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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.


7

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


7

First of all: you shouldn't give up on problems after 30 minutes. Take a break, try a different problem, maybe wait a few days and try again -- you'll gain a lot more from the problem if you struggle and solve it yourself. Having access to solutions can be helpful, but you don't want to find yourself relying on them. (There's a phrase that gets thrown around ...


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.


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 ...


5

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 ...


4

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 ...


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- ...


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 = ...


3

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


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.


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.)


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 $$ \frac{\Gamma(\alpha + \beta + 2n + 1)}{n! \Gamma(\alpha + \beta + n + 1)} \left(\frac{z-1}{2}\right)^n \sim \frac{\Gamma(\alpha + \beta + 2n + 1)}{2^n n! \Gamma(\alpha + \beta + n + 1)} z^n $$ ...


2

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 ...


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

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 ...


2

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


2

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 ...


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 ...


1

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 ...


1

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 ...


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 ...


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 ...


1

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 ...


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 ...


1

Get Thomas Apostol's book "Mathematical Analysis." I'm studying for a Modern Analysis qualifying exam, and finished a course in "baby" reals a year ago using Rudin and occasionally referring to Apostol on my own for help. Out of the ridiculous set of texts I've checked out from the library to help me study for my Analysis qual ("Baby" and "Mama" Rudin, ...


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.


1

Surely there are! It's a bit arrogant to even think that maybe another language has nothing good that you can't find in English... I wish I knew French, German, Russian, Italian, Polish, Japanese, Hungarian, Chinese, Dutch and all other languages... English is by far the most useful, but learning another language is always useful. Btw, Gauss learned Russian ...


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.


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.



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