New answers tagged

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$$ \int_0^\infty \frac{1}{1+x^{10}}dx=\frac{\phi\pi}{5} $$


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This one is a bit messy. $$ \int_0^\infty \frac{1}{(\sqrt5^x)^{2^{-(\sqrt5-1)}}+\sqrt5-1}dx=2^{\phi^{-3}}\cdot\phi $$


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Consider the sequence $1,2,2,3,3,4,4,4,...$ where $a_1=1,a_{n+1}\in{{a_n,a_n+1}}$, and $a_n$ is the number of times $n$ occurs in the sequence. Then if we assume that $a_n$ grows asymptotically as $\alpha n^\beta$, we get $\alpha=\phi^{1/{\phi^2}}$ $\beta=1/\phi$. I saw this is a textbook problem on asymptotic analysis. It turns out that for all $n$ ...


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$$ F_0=0, F_1=1 ; F_{n+1}=F_{n-1}+F_n; Fibonacc- numbers $$ $$\zeta(s)$$ is the zeta function $$ \prod_{n=1}^{\infty}\left[(-1)^{n+1}\phi F_n+(-1)^nF_{n+1}\right]^{n^{-(s+1)}}=\phi^{-\zeta(s)} $$


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Here is another one $$ \int_0^\infty \frac{1}{5^{\frac{x}{4}}+5^{\frac{1}{2}}-5^0}dx=\phi $$


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$$ \int_0^1 \frac{1+x^8}{1+x^{10}}dx=\frac{\pi}{\phi^5-8} $$


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You can do this with simple arithmetic You have 2000 elements The largest list you'll get after splitting is then 1000 elements Split again and get 500 Split again and get 250 Split again and get 125 Split again and get 63 (worst case) Split again and get 32 Split again and get 16 Split again and get 8 Then 4 Then 2 Then you compare once more And maybe ...


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Topology: Why you can't turn your shirt inside out while wearing it.


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How about the Liouville function, $$\lambda(n)=(-1)^{\Omega(n)},$$ versus the Möbius function, defined as $$\mu(n) = \begin{cases} (-1)^{\Omega(n)} && n \text{ is not squarefree} \\ 0 && n \text{ is squarefree} \end{cases}$$ They differ on infinitely many points, but they are also the same on infinitely many points.


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Here is a neat proof from Qiaochu Yuan's answer to this question: If $L$ is diagonalizable with eigenvalues $\lambda_1, \dots \lambda_n$, then it's clear that $(L - \lambda_1) \dots (L - \lambda_n) = 0$, which is the Cayley-Hamilton theorem for $L$. But the Cayley-Hamilton theorem is a "continuous" fact: for an $n \times n$ matrix it asserts that $n^2$ ...


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My favorite : let $k$ be your ground field, and let $A = k[X_{ij}]_{1\leqslant i,j\leqslant n}$ be the ring of polynomials in $n^2$ indeterminates over $k$, and $K = Frac(A)$. Then put $M = (X_{ij})_{ij}\in M_n(A)$ the "generic matrix". For any $N=(a_{ij})_{ij}\in M_n(k)$, there is a unique $k$-algebra morphism $\varphi_N:A\to k$ defined by ...


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$$\int_0^{\infty} \frac{1}{(1+x^\phi)^\phi}=1$$


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Divergent Series can be visual: from the Wikipedia showing that $(1-1+1-1+\dots)^2=1-2+3-4+\dots$


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Polynomials can describe geometric objects In high school we learn that some low order polynomials can describe geometric shapes: Basic shapes we all recognize ( as intro ) $$\begin{array}{rr}y=kx+m& (line)\\x^2+y^2 = r^2 & (circle)\\y = x^2+ax+b &( parabola)\end{array}$$ Cool properties consider the rotation ...


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The review of Perlman and Chaudhuri, Reversing the Stein effect, Statist. Sci. 27 (2012), no. 1, 135–143, MR2953500, says (in part), "The authors make their point in a rather unorthodox way using a made-up Star Trek episode. The starship Enterprise is lost in space and unable to move. To be rescued it is necessary to send a probe close enough to the ...


1

This is moderately common in Theoretical Computer Science. A lot of problems and algorithms in that field are posed and discussed in terms of mini stories, even in technical papers. In his celebrated paper introducing what is known as the "Arthur-Merlin Protocols", Laszlo Babai writes as if telling a fable about Camelot. Leslie Lamport, the founder of the ...


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One piece that comes to mind is Gromov's "Metric structures for Riemannian and Non-riemannian spaces." Everything from the numbering - a Gromov hallmark - to some of the colorful yet sometimes remarkably illuminating language - a wonderful example is 1.25.1/2, "If one feels disgusted by the spineless flexibility of arc-wise isometric maps, ..." - is rather ...


2

For outer measure, if $A \subseteq B$ and $m^*(A)<\infty$, then $m^*(B\setminus A) \geq m^*(B)-m^*(A)$ because $A \cup (B\setminus A)=B$, so $m^*(B) \leq m^*(A)+m^*(B\setminus A)$ by the property of outer measure that you mentioned.


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Desmos Graphing Calculator, is, yes a graphing calculator, but it is also really useful for geometric drawing. On their homepage they illustrate several examples of how versatile the tool is for drawing. The main drawback of the program is that you need to know the algebraic (Cartesian, Parametric, or Polar) equations behind your shape to draw it. On the ...


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The first time I was fascinated by mathematics was when I read Christian Goldbach's conjecture. From that day onwards, I am trying to decode the mystery of primes, which seem to be simple at first sight but are actually very difficult to understand. That's the beauty of mathematics.


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If you are referring to geometric sketch, I like to use Dr. Geo, a free software of mine. It is an interactive geometry and programming software. You can make very easily construction with the mouse or more complex iterative one with its programming API. Interactive geometric sketch designed with mouse and clic


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I'm quite partial to Scott Aaronson's 'Quantum computing since Democritus'. It covers many interesting ideas in logic, complexity and computing. It explains things clearly, but does not shy away from the mathematics when necessary.


1

Consider $\Sigma(m,n)$ to be the busy beaver function with $m$ states and $n$ colors. The following is then very hard to find: What is the smallest $m$ such that $\Sigma(m,2)>\Sigma(10,3)$? Since there is a color reduction method for reducing a $k$ state, 3 color machine to a $7k$ state, 2 color machine, it is known that $m\leq 70$. On the other ...


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The Meyers Serrin theorem that $ H=W $ (see their paper with exactly this equality as title) came only decades after people had proven all kinds of things with both $ H $ and $ W $, and is so elementary that you can ask second year students to prove it as exercise (extra credit, to be fair)


2

Various irrationality and transcendence results have already been posted, but it is interesting to see that the mere existence of transcendental numbers was not proven until the nineteenth century. Of course the existence of irrational numbers was already known in ancient Greece, but it took until 1844 before we first knew with certainty that there exist ...


1

How about this one: $$\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}=\frac{2}{\phi}-\ln \phi$$ There is an infinitely nested radical in the denominator. A finite one is also possible: $$\int_0^{1/16} \frac{dx}{\sqrt{x+\sqrt{x}}}=\phi-\frac12-\ln (\phi+1)$$ Not exactly a series, but might also be of interest: ...


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Donald Knuth's book, Surreal Numbers, an exposition of John Conway's work, is unconventional in that it is written in the form of a novel.


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Joe Roberts, Elementary Number Theory – the unconventional thing about this book is that it's done in calligraphy.


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Imre Leader This answer is slightly different in spirit to most of the others. As a lecturer at the University of Cambridge, Leader lectured my cohort in one course per year for four consecutive years in the undergraduate degree: our first course in being a mathematician (Part IA Numbers and Sets), our first course in non-groups abstract algebra (Part IB ...


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Kurt Gödel Personal story. When I heard that one can prove that it is impossible to prove that mathematics is consistent provided that mathematics is consistent, I was fascinated. Indeed, this is the beginning of my deep interest in mathematics. I thank Gödel for proving the interesting fact vaguely stated above which is by the way known as the second ...


1

Not everybody agrees on what are the standard/canonical notions of foliation and lamination, but in the context of dynamical systems (still with many exceptions and/or schools), many people uses the first for something that can be parametrized more or less as in the flow box theorem, and uses the second when there are some possible topological problems. ...


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"Obviously" $$(x^y)^z = x^{y\cdot z}$$ for $x,y,z \in \mathbb{C}$ such that given expressions are defined.


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Exponents $1,2, -1, -2,1/2$, and $-1/2$ are probably the most common, with the complexity of the implied polynomial equation $P(a,b,c)=0$ increasing in that order. As a random example, the height in the crossed ladders problem satisfies $h^{-1} = a^{-1} + b^{-1}$. For the altitude of a right triangle, $h^{-2} = a^{-2} + b^{-2}$. There must be many more. ...


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Here's another one. Let $ABCD$ be a trapezoid of area $c$ and let $O$ be the intersection point of its diagonals. If $a$ and $b$ are areas of triangles $AOD$ and $BOC$ (or $AOB$ and $COD$) then $$a^{1/2}+b^{1/2}=c^{1/2}.$$


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A statistical estimator is just a random variable for what we can measure. We hope this measurement is reliable, and so anything that means the probability distribution is "well-behaved" is a desirable property. Therefore we would want things like: Small variance for the estimator. If we know that the estimator has a large variance, that means that taking ...



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