# Tag Info

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See http://en.wikipedia.org/wiki/Lists_of_integrals#Definite_integrals_lacking_closed-form_antiderivatives. This link tells you a list of integrals not expressible in a closed form expression.

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I wouldn't think so. A complete graph should always have a spanning tree. Pick your favorite vertex $u$ from the vertex set $V$, infinite or not, and construct a giant star tree with $u$ as the center. That is, construct the tree $T = (V, F)$ where the edge set is $F = \{uv : v \in V \setminus \{u\} \}$. This has to be a spanning tree (and as pointed out ...

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In Weyl's lemma we do not need to assume that $u\in L_{\mathrm{loc}}^1(\Omega)$, but rather that $u$ is locally expressible as a linear combination of (finitely many) partial derivatives of locally integrable functions. Weyl's lemma is generalized in several directions: For non-constant coefficients, for arbitrary degree elliptic operators (even ...

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The book "Riemannian Geometry" by Wilhelm Klingenberg does include an infinite-dimensional setting from the start, if I remember correctly, that is, your manifold is modelled on any separable Hilbert or Banach space (for a Riemannian metric, you obviously need a Hilbert space though). The most extensive treatment that I know if is the Book "The convenient ...

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Answer 1. The closest thing to this construction I have seen is the Eilenberg-Watts theorem, which says that for any right exact functor $F\colon R$-mod$\to Ab$ that commutes with arbitrary direct sums, we have a natural isomorphism $F(-)\cong F(R)\otimes_R-$, where $F(R)$ is given its natural structure as a right $R$-module. The key observation to ...

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See the web page Student Projects in Differential Equations maintained by David Arnold at College of the Redwoods. I remember seeing this back in the late 1990s when I taught an ODEs course for a few years, and I was quite surprised just now to find that they're still available and still on-going when I did a quick search for it.

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This is not a textbook, but this unit from MIT might help. The unit is part of MIT's 18.02 course, which is their multivariate calculus course, with the first unit (the one linked) being about vectors and matrices. The units after that talk about calculus with matrices, and should introduce any notation you are having trouble with. Simply look through the ...

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My notes http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/10_thetas_equi.pdf linked-to from http://www.math.umn.edu/~garrett/m/mfms/ prove the result when dimension is divisible by 8. About history and priorities: my impression was that Kloosterman had "known" the cases of dimension 5 and above by 1926. In any case, there is a bibliography in that ...

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Every $C^1$ domain, and more generally a Lipschitz domain, is a Sobolev extension domain, meaning that Sobolev functions on it can be extended to Sobolev functions on $\mathbb R^n$. In particular, all embedding theorems for Sobolev spaces hold on such domains. When $p>n$, Morrey's inequality gives a continuous embedding of $W^{1,p}$ into $C^\beta$ ...

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Regarding Matlab, I'd refer you to The Elements of MATLAB Style by Richard K. Johnson. This book is reviewed by Loren Shure on her MathWorks blog. Beyond the many tutorials available might also look at this post on Good MATLAB Coding Practices that links to several free PDF guides for users that have already learned the basics. You can browse a huge list of ...

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I suggest two real classics. The first one is Introduction to Calculus and Analysis Volume I and II from Richard Courant. The first volume is based on $\mathbb{R}$, the second treats Analysis on $\mathbb{R^d}$. Metric spaces are not considered. If you are looking for lots of good examples from Analysis without (explicitely) using the concept of metric ...

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You might consider looking at "Analysis in Euclidean Space" by Kenneth Hoffman. The book concentrates attention on $\mathbb{R}^n$, and has some good exercises. Generalizations refer to normed linear spaces, generally $\mathbb{R}^n$ with some norm, or $\mathbb{C}$, the complex numbers. This is great preparation if you eventually decide to get into functional ...

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First, you use $n$ in your post twice, in different meanings. First $n$ denotes the dimension of the phase space, and second, $n$ means the order of the terms you keep. I will use $d$ instead of $n$ in the former case. A lot is known about the case $d=2$, although not everything. Assume that we are given polynomial system on the plane  \dot x=P(x,y),\\ ...

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Here's one way to look at this: Suppose $\nabla$ is the unique covariant derivative operator associated with the metric tensor $g$ on the surface $S$, i.e. it is the unique symmetric connection on $S$ with $\nabla g = 0$. It is well know that the geodesics $\gamma(t)$ of $g$ satisfy $\nabla_{\mathbf w}\mathbf w = 0, \tag{1}$ where $\mathbf w$ is the ...

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Introductory Functional Analysis and applications by Erwin Kreszig ?? By the way Brezis is very good in the sense the theory is kind of developed by asking you to do exercises. For measure theory Halmos ? But why don't you try this ? Prove all the propositions and theorems and corollaries by yourself without first looking at the explained text. That ...

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I would not say that "increasing 'messiness' as increasingly general cases are considered" is a "cost" of the continuum approximation or averaging procedure. The more general cases would be far messier still without the continuum approximation/averaging. There are branches of statistical mechanics in which discrete phenomena are modeled by discrete ...

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A lot of books which cover this are under the name "image processing". Gonzales and Woods' Digital Image Processing is the standard reference for image processing. Lim's Two-Dimensional Signal and Image Processing is also a good book, but is quite old (early 1980s). Some people I know also like Bovik's The Essential Guide to Image Processing. The ...

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This thread may help. The subject is called "Diophantine analysis" (or "Diophantine approximation(s)") and I enjoyed much Edward Burger's introduction "Exploring the Number Jungle". Other references may be found in Steuding's fine online course.

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I'm not entirely sure whether any books will directly tell you how to solve those particular problems, but if you're looking for a good number theory book. Hardy & Wright's An Introduction to the Theory of Numbers may be of good help (both in the continued fractions topic and in number theory in general.) Or for algebraic number theory, I would ...

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