...

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John von Neumann discovered some of the fundamentals of molecular biology back in the 1940s long before the field of molecular biology even existed. When von Neumann was developing his theory on universal constructors (UCs), machines that can build any possible physical structure including making copies of itself, he stumbled on a generic problem. The ...

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I agree with some of the comments that the likely basis for this fuzzy memory is the Dirac "prediction" of positrons (the first known instance of anti-matter). However, slightly before Dirac's 1926 publication of a relativistic wave equation with negative energy states (which he didn't initially believe were "physical", so calling it a "prediction" was only ...

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The Fermi-Pasta-Ulam numerical experiments in the 1950's that led to theory of integrable systems. One day the computer simulation was accidentally left running longer than intended, showing that a nonlinear wave system almost returned to its original state instead of thermalizing. The theory that developed from this is enormous and applications include ...

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Berry's Phase is a good example of mathematics uncovering new physics. In particular, a derivation in quantum mechanics assumed a one-dimensional domain in an integration. If the parameter space is higher-dimensional then the parameter domain can have nontrivial topology which ultimately leads to a nontrivial integral. This integral implies we can manipulate ...

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I believe the story mentioned in the question is the story of the $\Omega^-$ particle, conjectured to exist by Gell-Mann and Ne'eman (see https://en.wikipedia.org/wiki/Eightfold_Way_(physics) and https://en.wikipedia.org/wiki/Yuval_Ne%27eman) as part of a representation-theoretic approach to quantum mechanics, and eventually discovered by a team at ...

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The planet Neptune's discovery was an example of something similar to this. It was known that Newtons's Equations gave the wrong description of the motion of Uranus and Mercury. Urbain Le Verrier sat down and tried to see what would happen if we assumed that the equations were right and the universe was wrong. He set up a complicated system of equations that ...

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Quasicrystals. Aperiodic tilings of the plane and space were discovered by mathematicians, starting from Robert Berger's work on Wang tiles in the 1960's. Physical materials exhibiting these properties were found in the 1980's by Dan Shechtman, who won the Nobel Prize for Chemistry in 2011 for this work.

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Differential calculus was being developed by Leibniz and Newton in the 17-18th centuries at roughly the same time as Newton formulated his famous equations for mass, force and acceleration governing mechanical motion. I wonder what such laws would have looked like without having any differential and integral calculus available!

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Dirac explains here how special relativity led to the Dirac equations for the electron which predicts its spin, magnetic moment and the existence of the positron.

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Write down Maxwell's equations in a vacuum: $$\nabla \cdot \vec{E}=0$$ $$\nabla \cdot \vec{B}=0$$ $$\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$$ $$\nabla \times \vec{B}=\mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$$ Note the vector identity $\nabla\times(\nabla \times \vec{X})=\nabla(\nabla\cdot\vec{X})-\nabla^2\vec{X}$. Apply this ...

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Kepler's attempted to match the orbits of the planets to a nested arrangement of platonic solids. Eventually, his data led him to the mathematics of Kepler's Laws. Kepler wasn't impressed by his three laws, but Newton found them in his papers. (from my post elsewhere)

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