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A good refernce in my opinion is Theorem 27.3.9 in the book by Patrice Tauvel and Rupert Yu named Lie Algebras and Algebraic Groups.


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A similar question was asked here From the answers I suppose that he did not prove the theorem at all...


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To long for a comment but not a complete answer sorry, Iam interested in a publication that explains it more exhaustive as well , but hope this helps a bit) You mean something like I have proven $ P \to Q $ in system 1 and you want to transform it into a proof of $ A \to B$ in system 2 what you need is some dictionary that relates P and Q from system 1 ...


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Here are some more books which are informative and also provide fun! Combinatorics If you like Generatingfunctionology by Herbert Wilf then of course you will also like Concrete Mathematics by Graham, Knuth & Patashnik. With respect to the authors, especially Donald E. Knuth no comments are necessary to this book from my side ...


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Three additional recommendations for your extensive book list (considering your related questions :-)) Number Theory Books dedicated to counterexamples in number theory are not known to me, but since you pointed to number theory here are books with related information Unsolved Problems in Number Theory by Richard K. Guy. This book is ...


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I have no clean bijection as the one you are looking for. But there is a classical link between latin squares of order $n$ and distance-regular graphs on $n^2$ vertices and diameter 2. See Relevant keywords : partial geometries, strong regular graphs See articles and books by H. Van Lint, Peter Cameron. Chris Godsil and Gordon Royle "Algebraic Graph ...


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First Proof $a+b=x$ , $b+c=y$ , $c+a=z$ $\therefore 2 \left(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\right)=(x+y+z)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)-6\\=\underbrace{\dfrac{x}{y}+\dfrac{y}{x}}_{\geq 2}+\underbrace{\dfrac{y}{z}+\dfrac{z}{y}}_{\geq 2}+\underbrace{\dfrac{z}{x}+\dfrac{x}{z}}_{\geq 2}-3 \geq 3$ Second Proof ...


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A logician's reason: Every recursive function (and much more) is definable in the (first-order) structure $(\mathbb{N}, +, \times)$ but not in $(\mathbb{N}, S, +)$. This is due to Godel. To appreciate why this theorem is non-trivial try finding a formula $\phi(x, y, z)$ in the language of arithmetic that expresses $x^y = z$.


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I have compiled a list of undergraduate level university Math textbooks. Hopefully it is of help to you! http://mathtuition88.com/2014/10/19/undergraduate-level-math-book-recommendations/ I would like to suggest the above books (mainly for Pure Mathematics). Ideally, the motivated student is able to self study and obtain the knowledge equivalent to a 4 ...


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I'm sorry but AM-GM Inequality is all that I know, So, By using AM-GM Inequality for $ (a+b) , (b+c) , (c+a)$ and $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ We get $$\frac{(a+b)+(b+c)+(c+a)}{3} \ge\sqrt[3]{(a+b)(b+c)(c+a)}\tag{1}$$ $$\frac{\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}}{3} \ge\sqrt[3]{\frac{1}{(b+c)(c+a)(a+b)}}\tag{2}$$ Multiplying $(1)$ and ...


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If you want to use the power series way to do it(although I think it's overkill), by convex property of $f(x)=x^n$ we have $\frac{(a^n+b^n+c^n)}{3}\geq (\frac{a+b+c}{3})^n$ or equivalently $(a^n+b^n+c^n)\geq \frac{(a+b+c)^n}{3^{n-1}}$, where $n$ is positive integer. Therefore your left hand side(series expansion) is greater or equal to ...


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I have compiled a list of undergraduate level university Math textbooks. Hopefully it is of help to you! http://mathtuition88.com/2014/10/19/undergraduate-level-math-book-recommendations/ I would like to suggest the above books (mainly for Pure Mathematics). Ideally, the motivated student is able to self study and obtain the knowledge equivalent to a 4 ...


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Although not a movie regarding mathematics, I think Limitless by Bradley Cooper and Robert De Niro could be a good motivator movie. The movie is quite fascinating and entertaining. Just don't get lost in some of the fantasies they throw your way.


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Problem solving strategies, Arthur Engel. Problem-Solving Strategies is a unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. The discussion of problem solving strategies is extensive. It is written for trainers and participants of contests of all levels up to the ...


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Try Flatland: The movie. Its about a square who discovers that there are more than 2 dimensions.


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Instead of thinking of multiplication as repeated addition, we can define it as the unique commutative, associative operator that distributes over addition and has 1 as the identity. This suggests another approach to generalizing beyond multiplication, by asking what operator distributes over multiplication. Define the operator $$ a *_n b = \exp^n ...


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Many journals have a particular style for citations, but the style differs from one journal to another (or at least from one publisher to another). When authors are left free to select their own styles, even more variation results. I've come to prefer a style in which (1) authors' names are given in full, including their first names if I can find them, ...


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I've found Gillman's Writing Mathematics Well to be very useful.


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You could try Milne's Fields and Galois Theory, available for free on his website. It seemed relatively thorough last time I skimmed it.


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I don't think it's explicitly mathematical, but Peter Westergaard's An Introduction to Tonal Theory might be appealing (I haven't read it myself). There is also a blog which seems to have much about it: http://mathemusicality.wordpress.com/category/westergaardian-theory/


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Check out Contemporary Abstract Algebra by J.A Galian


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Alex Kasman's Mathematical Fiction website includes an extensive list of movies.


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It must be said, Stand and Deliver is a great mathematics movie - truly inspirational. I'm not sure about this, but perhaps the film based on Alan Turing, famous cryptographer - The Imitation Game? An Invisible Sign is also a fairly interesting film. I suggest looking at Wikipedia's list of movies on mathematicians.


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If one drops the identity and associativity axioms from the definition of group, we get the definition of a quasigroup, and if we retain the identity axiom, the structure is called a loop. A priori these need not satisfy any further composition conditions, but some well-known nonassociative quasigroups do satisfy such conditions. For example, the loop ...


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I could not get the above link to work, either from a browser or from an ftp client. This link works, at least as of October 17, 2014: http://www.math.mcgill.ca/barr/papers/gk.pdf


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Perhaps the father of modern infinite deserves to be quoted here: My theory stands as firm as a rock; every arrow directed against it will quickly return to the archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and ...


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"The Professor and his Beloved Equation" is a Japanese movie by Takashi Koizumi about the life of a number theorist and the joy of being a mathematician. Also "An Invisible Sign" could be interesting for you. It is about a math teacher and the role of numbers in her life.


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Let me begin with two examples. The first example is the famous Hilbert and Brouwer controversy. A part of this philosophical contest was on Cantor's theory of infinite numbers. Of course Brouwer as an intuitionist is against using infinite numbers in maths. On the other side we have Hilbert's famous sentence that "No one can expel us from Cantor's ...


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Others have done a good job of explaining how exponentiation can be generalized. I'm going to address the question of why mathematicians are not interested in these generalizations. I think the main reason why Knuth's up arrow has not taken off in the same way that, say, exponentiation has is that some of the very nice properties enjoyed by addition and ...


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Moneyball (2011) - Based on the true story of how the Oakland A's manager, Billy Bean, used a statistical approach to finding undervalued players, ultimately allowing him to create a competitive team with a fraction of the budget of teams like the Yankees.


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First thing that comes to mind when I hear the question is October Sky. It's the (slightly dramatized) true story of Homer Hickam and his friends "the Rocket Boys." They lived in a podunk coal-mining town in West Virginia when Sputnik went up, and when Homer (a high school student at the time) saw the satellite moving across the sky, he was inspired to ...


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I'm not a mathematician, but I suspect one reason is the fact that $(m^n)^p=m^{np}$ Which suggests there may not be lots of rich behaviour past the exponential operator. Another way to say the same thing might be that we don't need the next level notation because we can express it in terms of the exponential operator combined with the multiplication ...


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As others pointed out, we don't stop at exponentiation. We do stop at it in primary, secondary, and early secondary school though. Which I think is why your question arises. Simply we stop in school because teacher's have a hard enough time teaching multiplication, and they teach it incorrectly. You yourself may not understand multiplication when you say it ...


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Watch the movie $21$ with Kevin Spacey.. here (If you play poker you will like it very much)


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I saw 'Donald Duck in Mathemagic Land' many years ago, and I still remember it very fondly. I recommend it highly! And it is suitable for all ages!


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I think the best source to start an investigation on the notion of "equality", its history and its rules (also other basic laws of thought) would be here: http://en.wikipedia.org/wiki/Law_of_thought


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Take a look at Hard Problems. It follows the extremely gifted students who represented the United States in 2006 at the world's toughest math competition—the International Mathematical Olympiad (IMO). You can probably find the entire movie on Youtube.


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I would like to suggest "The Number $23$" starring Jim Carrey. It's not at all about math. It's about the 23 enigma which is only as relevant to mathematics as superstitions surrounding $7, 11$ or $13$. But it's the first time Jim has done a non-comedic role and it's worth a watch. But to defend myself: $23$ is a number and numbers are dealt under math. ...


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Try Numb3rs. http://en.wikipedia.org/wiki/Numbers_%28TV_series%29 Numb3rs (stylized NUMB3RS) is an American crime drama television series that ran on CBS from January 23, 2005, to March 12, 2010. The series was created by Nicolas Falacci and Cheryl Heuton, and follows FBI Special Agent Don Eppes (Rob Morrow) and his brother Charlie Eppes (David Krumholtz) ...


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It may not be your traditional math movie, but the original (NOT THE REMAKE!) of Cheaper by the Dozen is a well done adaptation of the book about an efficiency expert, and that entire process is highly mathematical/pattern recognition/problem solving. It inspired me as a child to think in such fashions.


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The usual method consists in writing that $a, b$ are the coordinates of a point in the plane, and use its polar coordinates, that is, $a=r\cos\theta$ and $b=r\sin\theta$. Then your equation becomes $$r\cos\theta\sin x+r\sin\theta\cos x=c$$ $$\sin(\theta+ x)=\frac cr$$ If $\left|\frac cr\right|\leq 1$, then the solutions are $$x=\arcsin\left(\frac ...


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Traveling salesman was a pleasant surprise for me. It's about group of mathematicians who prove P=NP.


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Equations like this can be solved by turning them into quadratic equations. For example, you can write $x=2y$, so that $$ \begin{align} \sin x &= 2\sin y\cos y\\ \cos x &= \cos^2 y - \sin^2 y \end{align} $$ the equation can then be written as $$ 2a \sin y\cos y + b(\cos^2 y - \sin^2 y) = c(\sin^2 y + \cos^2 y) $$ Dividing by $\cos^2y$, we get $$ ...


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Try A A Kirillov, A D Gvishiani: Theorems and Problems in Functional Analysis.


1

Try Chorin, Hald: Stochastic Tools in Mathematics and Science. It is based on a course at UC Berkeley which was designed following a survey of former students, where the basic question was: "What kind of stochastics you really need for your scientific work?"


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Try Jürgen Richter-Gebert: Perspectives on Projective Geometry. It shows the classical material in a modern way, and is written in an excellent pedagogical style.


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Try Larson, Bengzon: The Finite Element Method.


2

Try MESH by Beau Janzen and Konrad Polthier.


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Comment j'ai détesté les maths (How I came to hate maths), elected in France as the best 2014 documentary film. I really enjoyed it. You can see the trailer (with English subtitles) here. A movie that I haven't seen yet: The Imitation Game, about Alan Turing's life. See also the Wikipedia page on Films about mathematicians.


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Good examples are: Havil: Gamma Ziegler: Proofs from THE BOOK Schroeder: Number Theory in Science and Communication Arnold: Catastrophe Theory Tabachnikov: Geometry and Billiards Markowich: Applied Partial Differential Equations Pesic: Abel's Proof Harel: Computers Ltd.



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