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Does not seem likely. I am looking at Khinchin's little book. Quadratic irrationals have bounded ""elements" because they are eventually periodic. Meanwhile, Theorem 23 on page 36, bounded elements implies only finitely many convergents with error less than $c/q^2.$ Put that with Thue-Siegel-Roth, i think higher degree algebraic have unbounded elements. ...


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Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers Discrete Mathematics with Applications A First Course in Mathematical Analysis by David Alexandr Brannan Numbers and Functions: Steps to Analysis by R.P. Burn Mathematical Analysis and Proof by David Stirling The Foundations of Analysis: A Straightforward Introduction: ...


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Finally found it, myself! It was: Caluclus: Basic Concepts For High Schools by LV Tarasov (1982). Though the convrsation is not exactly between a teacher and his student but between the Author and the Reader. I can sleep peacefully today!


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Intuitively, you can think about localization as a kind of 'zooming in' process on the prime $p$. The ring $\mathbb{Z}_{(p)}$ is like the ring $p$, but it is relatively more '$p$-focused'. This can be intuited in several ways. One possibility is that one can understand the information contained in a ring as being expressed in terms of obstructions. Every ...


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Probably. Math Reviews shows Weil, André, Introduction à l'étude des variétés kählériennes. (French) Publications de l'Institut de Mathématique de l'Université de Nancago, VI. Actualités Sci. Ind. no. 1267 Hermann, Paris 1958 175 pp. MR0111056 (Reviewer: F. Hirzebruch) It doesn't show any 1968 publication by Weil with "kählériennes" in the title.


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After looking far and wide for such a textbook, I finally settled on Serge Lang's Calculus of Several Variables. It is rigorous, easy to read, and perhaps one of the best textbooks I have ever come across. Do give it a look.


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Some keywords and starting references: Spectral clustering Community detection


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Let us assume that $\Bbb{F}_p$ does not have characteristic three (so $p>3$ if you are, as seems to be the case, only interested in prime fields). Denote $$ T(a,b,c)=\sum_{x\in\Bbb{F}_p}\omega^{ax^3+bx^2+cx}. $$ Assume that $a\neq0$. Let $z\in\Bbb{F}_p$ be a parameter. For a fixed $z$ as $x$ ranges over $\Bbb{F}_p$ so does $x+z$. Thus $$ \begin{aligned} ...


1

The following is hardly deep, but it's another characterization of the property you're interested in. Claim: Let $(A_i)_{i\in I}$ be a family of algebras. For each $i$, the following are equivalent: $A_i$ embeds in $\prod_{j\in I}A_j$. There is a homomorphism (not necessarily an embedding) $A_i\rightarrow \prod_{j\in I} A_j$. For all $j\in I$, there is a ...


2

The question if this integral is zero, is one of the classical $NP$-complete Problems given in the book of Garey and Johnson, Computers and Intractability, 1979. p. 252. There called AN14 and with the upper limit of the integral $2\pi$ instead of $\pi$.


2

Yes, this is very old work. In 1935, Koethe proved that the modules of Artinian principal ideal rings are all direct sums of cyclic submodules. Of course, your example falls into this category. In fact, all of the proper quotients of a principal ideal domain are Artinian principal ideal rings (in fact they are also self-injective, hence quasi-Frobenius.) If ...


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This is a remark concerning points (e-g): your result is mentioned in the article R. B. WARFIELD, COUNTABLY GENERATED MODULES OVER COMMUTATIVE ARTINIAN RINGS, PACIFIC JOURNAL OF MATHEMATICS Vol. 60, No 2, 1975 right at the beginning, but without proof. However some pointers are given to articles about certain non-commutative rings in which the problem ...


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One can view the winding number as actually an element of the infinite cyclic group $\Bbb Z$. Each number corresponds to an equivalence class of paths in $\Bbb C\setminus\{0\}$ (so basically paths around the origin) modulo homotopy (which begin and end at the same point, i.e. form a loop). Hence the winding number is generalized by the notion of a ...


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I recommend "The Geometrical Language of Continuum Mechanics" by Marcelo Epstein. I still have some way to go to understand everything this book says, but he makes a strong case that differential geometry and the mechanics of a continuum dovetail nearly perfectly with each other. Epstein states (I quote) "...the presence of continuum mechanics as a ...


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As stated in my comment I am unsure of the distinctions that must be made between embedding undirected graphs and directed graphs. To me it seems that to embed a directed graph you just embed the underlying undirected graph then orient the edges (possibly "splitting" an edge in the the case our directed graph has an edge in both directions between two ...


4

For something that has a little bit of everything, check out Partial Differential Equations by Walter A Strauss It is a great intro to all of these topics. For more in depth references, I reccommend these to anyone studying this field: Partial Differential Equations- Lawrence C Evans Numerical Solution of Partial Differential Equations: An Introduction- ...


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The series for $1/\pi$ is proved in J. M. Borwein and P. B. Borwein, Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987. See also Motivation for Ramanujan's mysterious $\pi$ formula


1

A nice book focused on examples and treating much of the classical families of PDE step by step is the Partial Differential Equations of L. C. Evans. It seems to cover near all your syllabus, maybe without the discretization part. Whatever, it is typically a good first course companion. Hoping it will be of some help.


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The best choice might depend on which type of book you would prefer. In my opinion: If you want to privilege clarity, I would suggest you "Models for Probability and Statistical Inference: Theory and Applications" by James H. Stapleton: this is a relatively short but clear and comprehensive book on probability and statistical inference, with a lot of ...


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You can build an embedding inductively, since the complement of a graph in $\mathbb{R}^3$ is path-connected: Place your vertices at distinct points. Label the edges with some ordering $E_1, E_2,\ldots$, et cetera. Letting $\mathcal{G}_0$ denote the set of vertices, note that $\mathbb{R}^3 \setminus \mathcal{G}_0$ is path-connected. Thus we can represent ...


1

Assuming that Schnaderhuepfel are the (south?) German equivalent of limericks, I offer the following, which I heard from my father (but the misspellings are my own): Mir fehlt nur ein Hilfssatz, Dann bin ich ein Gauss. Doch den Hilfssatz, den Hilfssatz, Den krieg ich nicht raus.


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Huang, Chi-fu, and Robert H. Litzenberger. "Foundations for Financial Economics." Englewood Cliffs, NJ: Prentice-Hall (1988).


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Hint: Embed in $\mathbb{R}^3.$ Start by drawing it on the $x,y$ plane with crossings OK, but make each crossing only at a point, in a reasonably "nice" way. There are then only finitely many such crossings, and perhaps one can move the edges up/down by a small perturbation to eliminate the crossings. (This is only a hint since I don't know if it can be ...


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$$\int_1^{\sqrt[3]{3}} t^2\mathrm{d}t\cdot\cos\left(\frac{3\pi}{9}\right)=\ln(\sqrt[3]{e})$$ $$\text{Integral t squared dt,}$$ $$\text{from 1 to the cube root of $3$,}$$ $$\text{times the cosine,}$$ $$\text{of three pi over $9$,}$$ $$\text{equals log of the cube root of $e$.}$$ You can find some more here: http://www.trottermath.net/humor/limricks.html


6

Not sure without more context, but my best guess is the mathematician was Paul Erdős and technique you are talking about is the Probabilistic Method.


0

It seems that I have found the answer in this article. The main result (restricted to the interval $(0, 1)$) is that only $$ \{\sin(n \pi x)\}_{n \in \mathbb{N}} \quad \mbox{or} \quad \{1, \cos(n \pi x)\}_{n \in \mathbb{N}} $$ forms a periodic orthogonal basis in $L^2(0, 1)$. I'll leave this answer here for better search on this question.


0

I have an existence proof that you can make an asymptotically angularly isotropic three dimensional cellular automata rule that works in a cartesian lattice. It creates gliders which can move in (asymptotically) any direction at (asymptotically) any velocity. However the rule as written does not deal well with collisions between these gliders. But at least ...


2

There may be such a definition in the literature, but I doubt that it would do the alphabet justice. Can you think of a definition of the letter "A" that encompasses all of the examples below? (This figure is taken from Douglas Hofstader's book Surfaces and Essences.)


3

The book you want is Mark Levi, The Mathematical Mechanic: Using Physical Reasoning to Solve Problems.


1

I think most discrete math books are junk (e.g. things like Johnsonbaugh's book) aside from Knuth, Oren and Paschnik's Concrete Mathematics - they don't go into enough detail for getting useful things out of them. You're better off with some basic combinatorics book like Van Lint's "A Course in Combinatorics" + Wilf's Generatingfunctionology (free on ...


0

I agree that the best book to start off rigorous Set Theory is Naive Set Theory by Halmos. You can progress to more advanced texts thereafter. And there are two different constructions (that I know of that is) of the Real Numbers. One is the method of Dedekind cuts. In my opinion no book is better than the true classics. Look for Foundations of Analysis by ...


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For real numbers I like MAA's book by Henle Which Numbers Are Real, it's very didactic and has lots of exercises. The added advantage is that you can also go over constructions of other number systems with it when you are ready for them, complex, quaternions, etc. For Discrete Mathematics I like Rosen, well structured and well written, also lots of ...


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For set theory, the following is a good book to start out with: Naive Set Theory by Halmos, http://books.google.com/books?id=x6cZBQ9qtgoC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false, Then for discrete, Concrete Mathematics by Graham, Knuth, & Patashnik is good if you have a bit of math background. As far ...


3

Peter Woit, the author of the book "Not Even Wrong" and a blog by the same name, has been working on a book on quantum mechanics as described by representation theory. The latest draft may be found at the following link: Quantum Theory, Groups and Representations: An Introduction.


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Howard Georgi's "Lie Algebras in Particle Physics" is good, if more intended for the physicist going towards the math than vice versa. It should provide a lot of context, though, and there's a PDF version floating around on google. I'd say similar things about these two introductions to aspects of high-energy theory [1] [2]. I'll see if I can remember some ...


5

There's been a lot of work done on classifying manifolds up to homeomorphism, diffeomorphism, homotopy equivalence, etc. The basic question is whether a homotopy sphere in dimension $n$ is actually "equivalent" to $S^n$. I put "equivalent" in quotes because there are many different categories in which you ask the question: smooth, topological, PL, etc. More ...


3

According to this paper (which I just found with a Google search, so it'd probably be useful to look up the citation it lists), the Schur multiplier of $GL_n(\mathbb{F}_p)$ is trivial for $p\not = 2$. The paper mentions that it was "computed only recently" (the paper it lists was published in 2008), though that might just be the general case rather than $p = ...


1

Yes, the Feferman-Levy model is described in Jech The Axiom of Choice in Chapter 10 (Theorem 10.6), as well you can find nice detailed accounts in both Arnie Miller's papers about Dedekind-finite Borel sets and Long Borel Hierarchies (both of which you can find here), as well Ioanna Dimitriou M.Sc. and Ph.D. thesis (both of which appear here). The idea is ...


8

Yes, the reference is not correct. The correct reference (which can be found in the original Russian edition) is to Shilov's Mathematical Analysis: Special Course.


1

You may want to spend some time looking into the history of this subject, and to see how this Math was originally used and how it evolved. Most of the subject has its roots in convolution integral equations on the real line, and the important techniques of Complex Analysis to solve such problems through factoring holomorphic functions originated with Norbert ...


0

Here is the list of examples we have which were directed by the comments thus far: $PSL_n(K)$ for when $K$ is an infinite field and $n\geq 2$ 1 The finitary alternating group $A(\kappa)$ for any infinite cardinal $\kappa$ 2 REFERENCES This entry at Groupprops shows that $PSL_n(K)$ is actually simple for all $n\geq 2$ and any $K$ except for $PSL_2(\Bbb ...


1

Here is a very subjective list: Arnold, Mathematical Methods of Classical Mechanics This book treats classical mechanics from mathematical point of view and introduces a wealth of various mathematical concepts (such as a differential manifold or differential form) in a right context. The main mathematical tool is ODE and it is really inspiring to see how ...


0

A few people have mentioned, if you study abstract algebra, then you know that integers form a ring under addition and multiplication. If you go on further to study some category theory then you will see that it is not surprising because, to put it vaguely, you can try to find bijections between the sets of different algebraic structures. Category theory ...


0

The book "Banach Algebra Techniques in Operator Theory" written by Ron Douglas, has an excellent chapter on Toeplitz operators at the end of the book. It's where I started when I began my PhD research. He also has a follow up book "Banach Algebra Techniques in the theory of Toeplitz Operators." You will need some background in working with Hardy Spaces. ...


1

You can find this in the classical books on Lie algebras, e.g. N. Jacobson's book. Since $\mathfrak{gl}_n(K\simeq \mathfrak{sl}_n(K)\oplus K$ is a reductive Lie algebra, and $\mathfrak{sl}_n(K)$ is simple (if the characteristic of $K$ is zero, or not dividing $n$), we know all the ideals. Recall that a simple Lie algebra has only the trivial ideals. The ...


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It is "almost" legitimate to ask : << However, its not at all clear how to formalize the various chain rules. For example, the single-variable chain rule says: $$\forall x\left[\frac{\partial z}{\partial x} = \frac{\partial z}{\partial y}\frac{\partial y}{\partial x}\right]$$ The problem is that $y$ is being used both as a variable symbol, and also as ...


1

I'm going to agree with Bryant in the mentioned link and recommend O'Neill's Elementary Differential Geometry. It is a gentle enough introduction to differential geometry, uses the common language and will prepare you for the usual problems in $\Bbb R^3$ while giving you a hint of what comes next. It may be profitably followed by his second book and/or ...


1

In my opinion the best Differential geometry book is John M. Lee - Introduction to Smooth Manifolds followed by Loring W. Tu - Introduction to manifolds and Jeffrey M. Lee - Manifolds and Differential Geometry. For connections and Riemannian Geometry look also John M. Lee - Riemannian Manifolds: An introduction to curvature.


2

For your second question, by the second formula under "Definitions" on the wikipedia page, the term with the largest power $n$ in $P_n^{(\alpha,\beta)}(z)$ is $$ \frac{\Gamma(\alpha + \beta + 2n + 1)}{n! \Gamma(\alpha + \beta + n + 1)} \left(\frac{z-1}{2}\right)^n \sim \frac{\Gamma(\alpha + \beta + 2n + 1)}{2^n n! \Gamma(\alpha + \beta + n + 1)} z^n $$ ...



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