# Tag Info

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My advice would be: $\bullet$ Do many calculations $\bullet \bullet$ Ask yourself concrete questions whose answer is a number. $\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!) $\bullet \bullet \bullet \bullet$ Beware the illusion that nice general theorems are the ultimate goal in ...

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Somewhat self promotion but Dave Gale and Colin Beveridge record the Wrong, But Useful Podcast on an approximately monthly basis. http://www.flyingcoloursmaths.co.uk/category/podcasts/ Also can be found on iTunes.

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May be it is not the most generalized theory, but I think Lagrangian approach had the most general impact on science. Modern physics, modern chemistry, are all based on such approach. But that is not math, or was it ?

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I would have to say the invention of set theory by Cantor. At first, there was a huge amount of resistance to it, which, if anything, is a very good measure of how big of a leap it was at the time. Grothendieck, in the introduction to EGA, says: «Il sera sans doute difficile au mathématicien, dans l’avenir, de se dérober à ce nouvel effort ...

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Perhaps Claude Shannon deserves a mention for his work, such as defining Entropy in the 1948 paper "A Mathematical Theory of Communication" and generally spurring the field of information theory. He created very general models for communication and gave some profound results about them. I don't know much about Shannon's other contributions (besides that ...

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You may want to check this site for over 46.000 free ebooks. Copyrights of the books contained in this site are expired so you can freely download books. You can download the books in PDF, EPUB or LATEX. main site : https://www.gutenberg.org/wiki/Main_Page math bookshelf : https://www.gutenberg.org/wiki/Mathematics_%28Bookshelf%29

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In the field of optimal control, there is a set of lecture notes by Hector Sussmann with source code located here.

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Rob Beezer's "A First Course in Linear Algebra" represents the future of OER textbooks for math, imho. His MathBook XML production flow allows a single source input (written in xml) to output in multiple formats (right now pdf-via-LaTeX and html, but the future could include more). To compete with commercial textbooks, both a quality book and a quality ...

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I'd like to introduce my favorite example by citing a question I found in an email in the net: When I asked many people what is the most difficult areas of math to understand, and also what is the most important unsolved mathematical problem, they always responded with two words: Langlands Conjecture (or Langlands philosophy or Langlands ...

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Under the $L^p$ metric, the completion of the space of continuous functions from $\mathbb{R}^n$ to $\mathbb{C}$ is $L^p(\mathbb{R}^n)$ for any $p$ such that $1 \le p < \infty$.

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Riemann's epoch making paper on Number Theory which connected two seemingly unrelated areas of Mathematics, Number Theory and Complex Analysis.

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The book Math on Trial: How Numbers Get Used and Abused in the Courtroom by Leila Schneps and Coralie Colmez has many examples.

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After the 2000 census, the state of Utah sued the Federal Government arguing that it should be given the delegate that went to the state of North Carolina. This document references some of the applied mathematics that went into making this decision. http://www.ncssm.edu/courses/math/Talks/PDFS/VotingAndPower.pdf

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There was an acerbic dispute between David Hilbert and L.E.J. Brouwer in the beginning of the 20th century. Basically, the former found the latter's insistence on constructivist methods ever more annoying. The animosity had eventually led to Brouwer's isolation within the scientific community. Brouwer survived his exile by almost forty years and died at an ...

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$\sqrt{2}$ is usually read as "root two". The degree of root should be mentioned though (like "square root two"). $\tan^{-1}$ is used for $\arctan$ mostly in Physics and Electronics. $j$ is used for $\sqrt{-1}$ in Electronics, while $i$ is used for it in Mathematics. Dots are used for time derivative in Pyhsics and Control Engineering (e.g.; ...

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We don't seem to have 15 yet: ${2\cdot 2 + 2 \choose 2} = 15$ If we're going to use $\Gamma$ and the like, we can get $30$ too: $2 \cdot {\Gamma(2+2) \choose 2} = 30$ No $19$ or $27$ or $29$ yet either (though this keeps getting more absurd): $\lfloor{2+\sqrt{2}}\rfloor^{\lfloor{2+\sqrt{2}}\rfloor} = 27$ $\lfloor \mathrm{Exp}(2)\rfloor \cdot ... 2 A slightly more creative way for 12 and 17:$|2+2\sqrt{-2}|^2 = 12|(2+2)!+\sqrt{-2}|/{\sqrt{2}} = 17$1 Are you allowed$.\dot 2=\frac 29$? That opens up a host of possibilities - for example$2+2+\sqrt {\cfrac 2{.\dot 2}}=7$1 this is for 24 and 26 $$22+2\Gamma (2)=24$$ $$(2+2)!+2\Gamma (2)=26$$ 3 The following is a complete list of the rationals obtainable by the five binary operations you gave, from a list of four twos. There are numerous expressions for many of the results and several invalid expressions (like$(2-2)^{2-2}), but only the first valid expression my program found is shown below. $$\begin{array}{ll} -4194302 & 2-2^{22} \\ -482 ... 5 Though I suspect this may be pushing common functions.. 2^2 + 2 + \Gamma(2) = 7 22/{\sqrt{2}^2} = 11 \int_{2/2}^{22}dx =21 \int_{2-2}^{22}dx =22 (2+2)! + \sqrt{2+2} = 26 3 \left\lfloor \exp\left(2^2\right)\right\rfloor+2+2=11 1$$7=\left\lfloor e^2 \right\rfloor+2(2-2)=2.2.2-\lg211=\left \lceil e^2 \right \rceil +2+2/2=\frac{22}{2}\lg217=2^{2^2}+\lg2$$2 I'm pretty sure that 7 is impossible. In fact, 11,13,15,17,19 and every number from 20 to 30 is imposssible, too. A=Possible results with two twos: 0,1,4 B=New possible results with three twos: 3,6,8,16 C=New results of an operation with a two and an element of B: 5,9,10,12,14,18,32,36,64,256 Operating two elements of A gives ... 6 14=2^{2^2}-2and18=2^{2^2}+2$$13=\frac{22}2+2$$24=\frac{(2^2)!}{\frac22}$$20=\sqrt{{22}^2}-2=(2^2)!-2^2$$22=\sqrt{\left[(2^2)!-2\right]^2} 3 0=2+2-2-2 uses four twos not three. 10=\cfrac {22-2}2 0 I believe that one on the list shopuld be the Conjugacy Class Equation given by$$|G| = |Z(G)| + \sum_{x_a\notin Z(G)}|Cl(x_a)|$$Where G is a group, Z(G) is its center. Considering the following conjugacy representation$$\begin{align}\mathcal{I} : G &\longrightarrow \mathcal{P}(G_0)\\g &\longmapsto \mathcal{I}_g : G_0 &\rightarrow ... 0\exp(\pi\sqrt{163})\approx 262537412640768743.99999999999925$0 One of the most beautiful formulas in combinatorics: Cayley's Formula:Number of labelled trees on$n$vertices$=n^{n-2}$And here are some interersting, yet not-so-popular combinatorial identities: Notations:$F_n = n^{th}$Fibonacci number$H_n =n^{th}$Harmonic number;$H_n = 1+ \frac{1}{2}+\frac{1}{3} +...+\frac{1}{n}$$$\sum_{n \ge 1} ... 0 Möbius inversion formula may be an example. Also Ramanujan's Partition Congruences were surprising to me when I first saw them. 1 I find the Young-Frobenius identity, found on p. 8 here, surprising: A partition \lambda\vdash n of an integer n\geq0, i.e. a sequence (\lambda_{1}, \cdots,\lambda_{k}) with \lambda_{1}\geq \cdots \geq \lambda_{k}>0 and \lambda_{1} + \cdots + \lambda_{k}=n, can be identified with a diagram consisting of k left-justified rows of boxes, where ... 6 This may be a regional thing, but when I started studying at a British university, so many of the lecturers wrote multiplication as a single . (full-stop). This got really confusing when, after having studied in the states I was used to$$ 0.5 + 0.5 = 1$$whereas here it meant$$0.5 + 0.5 = 0 + 0 = 0$$3 Nonlinear analysis is a very large field, and you'd be hard-pressed to find a resource that deals with its many methods in a comprehensive manner. Most resources I know of deal with a subset of methods or methods applied to particular situations (nonlinear elliptic equations, for example). That said, if you're only looking for an introduction to the subject ... 7 Calculus I,II,III: The 'dx's in integrals and derivatives are just notation to help keep track of the important variables in a given problem, and they're otherwise meaningless in isolation. Real Analysis: In fact, when the integration variable is unambiguous we may as well dispense with the differentials altogether and just denote the integral of f over ... 1 Fix \alpha, \kappa\in \Bbb R. We (or maybe just I) say that \alpha is \kappa-approximable if there exists an infinite sequence of rationals (p_n/q_n) in lowest terms such that$$\left|\alpha-\frac{p_n}{q_n}\right| < \frac{1}{q_n^\kappa}.$$Define the irrationality measure \mu of \alpha to be$$\mu(\alpha) = \sup\{\kappa : \alpha \text{ is } ... 0 Almost all real numbers contain in their binary expansions the contents of all books ever written an infinite number of times. 3 The Stacks Project and the CRing project are good for learning algebraic geometry and commutative algebra, respectively: Stacks Project: http://stacks.math.columbia.edu/ CRing Project: http://people.fas.harvard.edu/~amathew/cr.html 1 When we list seasons it goes, summer, autumn, winter, spring, summer, autman, and one says seasons occur cyclically. In the group denoted (by those without broken pieces of chalk) as$\mathbf{Z}$the elements are (half of them) go like this 1,2,3,4, etc without ever repeating and yet it is called the infinite cyclic group! 17 Let$[a_0; a_1, a_2, \ldots]$be the continued fraction expansion of$x$. Then there exists a constant$\kappa\approx 2.685452001$with the property that, for almost all$x$, $$\lim_{n\to\infty} (a_0a_1\ldots a_n)^{1/n} = \kappa.$$ (Many properties about continued fraction expansions are analogous to properties about the normality of base-$n$expansions. ... 5 Let$s_n(x)$be the sum of the$n$digits following the decimal point of the decimal expansion of$x$; for example$s_3(\pi) = 1+4+1 = 6$. Then for almost all$x$, the digits$s_n(x)$are uniquely determined. And for almost all$s$the average$\frac{s_n(x)}{n}$has a limit as$n$approaches infinity. Furthermore for almost all$x$that limit is ... 4 Here are a few I've found over the years. I've included some brief comments on the first four, which I've personally used in my classes. Stitz and Zeager Precalculus materials Outstanding. Includes a lot of ancillaries like answers and youtube videos. CCL Mooculus Very, very good calculus MOOC offered at Ohio State University. Text is freely available ... 2 Almost all real numbers are normal in all the bases. 8$\forall x\in\mathbb R$: Almost all real numbers are different from$x$. 35 Almost all real numbers are transcendental. Almost all real numbers are not constructible. Almost all real numbers are not computable. Almost all real numbers are normal. Almost all real numbers are not definable. Almost all real numbers are not periods. 7 Almost all real numbers are irrational. 4 The inconsistency between the reading of "Negative" versus "Minus" has, in my opinion, been a thorn in the side of every teacher and student since their acceptance. 4 The jump form length/area/volume to the abstract measure theory, that encompasses this basic measures and much stranger measures and is the essential tool of probability theory. 3 Ramsey theory, beginning with Ramsey's theorem itself, or before that with the pigeonhole principle, and generalized by Erdős and Rado et al. to transfinite cardinal and ordinal numbers, order types, graphs, etc. 2 $$\bigcup \{ A ,B, C\} = A \cup B \cup C$$ or $$\bigcup \mathcal A$$ where$\mathcal A $is a set of sets. It should be $$\bigcup_{X \in \mathcal A} X.$$ This can come up in topology sometimes (unioning over a collection of open sets), and can get rather confusing if you start doing things (edit:) like$\left(\bigcup \mathcal A\right) \cup U \cup V$... 5$\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R \subset \mathbb C \subset \mathbb H$If$f \colon X \rightarrow Y$is a map and$A \subseteq X$, then people often write$f(A)$to denote$f'' A := \{ y \in Y \mid \exists x \in X : f(x) = y \}$, which can be quite confusing in cases where$A \in X$. "Canonical" ...$(a,b,c) = ((a,b),c) = ...

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