# Tag Info

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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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Hint: for Pick's Theorem ($AREA = I+\frac{B}{2}$-1) if there are more integer points contained in the polygon ( I ), then there are more chance to find some rectangles...so you have to place the polygon in order to have the least number of lattice points that are also in the polygon's perimeter ( B )...

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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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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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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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For Matlab I prefer "Introduction to Matlab" by Ross Spencer and "Applied mathematical problems with Matlab" by D. Xue and Y. Chen (but the last book is with problems that need to be solved).

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Matlab help is fantastic. Seriously,I've met three or four tools/products that had really good built-in help/documentation and Matlab was one of them. Basic programming and basic tasks are covered very well. Many toolboxes also have the same high-class documentation. I never felt that I need book when I was doing Matlab.

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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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http://www.math.uh.edu/~minru/4350-11/geodesic.pdf See pages 3 and 4 of this PDF!

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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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For all $n\geq2$, there exists $x,y,z$ positive integers such that $\cfrac{4}{n}=\cfrac{1}{x}+\cfrac{1}{y}+\cfrac{1}{z}$ if and only if $n$ has no common factor with $\varphi$ and $$\cfrac{4}{n}=\cfrac{\log(\pi^{\varphi})}{\log(\sqrt{\pi^{\frac{n}{2}\varphi}})},$$ where $\pi$ is any constant$>1$ and $\varphi$ the sum of $x,y,z.$ The $K$ case is easy ...

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Not every continuous time minimal Markov chain has the Feller property. See a counterexample.

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Might be a bit late, but one reference that looks nice is Differentiation Under the Integral Sign Harley Flanders The American Mathematical Monthly , Vol. 80, No. 6 (Jun. - Jul., 1973) , pp. 615-627 Published by: Mathematical Association of America Article DOI: 10.2307/2319163 Article Stable URL: http://www.jstor.org/stable/2319163 Avaiable at jstor

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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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The theory of force directed drawing algorithms(further theory of Tutte's barycenter method ) is the way to solve it if is planar...if is not you drop in the previous answer. BtW cool game :)

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Unless you are a mathematician interested in the calculus of variations approach, before going into optimal control and estimation I'd start with a more introductory and less specialized text on classical control such as Feedback Systems An Introduction for Scientists and Engineers Karl Johan Åström Richard M. Murray There are many other good textbooks for ...

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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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Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts) [Hardcover] Gerald B. Folland (Author) http://www.amazon.com/Real-Analysis-Techniques-Applications-Mathematics/dp/0471317160

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I've used Elstrodt's "Maß- und Integrationstheorie" a lot, and believe it to be an excellent introduction to the topic. The book is german, though - I don't know if there's an english translation.

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Gripenberg, Londen, Staffans, Volterra Integral and Functional Equations (Encyclopedia of Mathematics and its Applications) Zemyan, The Classical Theory of Integral Equations, A Concise Treatment, Birkhauser DiBenedetto, Partial differential equations, chap.4

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Here's a link to a thorough and authoritative source: http://www.cs.utexas.edu/users/inderjit/public_papers/HLA_SVD.pdf Inderjit Dhillon is a leading expert on eigenvalues.

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You might want to look at Statistical Inference by George Casella and Roger L. Berger. It's the book I used for my graduate inference class.

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It appears that the transition semigroup need not be Feller. An example: Denote the non-negative integers by $\mathbb{N}$. Define, for $i,j\in\mathbb{N}$, $$q(i, j) =\begin{cases} 0 &\mbox{ if }i =0\\ i^2(\delta_{i-1, j}-\delta_{i,j}) &\mbox{ otherwise.}\end{cases}$$ Then the backward equations are $$p_t'(0,0) = 0$$ and $$p_t'(i, i-1) = ... 1 For an LPV approach based on a set of linear models defined on a grid of parameter values you should look at: F. Wu, Control of linear parameter varying systems. Ph.D. Dissertation, University of California at Berkeley, 1995. (http://www.mae.ncsu.edu/wu/paper/PhDthesis.ps) For an LPV approach where the system can be modeled using a linear fractional ... 0 Two fantastic books that every mathematician should know are Counterexamples in Analysis by Olmsted Counterexamples in Topology by Steen and Seebach, already mentioned in the comments. 0 Both of the following books have good treatments of these ideas: Sobolev Spaces on Riemannian Manifolds by Emmanuel Hebey Some Nonlinear Problems in Riemannian Geometry by Thierry Aubin 2 Let X=D^{-c}H with obvious notations and x\gt0, then [X\gt x]=[H\gt xD^c] hence, by independence of H and D,$$P(X\gt x)=E(\exp(-\lambda xD^c)).$$Since the distribution of D is known, at least in principle this expectation can be computed. If D is uniform on (0,1), the change of variable v=tu^c yields$$ ...

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Hint: You need to apply the method of bivariate transformation of random variables. You can define $$X \sim U(a,b)$$ and $$Y\sim \exp(\lambda)$$ (to simplify notation, $X$ stands for your $d_{mn}$ and $Y$ for $h_{mn}$) and then define $$U=X,\qquad V=\sqrt{(1/X)^a}\cdot Y$$ Solving for $X$ and $Y$ you find that $$X=U, \qquad Y=\frac{V}{\sqrt{(1/U)^a}}$$ from ...

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Basically, you are searching for Euclid's Elements Book V which covers the abstract theory of ratio and proportion. In modern setting, see David Hilbert, The Foundations of Geometry (1902) : §15. AN ALGEBRA OF SEGMENTS. All the development of geometry is reviewed into Francis Borceux, An Axiomatic Approach to Geometry. Geometric Trilogy I (2014); see ...

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You can find it in most of books an analysis of functions of several variables. Let me mention Apostol, Bartle, Marsden & Hofman and Spivak.

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For combinatorics, Kenneth Bogart's Combinatorics through Guided Discovery is outstanding. For self study and guided exploration.

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The lecture notes by William Chen cover much of the starting undergraduate curriculum in math, and are very readable.

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You could try Precalculus (Stewart, Redlin, Watson). It barely touches on number theory, and abounds more in exercises than in proofs, but given your context (elementary concepts, not too rigour, self study) it might be useful.

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I would recommend you to try Math Olympiad Books, as they will be very interesting along with providing solid concepts and problems.

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Try this book: http://www.amazon.co.uk/Fundamentals-University-Mathematics-Albion-Applications/dp/1898563101/ref=pd_sxp_grid_pt_0_1 It covers the topics you listed above and many more and is written in a way that's meant to prepare one for university/college study.

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There are unsolvable puzzles. Untangle gives you a graph, i.e. a set of vertices (points) with edges (lines) between some of them. You are asked to find an embedding of the graph into the two-dimensional plane such that no two edges cross each other. A graph where that is possible is called a planar graph. Not all graphs are planar - the smallest non-planar ...

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This may not be exactly the type of reference you're seeking, but G. H. Hardy's book Orders of Infinity may be of interest. Wikipedia's definitions of "degree" are defined by a function's behavior near infinity, and therefore do not bound the number of real roots for an arbitrary function. If $b > 0$ and $c$ are real numbers, the function $$f(x) = c + ... 0 I like Topology and Geometry by Glen E. Bredon, especially as a reference for cohomology and homology products. I second Weibel's book on homological algebra, especially for the universal coefficient theorem. 1 That's a tall order! I would wager that very few people understand Freedman's proof completely. I found this series of video lectures that he gave to be quite helpful. In particular, it will give you an idea about the sort of mathematics that is involved. Also, Freedman and Quinn's book on 4-manifold topology is great! Even if you aren't to the point ... 1 Start by reading Milnor's "Lectures on h-cobordism theorem", since Freedman's proof is a very difficult variation on Smale's proof in higher dimensions. 0 Look for solutions of 1+y^n=z^n, i.e.$$z = (1+y^n)^{1/n}= \sum_k a_k y^{nk}, where $a_k = \binom{1/n}{k}$. If $p > n$, then $v_p(a_k) \geq -v_p(k!)$, hence this serie converges for $v_p(y^n) > (p-1)/p$.

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Not sure what you consider to be elementary but it can be solved with Schur's theorem. See: http://math.mit.edu/~fox/MAT307-lecture05.pdf It is theorem 4 in that paper.

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The basic point of manifold theory, the point which we make great effort to affirm in texts and elementary treatments, is that when you do calculus on a manifold it is the same as it is in $\mathbb{R}^n$ locally. If you write the formula for functions from one manifold to another in terms of coordinate charts then the manner in which you differentiate is ...

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A good choice for a type of ring "sharing many properties with $\Bbb Z$" would be any principal ideal domain, and there is a simple classification theorem for finitely generated modules over such rings. You've picked an even narrower subclass of finitely generated $\Bbb Z$ modules: that of the finitely generated free $\Bbb Z$-modules. Quite nicely, the ...

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I recommend the beautifully illustrated N. Saveliev's Lectures on the topology of 3-manifolds.

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You can try to read Hatcher's book. It does not require a lot of background and covered a lot basic material. However, I do not know if Heegard diagrams and Haken manifolds is covered there as I never finished reading it. Real experts in the forum should be able to give better recommendations.

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Two possibilities: Take what you have learned for granted (yes, there are holes; but what you got explained by vigorous handwaving can be proved rigurously). Dig down to the formal proofs of new material as you need it, perhaps filling in whatever underlying material makes you uneasy. Start from the beginning, fill in any and all holes. As a matter of ...

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To understand the basics of most of the math used in engineering, I suggest you start with real analysis, then move on to complex analysis, functional analysis an measure theory. Real analysis might be a bit hard a first, if you're not used to finding and writing rigouros proofs, but I'll get easier afterwards.

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If you also require the obvious compatibilities, you get an enriched functor.

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For many reasons, I would suggest Weibel's "Intro to Homological Algebra", because it puts things like "group cohomology" into a somewhat larger context, enabling comparisons to other things... Perhaps no reason to be a complete slave to the ordering of topics therein... but to see that "group cohomology" consists of the right-derived functors of the ...

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