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A lower dimensional surface always has Lebesgue measure zero. Indeed, if $j < k$, and $\mathcal H^j(E) < \infty$, $\mathcal H^k(E)=0$. This is a straightforward consequence of the definition of the Hausdorff measure; if this isn't obvious, you should try proving it yourself. On the other hand $\mathcal H^n = \mathcal L$. It's not so hard to see that $\... 1 After these references it really depends on what type of commutative or homological algebra you intend to work in. However, some of the most widely useful general references are as follows. Almost everything in the book Bruns and Herzog is lingua franca and can't be skipped. Most of the chapters of Weibel's Homological algebra. Here its I guess okay to ... 2 It's not clear to me what needs to be cited away. As far as I can tell, here's an answer. Consider the vector field$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 ... 0 The Matrix Cookbook is the most comprehensive one. Can't post the link due to my missing rep., but it's first hit on Google. See these, too: Matrix Calculus - Colorado and Matrix Algebra - LMU - German. 2 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 ... 3 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 different candidate definitions of manifolds competed ... 5 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. 1 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. 0 http://www.math.uiuc.edu/K-theory/ is one site which contains some preprints, some survey articles. I thought this may be useful for other users, which is why i made it as an answer. 0 Here are three rather complementary recommendations: Introductory Mathematical Statistics: Principles and Methods by Erwin Kreyszig This classic is an elementary introduction which presents in the first part Descriptive statistics, followed by Probability Theory and the main part is Statistical Inference. E. Kreyszig has the talent to make ... 1 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. 2 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. 0 This is a very long comment, and not an answer! I shall write only about the Grothendieck cohomology of sheaves of Abelian groups over a topological space! Let$X$be a topological space; one can define \begin{equation*} \forall U\subseteq X\,\text{open},\,\mathcal{C}(U)=\{f:U\to\mathbb{R}\,\text{is continuous}\} \end{equation*} where over$\mathbb{R}$... 0 Zorich, "Mathematical Analysis". Intuitive explanations, physical connections; in short, a human touch'' of a caring teacher. 1 For olympiads Number Theory this is a must-"Number Theory-Andrescu Titu"-https://blngcc.files.wordpress.com/2008/11/andreescu-andrica-problems-on-number-theory.pdf And for Geometry this one-"Coxeter-Geometry Revisited"-http://www.aproged.pt/biblioteca/geometryrevisited_coxetergreitzer.pdf I have another good book,but I have no idea if it's available ... 1 So I'm not sure about references specific to Hauptmoduln (except specialized research articles), so perhaps you would be better served by reading up on modular functions, forms and curves. I suggest Milne's online notes which are excellent, or Diamond and Shurmon's book which is also excellent. As far as your specific and interesting question regarding the ... 5 Here is a proof involving abstract linear algebra. View$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$is ... 2 Take a look at the World War 2 Education Manual EM310 Solid Geometry. It can be ordered on Amazon for a very reasonable price. Content is explicitly designed for self-instruction, with answers provided--consider the intended audience. It covers the topics you want and does so without analytic geometry: is intended to be in the curriculum after plane ... 0 I recommend you "A COURSE ON GROUP THEORY" by John S. Rose. You can find it here. Unfortunately, it is not well written, however it contains plenty of results. Moreover, it is not introductory and it focuses on the notion of group actions on sets and on groups, see Burnside Theorem and else. (I don' t know if you can find it free in online form) 2 I think one of the best and most complete book I know on (especially finite) group theory is Antonio Machì's http://www.springer.com/us/book/9788847024205 (original language: italian), despite the word "Introduction" in the subtitle. The author is also the translator of Herstein's much more elementary$\textit{Algebra}$. There are also many exercises and ... 1 You may be interested in "van Kampen diagrams". I once wrote out a math.SE answer about them here. Basically, diagrams over a presentation$\mathcal{P}$are "cyclic" words - words written on a circle equal to the trivial word in$\mathcal{P}$. The boundary of a diagram is a cyclically reduced word. (My references are the same as Paul Plummers last two, as ... 0 If you are still looking, here is a nice reference: Calculus of variations and harmonic maps-By Hajime Urakawa, see chapter 4, proposition 3.14 (page 146). It is a very interesting book in general. 3 I was quite surprised to find that "proexample" is actually a word. Google turns up, for example (or should that be "for proexample"?) Corcoran, J. 2005. Counterexamples and proexamples. Bulletin of Symbolic Logic 11(2005) 460. Are you that John Corcoran? If so, you seem to be the main published user of this word, so you are the one who should be ... 1 I would be somewhat surprised if notes that focus on cyclically reduced words exists, as I doubt they are interesting to that degree, and I would be surprised if there was much to say. (Maybe some notes outlining the basics are around, from a class or something) Both Combinatorial Group Theory books (the ones by Lyndon and Schupp, and the other by Magnus, ... 4 To directly answer your question: Andrej meant "monomorphism". Since the objects of a topos don't typically consist of (raw, unstructured) sets, the usual notions of injectivity and surjectivity don't apply. But for convenience, people still talk about "injections" and "surjections" in those contexts (and mean monomorphisms and epimorphisms, respectively). ... 1 Here's something I tried, thanks to Adam's comment for the basic idea for$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: $$\forall 0 \neq a \in \mathbb{C}, 0 \neq b \in \mathbb{C}, 1 < m \in \mathbb{N} . \exists x . |a e^{... 2 Answer to part 1: If f is an isomorphism of algebraic varieties, it must be a regular map (which is therefore analytic, as polynomials and their appropriately-taken quotients are analytic). Similarly, it's two-sided inverse must also be regular and thus analytic, and so we have that f is also an isomorphism of analytic varieties. Therefore it must be a ... 0 Given any real analytic function in a neighborhood of 0, we get f(A) defined as long as some induced norm for A is smaller than the radius of convergence for the Taylor series of f. Or, of course, if the radius is infinite, as in e^x. However, the Cayley Hamilton Theorem says that f(A) can be rewritten as a polynomial in A, of degree no ... 0 The claim is false, but for a sort of stupid reason. Let's take M to be the plane, and S to consist of two circles: the unit circle C at the origin, and the circle Q of radius 1/4 centered at (-1/4, 0); note that Q contains the origin. Now at the point ( (0,0), 0) of NS, the exponential map is just lovely. On the other hand, the point (... 1 There is indeed a standard definition like this, known as Gromov-Hausdorff convergence. You can learn much more by looking up this term; let me just briefly state the definition. First, if A and B are nonempty subsets of a metric space X, define d_H(A,B)=\max(\sup_{a\in A} d(a,B),\sup_{b\in B}d(b,A)) (this is called the "Hausdorff metric" on ... 1 Using Theorem (Hölder's inequality). Let (S, Σ, μ) be a measure space and let p, q ∈ [1, ∞] with 1/p + 1/q = 1. Then, for all measurable real- or complex-valued functions f and g on S, {\displaystyle \|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}.} \|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}. you can see that$$|\mathcal{K}u(y)| = |\int k(x,y)u(x)d\mu(x)|\leq \|k(x,y)\|\|... 2 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: $$a^x \equiv b \bmod p^s \implies a^x \equiv b \bmod p^{... 0 The theory of wavelets is a broad field, but looking at your background and the paper you mention I think the book Funtions, Spaces and Expansions by Ole Christensen covers everything you need to know. The prerequisites for reading this book are linear algebra and elementary analysis. The book introduces basic concepts as normed spaces, Hilbert spaces and ... 0 I'm not sure if I understood correctly what you mean, but if a function \;\alpha\; on an interval \;[a,b]\; is not of bounded variation then we there always will exist continous functions on that interval that are Riemann-Stieltjes integrable with respect t \;\alpha\; , for example \;f(x)\equiv1\;, since for any partition of that interval we get$$\... 1 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 ... 2 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 ... 3 I don't know if you'd call this a formal definition, but a counterexample to $$\forall(x \in X, y \in Y)P(x,y)$$ is a basically a substitution$(x:= x_0, y:= y_0)$together with a proof that $$(x:= x_0, y:= y_0) P(x,y) \rightarrow \bot.$$ By the way, we can think of a substitution like$(x:= x_0,y:= y_0)$as being a bit like an ordered tuple, in this case$(...

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In my experience the author’s symmetric is more often called a semimetric, though I have seen the auther’s term used before. For example, Gary Gruenhage uses it in Definition $\mathbf{9.5}$ of ‘Generalized Metric Spaces’ in Handbook of Set-Theoretic Topology, K. Kunen & J.E. Vaughan, eds., though he uses the usual term semimetric space for a set endowed ...

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Maybe Mercio has answered enough... meanwhile, I am getting a better idea of what you want: here is the Lagrange method for finding the cycle of neighboring reduced forms in the $SL_2 \mathbb Z$ of $x^2 + xy - 48 y^2.$ Look at the absolute values of my "delta" numbers at the right of each line, these are the partial quotients in a continued fraction. EDIT: ...

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Let $\epsilon = \Delta \mod 2 \in \{0 ; 1\}$ and $t = \frac 12 (\sqrt \Delta +\epsilon)$. $t$ is a quadratic algebraic number and it satisfies the equation $t^2 - \epsilon t - \frac 14(\Delta-\epsilon)$. Consider $t$ together with his conjugate $\overline t < 0$ (because $\Delta>1$) and look at what happens during the continued fraction process. If $(... 20 First a joke: I don't know what a counterexample is, but I can recognize one when I see one. In a first order context, something like the following begins to capture the notion. Let$T$be a theory over the language$L$. and let$\phi$be the sentence$\forall x_1\dots\forall x_n\psi(x_1,\dots,x_n)$. Then a counterexample to$\phi$in the context$T$is ... 5 You can think of our knowledge of math as divided into (1) knowledge about mathematics itself—theorems of algebra, analysis, number theory, topology, etc.—and (2) knowledge about how to do mathematics—paradigmatic cases, heuristics for approaching a problem, counterexamples, etc. Counterexamples are part of our knowledge of how to do ... 0 To use proof by counterexample, the original statement has to be of the form "For all x, statement$A_x$is true". (1) Thus, showing that for some x, the statement$A_x$is false, proves that the original statement is false. A formal definition of a counterexample could be "an example that proves a statement false" or something. This is a rigorous way of ... 10 A counterexample is a special case of a general claim: a case that shows the claim to be false. This is really just a matter of common sense and everyday logic that applies far beyond mathematics. It isn't usually regarded as something that needs formal definition. Once you have a counterexample, you know that the general claim isn't true, and that's the end ... 1 You should consider carefully whether you need to satisfy$\sum a_{i}u_{i}=Ua=v$exactly, or whether you can live with a solution where$\| Ua-v \|_{2}$is small enough (and then of course you have to decide what's a small enough error.) You should also consider carefully what property of the vector$a$you want to optimize. If (as another answer has ... 1 First of all, by using something like$f(x)=\arctan x$(which decreases distances), we can map our set bijectively to a bounded set. For such a set$A\subseteq [a,b]$, we can now apply the map$A\to [0,|A|]$,$x\mapsto |A\cap[a,x]|$, which has the required properties. 1 If we have a linear system$A x = b$with infinitely many solutions, then finding the solution with the minimum$2$-norm is the least-norm problem $$\begin{array}{ll} \text{minimize} & \|x\|_2^2\\ \text{subject to} & A x = b\end{array}$$ Assuming that$A\$ has full row rank, the least-norm solution is $$x_{\ln} := A^T (A A^T)^{-1} b$$ Including ...

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Have you consulted Peter Morter's book "Brownian Motion"? Chapter 8.3. It is explained there that Martin Capacity is zero if and only if the F-capacity is zero (where F is the potential kernel). I guess it only matters whether capacities are positive or zero, so this migh be helpful.

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Trolling Euclid by Tom Wright is my favorite book about unsolved problems. Very nice read.

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reference on the subject Isospectral vs Isometry of the problem of Carolyn Gordon, David L. Webb and Scott Wolpert. The problem is from an article by Marc Kac and is known as "can you hear the shape of a drum". The negative solution in high dimension was found by Milnor before Kac's paper, but getting examples for Kac's problem about planar domains was ...

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