# Tag Info

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We have found a proof and formalized it using the Coq proof assistant, the details are here: https://hal.inria.fr/hal-01332044

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Homotopy groups have the nice property that a fibration $F\hookrightarrow E\to B$ induces a long exact sequence on homotopy: $$\cdots\pi_i(F)\to \pi_i(E)\to \pi_i(B)\to \pi_{i-1}(F)\cdots$$ This is unheard of in homology where the situation is considerably more complicated. In the simple case of product fibrations the Künneth formula is sufficient but for ...

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Edward Packel's brief text The Mathematics of Games and Gambling is a basic introduction to the mathematics needed to analyze gambling and game activities. Packel focuses on the mathematics. The book is not intended as a discussion of gambling strategies. https://www.amazon.com/Mathematics-Games-Gambling-Mathematical-Library/dp/0883856468/ref=sr_1_1?ie=...

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This might be too advanced, but as far as I am aware, Dubins and Savage's "How to Gamble if You Must" is the established classic in the field of martingales (a topic in probability theory) as applied to gambling: https://www.amazon.com/How-Gamble-You-Must-Inequalities/dp/0486780643

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The idea of the proof is this: Let $P$ be a rational point on $C$. Then look at all the lines through $P$. Each such line intersect $C$ in one more point. This lets us define a rational map $\mathbb P^1 \to C$ since lines are parametrized by $\mathbb P^1$ (it is rational because the tangent line only meets $C$ at one point). The map is well defined and ...

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Riemann surfaces is a very standard topics in math, then you can find a lot of books talking about Riemann surfaces under different point of views. I can suggest you: -Compact Riemann surfaces - J.Jost, -Riemann Surfaces - S.Donaldson, -Riemann Surfaces - Farkas and Kra, -Algebraic curves and Riemann surfaces - R.Miranda The first is pretty analytical,...

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I'm very fond of Forster's book Lectures on Riemann Surfaces. Check it out. There is also a lovely book by Phillip Griffiths called Introduction to Algebraic Curves.

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The standard undergraduate text for real analysis is Rudin's Principles of Mathematical Analysis (affectionately referred to as "Baby Rudin" since he wrote it when he was quite young). Another text I enjoyed was Serge Lang's Undergraduate Analysis. I think they're both have their pros and cons but are ultimately both fine books for first learning some ...

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'Diophantine Geometry: An Introduction', by Silverman and Hindry, contains a proof in section A.4.3 (Page 74). A few minor steps are skipped, but it should be fairly easy to follow.

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In fact one can! First, if it's all the same to you, let's assume that the base field is algebraically closed so we can say a bit more about representation theory. One should first note that this representation is just $V^* \otimes_{\mathbb{C}} V$ so you can completely describe its character. Second one even has an explicit description of the copy of the ...

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See in this book (page 7-8): https://books.google.it/books?id=9YuDAwAAQBAJ&pg=PP1&dq=linear+functional+analysis+Cerda&hl=it&sa=X&ved=0ahUKEwjiufnZ777NAhWLVRoKHeOCCTYQ6AEIJTAA#v=onepage&q=linear%20functional%20analysis%20Cerda&f=false and for regular partition of unity, you can consider this lemma: Proof of regular version of ...

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Yes, the probability is zero since it equals the symmetric group with probability one: If $Q_d(N)$ denotes the set of degree $d$ polynomials with coefficients $|a_i|\le N$ with Galois group not equal to $S_d$, then $$|Q_d(N)|\ll N^{d-1/2}\log N$$ This bound is sufficient to prove your result. This was proven by Patrick X. Gallagher. EDIT: Gallagher ...

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A. Weil, The field of definition of a variety, Amer. J. Math. 78 (1956), 509-524. Theorem 3. J-P. Serre, Groupes algébriques et corps de classes, Publications de l'institut de mathématique de l'université de Nancago, VII. Hermann, Paris 1959. Ch. V-20, Prop. 12, Cor. 2. There is an English translation of Serre's book: Algebraic Groups and Class Fields ...

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This is one of the books that started the collection "Projeto Euclides" at IMPA that has produced many first class books in Mathematics, to be used in Brazil. A link to the collection is here: http://www.impa.br/opencms/pt/publicacoes/projeto_euclides/ There are NO translations to English or any other language for that matter, in fact, even the Portuguese ...

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I found the following resources: http://www.educatorstechnology.com/2012/10/8-great-youtube-channels-for-math.html http://www.avatargeneration.com/2012/10/learning-math-with-youtube/ http://www.freetech4teachers.com/2012/04/seven-youtube-channels-not-named-khan.html#.V2pEdbh9600 TV show called Dara O'Braian's School of Hard Sums (highly recommended) http://...

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For a precise statement and a proof thereof, see Theorem 2.6 of this paper.

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Two possibilities: Infinite Dimensional Analysis: A Hitchhiker's Guide by Charalambos D. Aliprantis and Kim Border Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity by Claude Berge

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I don't know what you're getting at by mentioning you're in economics, but if you have some first year analysis then Munkres' book is a fantastic introduction.

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One should use official sources. The DOI is dx.doi.org/10.1080/00927877708822224. I shall not discuss debatable alternatives.

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It seems like you miscopied the text in your question. It actually says that: An odd composite number $2n + 1$ is in the sequence if and only if multiplicative order of $2 \pmod{(2n+1)}$ divides $2n$. So Chandler's theorem states that, if $2^k \equiv 1 \pmod{(2n+1)}$, then $k | 2n$.

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I think the earliest references on almost perfect numbers are due to Cross, and Jerrard & Temperley: R. P. Jerrard and Nicholas Temperley, Almost perfect numbers, Math. Mag. 46 (1973), 84–87. MR 0376511 (51 #12686). DOI: 10.2307/2689036 James T. Cross, A note on almost perfect numbers, Math. Mag. 47 (1974), 230–231. MR 0354536 (50 #7014). DOI: 10.2307/...

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Helpful Websites Khan Academy MathInsight Paul's Online Math Notes Free Textbooks Fundamentals of Calculus by Ben Crowell MIT's calculus course textbook Community Calculus's Books Wikibooks (incomplete) The MOOCulus Textbook There are more free and open source textbooks listed here. For more on non-free textbooks, see this question.

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There is a theorem by Taunt (in "Remarks on the Isomorphism Problem in the Theories of Construction of Finite Groups"), which gives a nice characterization if $\gcd(|N|,|H|)=1$. I'm citing from "Construction of Finite Groups" by Besche and Eick. Let $N,H$ be finite (soluble) groups with $\gcd(|N|,|H|)=1$. Furthermore let $\psi_i:H\rightarrow Aut(N)$ for $... 2 Based on your needs you can choose a proper book. Usually, there are two kind of books here. Those that emphasize on the theory and those that emphasize on the application. Often, engineering students are not that much into the theory and they use books which care more about application. Books like Spivak have an emphasis on the theory. If you want to ... 2 This is a personal recommendation. There are probably many things that you could do/read/study that would help you in your journey into the world of pure mathematics, but I have one suggestion. I suggest getting a good book on abstract algebra. A beginning book on abstract algebra will usually cover the topic of groups. A group is a set with a (binary) ... 1 Yes, this still holds, and I would prove it in a different (maybe equivalent) way. The key is the existence of$\psi \in C_0^\infty(B_3)$with$\psi \equiv 1$on$B_1$. Then, for all$\varepsilon > 0$, there is$\phi_\varepsilon \in C_0^\infty(B_2)$with$\phi_\varepsilon \equiv 1$on$F$and $$\int |\nabla \phi_\varepsilon|^2 \, \mathrm{d}V \le cap_2(F,... 0 The von Mangoldt function can be generated at the pole s=1 of Riemann zeta with the formula:$$\Lambda(n,s)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$So you can instead of s \rightarrow 1 let s be complex number so that you have: s=a+ib$$\Lambda(n,a,b)=\lim\limits_{s \rightarrow a+ib} \zeta(s)\sum\limits_{d|... 0 You have the following result: Let$U$be another open set with$\bar U \subset V$. Then, there is$C > 0$, such that for all$\Omega \subset U$we have $$\operatorname{cap}_2(\Omega, V) \le C \, \operatorname{cap}_2(\Omega, \mathbb{R}^n).$$ The proof is by smooth truncation: You take$\varphi \in C_0^\infty(V)$with$\varphi = 1$on$U$. If you than ... 2 Not exactly synonyms order theory is broader than lattice theory. Lattice theory studies the algebraic structure, lattices, whereas order theory is the general study of binary relations, including partially ordered sets, total orders, and lattices. For a book on LT, I recommend 'Lattice Theory with Applications' by V. K. Garg. 4 I learned the basics from Introduction to Lattices and Order, 2nd Edition, by B. A. Davey and H. A. Priestley. Its not bad. To answer your title question: technically, order theory is more general than lattice theory. In practice you don't really study one without also studying the other. 2 As has been pointed out in the comments, linearity of expectation does not require independence. Thus, the expected number of duplicates, which is$n-m$plus the expected number of empty bins, is$n-m$plus$m$times the probability for a given bin to be empty: $$E[D]=n-m+m\left(1-\frac1m\right)^n=n+m\left(\mathrm e^{-n/m}-1\right)+O\left(\frac nm\right)=\... 2 The fibers are precisely the orbits of the natural action of SL_2(\mathcal{O}_K) on \mathbb{P}^1(K). See this note by Keith Conrad for details. 0 The best introduction to affine geometry I know Vectors and Transformations in Plane Geometry by Philippe Tondeur. Using nothing more then vector and matrix algebra in the plane, it develops basic Euclidean geometry with the transformations of similarities and isometries in the plane as completely and clearly as any book I've seen. It also gives an ... 0 Elaborating on the answer: Without loss of generality, we can assume that an inner measure of A is zero(i.e. P_*(A)=0) and outer measure is positive(i.e., P^*(A)>0). Let A_1 be such measurable set that A\subset A_1 and P(A_1)=P^*(A). For C=(A\cap X)\cup (A^c \cap Y)(X,Y \in \cal{B}) we set P_1(C)=P(A_1\cap X)+P(((A_1)^c) \cap Y). Then ... 3 Have you looked at the book Simplicial homotopy theory by Goerss and Jardine? They have a 60-pages Chapter 1 on simplicial sets, at the end of which they define the model structure on \mathsf{sSet}. IMHO the book is quite good. I'm not sure what you mean by "doesn't take the top down model structure approach", but definitions don't fall from the sky and ... 0 Mas-Colell, Whinston, and Green is the bible for first-year graduate micro. If you want more focus on the technical stuff on the mathematical side, you may want to consider: Real Analysis with Economic Applications by Efe Ok Mathematics for Economists by Simon and Blume 1 A very good modern text on design experiments is the one by Gary Oehlert. It is available in hard back, but the author has posted a PDF online, so you can look at it for free. I have never taught out of this book, but I have used it for examples and as a reference. Data are limited in scope, but real. A traditional treatment of DOS is included in the Ott &... 1 I like Set Theory: An Introduction to Large Cardinals by Frank Drake. Is old by good. Discusses what happens without/with Choice and according to Andreas Blass: ... deserves your attention even if you're not particularly interested in large cardinals. Despite its subtitle, it contains very nice presentations of a lot of general set-theoretic background ... 1 Just to make it possible to "close the case": The article is possibly Baker, I. N., and Rippon P. J. "A Note on Complex Iteration." The American Mathematical Monthly 92.7 (1985): 501-04. Web. This is online, for instance via JStor. (Access possible using an account of an affiliated university or organiszation) 3 You could look at Hogg, Mckean and Craig's Introduction to Mathematical Statistics. But Casella and Berger's Statistical Inference is suitable as a first text for both undergrads and graduate students with basic mathematical maturity. 4 An introduction to the theory of cardinals is given in any set theory text. For decades, the standard introduction to basic set theory was Paul Halmos' Naive Set Theory , which is written in Halmos' concise, clear style with tons of problems to chew on. The book's recently been reissued in a nice inexpensive paperback, so that's really a good choice. My ... 3 At the most elementary level you might start with E. Kamke's Theory of Sets. Note that \aleph_1 is defined to be the cardinality of the set of all countable ordinals, and \aleph_2 is defined to be the cardinality of the set of all ordinals of cardinality \le \aleph_1, and so on through all \alephs of finite index. And \beth_0 is the same as \... 2 Notice that$$ f(x,y)=yM_{\mathbf P}(\log(x/y)), $$where$$ M_{\mathbf P}(t):=\int_{[0,1]}e^{tq}\,\mathbf P(dq),\qquad t\in\Bbb R, $$is the moment generating function of the probability measure \mathbf P. The function M_{\mathbf P} is necessarily continuous, \log M_{\mathbf P} is convex (by Holder's inequality), and M_{\mathbf P}(0)=1. The ... 1 The reason Euler's numbers are "lucky" is that the corresponding discriminant D=1-4A has class number 1. In the language of quadratic forms, this means that, up to equivalence, there is a unique binary quadratic form of discriminant D - i.e. f(x,y) =x^2-xy+Ay^2. This is the approach taken by @WillJagy in his answer. In the language of number fields,... 2 I like Buell. Much of what you would want is also in Dickson's Intro, about 1929. Plus, to be realistic, my answers here, as i generally use the quadratic form methodology. For your question the answer is all primes such that (\Delta|p) = 1, with \Delta = 1 - 4 k. Also k itself, when that is 2. The overall theme is that any such prime is represented ... 1 Ah, at least one empirical pattern which can be built recursively (if that heuristic holds...) Let$$a(n) = \sum_{k=0}^n \frac 1{2^{2^k}} $$and let c(n) be the list of partial denominators of the continued fraction (="entries" of the cf) of a(n) - written as a string because we have only few numbers and can nicely compact the notation to focus on the ... 8 Shallit's paper, which you've cited, gives a simple algorithm for generating these coefficients. It works for \sum_{k=0}^{\infty}u^{-2^k} for integer u\ge 3; but he notes that it also works for u=2 (your case) with a slight modification. Start with$$B_1=[1,3].$$Then repeatedly apply the following rule: B_{n+1} is generated from B_n by appending ... 0 Since the ratio test can be applied, we know there exists the limit L. Now we can say a_n=a_{n+1} for n\to\infty. If you are just looking for a close approximation, take wolfram alpha https://www.wolframalpha.com/input/?i=sum+%281%2F2^2^k%29 Note that after k=3 nothing dramatic happens anymore, since the fraction gets so small very quickly I don't ... 1 Let X_1 and X_2 have the discrete topology. Then X=X_1\times X_2 has the discrete topology, so \mathcal{B}_X=P(X). But if X_1 and X_2 are large enough \mathcal{B}_{X_1}\otimes\mathcal{B}_{X_2} will not be all of P(X). Here's one way to prove this. For S\subset X and x\in X_1, let S_x=\{y\in X_2:(x,y)\in S\}. Now let$$\mathcal{A}=\{... 1 The Gauss-Green theorem (the basis of the Green identities) states that if$u,v$are sufficiently smooth on a nice domain$\Omega$then $$\int_\Omega \frac{\partial u}{\partial x_j} v \, dx = - \int_\Omega u \frac{\partial v}{\partial x_j} \, dx + \int_{\partial \Omega} uv \nu_j dS$$ where$\nu_j$is the$j\$th component of the external normal unit vector and ...

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