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[2016-07-25]: Section Differential Geometry added. Although OP narrowed down the post, there are still many more important historical facts which should be addressed to adequately answer the question, than I can give in this answer. Nevertheless here are some aspects, which might be interesting. At least we will see, OP is right when he thinks that many ...


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As commented, the actual answer is real analysis. Now, you say you want to know why calculus works. There are proofs in most calculus books, you know. Those proofs do explain why things work. Finally, Spivak Calculus is an excellent calculus book, with much more emphasis on proofs than usual.


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Here is a proof involving abstract linear algebra. View $A$ as an operator on an $n$-dimensional vector space $V$. If $B$ is a right-inverse to $A$, then $A \circ B = \text{id}_V$. Thus, $B$ is injective, and an injective map from a finite-dimensional vector space to itself must be surjective, so $B$ is both injective and surjective, implying that $B$ is ...


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To directly answer your question: Andrej meant "monomorphism". Since the objects of a topos don't typically consist of (raw, unstructured) sets, the usual notions of injectivity and surjectivity don't apply. But for convenience, people still talk about "injections" and "surjections" in those contexts (and mean monomorphisms and epimorphisms, respectively). ...


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I was quite surprised to find that "proexample" is actually a word. Google turns up, for example (or should that be "for proexample"?) Corcoran, J. 2005. Counterexamples and proexamples. Bulletin of Symbolic Logic 11(2005) 460. Are you that John Corcoran? If so, you seem to be the main published user of this word, so you are the one who should be ...


2

Answer to part 1: If $f$ is an isomorphism of algebraic varieties, it must be a regular map (which is therefore analytic, as polynomials and their appropriately-taken quotients are analytic). Similarly, it's two-sided inverse must also be regular and thus analytic, and so we have that $f$ is also an isomorphism of analytic varieties. Therefore it must be a ...


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When $p$ is a small prime, finding a solution to $a^x \equiv b \bmod p$ is correspondingly easy (assuming that $a,b$ are not divisible by $p$). For example, $2^x \equiv 28 \bmod 3$ holds for any even power $x$. It is also clear that if $a^x \equiv b \bmod p^s$ for some integer power $s\ge 1$, then: $$ a^x \equiv b \bmod p^s \implies a^x \equiv b \bmod p^{...


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Take a look at the World War 2 Education Manual EM310 Solid Geometry. It can be ordered on Amazon for a very reasonable price. Content is explicitly designed for self-instruction, with answers provided--consider the intended audience. It covers the topics you want and does so without analytic geometry: is intended to be in the curriculum after plane ...


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Here is a very naive explanation with all technicalities skipped. Let $X$ be a simplicial set (this is a type of discretisation of a CW complex which is a generalization of a simplicial complex). In his paper Kan constructs out of $X$ another simplicial set, which he denotes by $GX$. The set of $i$-simplices of $GX$ is the free non-abelian group with ...


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One source (which I have not read the whole article) might be the survey article "History of Homological Algebra" written by C. Weibel, http://www.math.uiuc.edu/K-theory/0245/survey.pdf.


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I think one of the best and most complete book I know on (especially finite) group theory is Antonio Machì's http://www.springer.com/us/book/9788847024205 (original language: italian), despite the word "Introduction" in the subtitle. The author is also the translator of Herstein's much more elementary $\textit{Algebra}$. There are also many exercises and ...


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It's not clear to me what needs to be cited away. As far as I can tell, here's an answer. Consider the vector field $X$ that generates the flow. Because the flow must necessarily preserve the boundary components, if there are boundary components, they must be circular orbits. So there are none. If $X$ has no zeroes, then $\chi(M) = 0$; this follows either ...


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Its great that you are interested in foundations of calculus (normally even book authors and teachers are not so interested in teaching foundations of calculus to students who are learning calculus for the first time at an age of 16 years or so). The teaching of calculus follows almost the same pattern in most countries: First one is taught calculus as a ...


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As a child, I learned logic with the book "The Game of Logic", by Lewis Carroll. This does not use the modern symbols, but I think that it is a great book to introduce the basic concepts.


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I suggest Spivak's Calculus on Manifolds. I just finished reading it, and I feel that I was really enlightened. The text is terse but efficient (150 pages), but there are a plethora of exercises to help out. I did not find the exercises particularly difficult compared to Rudin but I certainly learned a lot more than I did in Rudin's book.


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For olympiads Number Theory this is a must-"Number Theory-Andrescu Titu"-https://blngcc.files.wordpress.com/2008/11/andreescu-andrica-problems-on-number-theory.pdf And for Geometry this one-"Coxeter-Geometry Revisited"-http://www.aproged.pt/biblioteca/geometryrevisited_coxetergreitzer.pdf I have another good book,but I have no idea if it's available ...


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After these references it really depends on what type of commutative or homological algebra you intend to work in. However, some of the most widely useful general references are as follows. Almost everything in the book Bruns and Herzog is lingua franca and can't be skipped. Most of the chapters of Weibel's Homological algebra. Here its I guess okay to ...


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There is indeed a standard definition like this, known as Gromov-Hausdorff convergence. You can learn much more by looking up this term; let me just briefly state the definition. First, if $A$ and $B$ are nonempty subsets of a metric space $X$, define $d_H(A,B)=\max(\sup_{a\in A} d(a,B),\sup_{b\in B}d(b,A))$ (this is called the "Hausdorff metric" on ...


1

Using Theorem (Hölder's inequality). Let (S, Σ, μ) be a measure space and let $p, q ∈ [1, ∞]$ with $1/p + 1/q = 1.$ Then, for all measurable real- or complex-valued functions f and g on S, ${\displaystyle \|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}.} \|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}.$ you can see that $$|\mathcal{K}u(y)| = |\int k(x,y)u(x)d\mu(x)|\leq \|k(x,y)\|\|...


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You may be interested in "van Kampen diagrams". I once wrote out a math.SE answer about them here. Basically, diagrams over a presentation $\mathcal{P}$ are "cyclic" words - words written on a circle equal to the trivial word in $\mathcal{P}$. The boundary of a diagram is a cyclically reduced word. (My references are the same as Paul Plummers last two, as ...


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I would be somewhat surprised if notes that focus on cyclically reduced words exists, as I doubt they are interesting to that degree, and I would be surprised if there was much to say. (Maybe some notes outlining the basics are around, from a class or something) Both Combinatorial Group Theory books (the ones by Lyndon and Schupp, and the other by Magnus, ...


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So I'm not sure about references specific to Hauptmoduln (except specialized research articles), so perhaps you would be better served by reading up on modular functions, forms and curves. I suggest Milne's online notes which are excellent, or Diamond and Shurmon's book which is also excellent. As far as your specific and interesting question regarding the ...


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Here's something I tried, thanks to Adam's comment for the basic idea for $p \ge 3$. This answer is missing some details, comments suggesting improvements are welcome, as would be other answers that fill in the gaps. It also relies on the fact (?) that: $$\forall 0 \neq a \in \mathbb{C}, 0 \neq b \in \mathbb{C}, 1 < m \in \mathbb{N} . \exists x . |a e^{...


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I'm not sure you need to use removal lemma to prove your claim, you can prove it using simple combinatoric argument: every edge in the graph can participate, at most, in $n-2$ triangles. so, removing any edge will reduce the number of triangles in the graph by at most $n-2$ triangles. If you remove $\epsilon n^2$ edges, it will reduce the number of ...



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