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0

It's not a book, but I found Timothy Gowers' address "The Importance of Mathematics" quite inspiring as a graduate student. I wish I had watched it earlier in my career.


0

Perhaps the best book for this is Johnson (Amazon). Also very good is the classic Magnus, Karrass and Solitar (Amazon), which has a large number of exercises for practice.


1

I presume you must have already taken a basic course into calculus and analysis, so the book Analysis In Vector Spaces-A course in advanced Calculus can be a good place for you to start with. The chapters $3$ to $5$ which I have read were very well-explained ...


0

It depends what your level is and what you're interested in. I think a book that's not about maths but uses maths is probably more interesting for most people. I've noticed this in undergraduates as well: give someone a course using the exact same maths but with the particulars of their subject area subbed in, and they'll like it much better. Examples being ...


1

Disclaimer: The following proof is not completely elementary and uses methods from complex analysis as well as the Fourier transform, but at least it does the job: Assume that the polynomials are not dense in your $L^{2}$-space. This implies the existence of $f\neq0$ in your $L^{2}$space for which $$ \int_{\mathbb{R}}\frac{f\left(x\right)\cdot ...


1

The following two papers might be useful, but note that their focus is not on finite sets. However, you might be able to find something more appropriate by searching for papers that cite one of these two papers. Edward O. Marczewski [Szpilrajn], Concerning the symmetric difference in the theory of sets and in Boolean algebras, Colloquium Mathematicum 1 #3 ...


3

Here is another bunch of texts. Like the ones suggested by Mathemagician1234, they are not general texts. The level of formality is variable. Classical mechanics F. Scheck, Mechanics, Springer, 2010. Although not specifically geared toward mathematicians, it makes use of mathematically advanced tools. I consider it the best book on classical mechanics ...


2

At Oxford we used http://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710 for the first year of the Physics Course.


0

You need the Banach space to be separable, I think. You can find it as Theorem 4.6 in these notes by Piotr Hajlasz.


1

Lemma 1.4.a in: J. M. F. Castillo and M. Gonzalez, Three-space problems in Banach space theory, Springer Lecture Notes in Math. 1667, 1997. Proof. Let $E$ be a Banach space and let $\{x_i\colon i\in I\}$ be a dense subset of the unit ball of $E$. Define a map $Q\colon \ell_1(I)\to E$ by $$Q ( (a_i)_{i\in I} ) = \sum_{i\in I} a_i x_i\quad (a_i)_{i\in ...


6

Since no one has mentioned it, I must add V.I Arnold's excellent "Mathematical Methods of Classical Mechanics". As the title suggests, the book focuses on classical mechanics, which is always a good start for a physics education. I'd suggest reading this or other good classical mechanics texts before moving on to various quantum theories, as it's difficult ...


2

Try Principles of Physics by Resnick, Halliday and Walker


21

There are now quite a few excellent ones, but most of these are pitched at fairly sophisticated readers-graduate students or professional mathematicans. The thinking according to such textbooks, of course,is that the readers are very far along in thier mathematical training and are ready to use that mathematics to learn physics at a very high level. Whether ...


1

Try The Theoretical Minimum: What You Need to Know to Start Doing Physics and Quantum Mechanics: The Theoretical Minimum.


0

I agree with Timbuc, the Schaum's calculus book has helped me with having many solved problems and explanations. I have seen a geometry addition being sold on amazon and ebay for a pretty good price


0

Try C. Wells, "Some applications of the wreath product construction", Amer. Math. Monthly 83 (1976), no. 5, 317–338. While I cannot say whether your particular application occurs in that paper, I've found it to be a good source on wreath product stuff more generally, including some work of Kaloujnin-Krasner, so yours might be in there.


1

The best introduction is Raymond Elementary Introduction to the Theory of Pseudodifferential Operators CRC Press.


1

I think you'll also enjoy A Hilbert Space Problem Book by Paul Halmos. The Preface of this book is interesting, and the book is written in the Halmos style. Not much else needs to be said.


3

This quickly got longer than I thought it would be, and I'm sure I inadvertently skipped something. If you have any questions, feel free to ask. Definition: (Lie Algebra) A Lie algebra is a real vector space $V$ equipped with an operation $$[\cdot,\cdot]:V\times V\to V,$$ called the bracket of the Lie algebra, such that $[\cdot,\cdot]$ is ...


2

Well, as far as your actual example is concerned, your proposed definition seems quite sensible to me. You are trying to translate the following definition into natural language (more or less): given groups $(G,*)$ and $(H,\diamond)$, $$ \mathbf{Grp}((G,*),\ (H,\diamond)):=\{((G,*),\ f,\ (H,\diamond)):\ (G,\ f,\ H)\in \mathbf{Set}(G,\ H)\ \wedge$$$$\wedge\ ...


5

Just to state my opinion: when you get to study functional analysis it is almost certainly that you are going into research, one way or another. At such level, "rich of complete, step by step, rigorous, and enlightening solutions" is probably not a good idea. When doing research, you won't find "complete, step by step, rigorous, and enlightening solutions" ...


2

Check these books: http://www.amazon.com/Problems-Variables-Lebesgue-Integration-Applications/dp/0070602212 http://www.amazon.com/Problems-Solutions-Undergraduate-Analysis-Mathematics/dp/0387982353/ref=sr_1_sc_1?s=books&ie=UTF8&qid=1411897062&sr=1-1-spell&keywords=Problems+and+Solutions+for+Undergraduate+Analysis+Rami+Shakarch ...


3

You can check these books: http://www.amazon.com/Problems-Analysis-Second-Charalambos-Aliprantis/dp/0120502534/ref=sr_1_1?s=books&ie=UTF8&qid=1411894207&sr=1-1&keywords=problems+in+real+analysis And ...


5

This is what I recommend to students learning analysis as a good companion: http://minds.wisconsin.edu/handle/1793/67009


1

As Youngsu mentioned above, there is in general no relation between the primary components of $I$ and $J$, or the numbers $r$ and $s$ (and Youngsu has given the example of $I = m$ in a local ring for a case where $r < s$). For an example with $r > s$: take $R = k[x,y]$, $I = (x) \cap (x,y)^2 = (x^2,xy) \supseteq J = (x^2)$. On a positive note, there ...


0

It depends on the topics you want to do also, but I found Applied Stochastic Processes by Lefebvre pretty decent for some probability theory topics.


0

here are some but not all solutions. http://www.oberlin.edu/math/faculty/wilmer/327/Solutions/HWSolutions-327-S06.pdf good luck!


1

Looking through the slides you linked, it looks like they can all be done without working in coordinates. Do you perhaps have specific identities in mind for which you mean ' possible to argue without coordinates.' You mentioned Cartan's magic formula can be proved without working in local coordinates, and so can $\mathcal L_{[X,Y]}\alpha=\mathcal ...


0

Calculus for Business, Economics, and the Social and Life Sciences is very accessible (and colourful!).


2

Maybe you'll find the Princeton Companion to Mathematics helpful. It has some nice and accessible surveys of mathematics, both past and present.


1

There is something to be said for Calculus in Context by Callahan et al. It emphasizes applications in the sciences. Students who will not use calculus in science or engineering courses that they take later should learn why calculus is important and considered a great achievement, rather than just learning to chant "n x to the n minus one", as in the ...


1

For a different take on linear algebra, try Practical Linear Algebra, A Geometry Toolbox by Farin and Hansford. For geometry, try Geometry: Euclid and Beyond by Hartshorne.


2

There is no conventional symbol for the set of radical expressions. If you find yourself needing one, just define one and be explicit about it.


0

Here are some of my recommendations: Elements Of The Differential And Integral Calculus by Granville Calculus Books by James Stewart


1

From my side Thomas calculus is best book for calculus which will give rigorous explanations of differentiation using geometry. It starts from the basics of differentiation and goes on to advanced level like vector calculus. Not only geometrical approach but it also provides writing exercises. A hallmark of this book has been the application of calculus to ...


1

I prefer sticking to the classical context for the first round of dealing with the spectral theorem; in particular, I would use Riemann-Stieltjes integrals instead of Borel measures. Once you have the Riemann-Stieltjes version, it is a fairly trivial matter to extend to the measure theoretic, when you want it. I highly recommend this text for self-study at ...


0

Take a look at this list, which is pretty solid and also covers more advanced material and other areas of economics (mathematical methods, macro, and econometrics). The standard micro theory texts are: Mas-Colell, Whinston, and Green's Microeconomic Theory. There are pdf solutions floating around online. This is the standard text in first year of graduate ...


2

Tikz is a popular drawing tool for $\LaTeX$. It is fairly dificult to get going, but the manual is very valuable, and there is also a tex SE site. With a lot of valuable information. What you can do with it is endless, but it will take time. Note that it is also possible to make simple figures/diagrams in programs such as maple, mathematica and the like and ...


2

You have the right ideas. Presumably, $F$ should be any subset of $\Bbb R^n$; what you're describing there is Carathéodory's criterion. Also, I think you have some extra words in you definition of Lebesgue measure; I would write $$ \lambda^{*}(E) = \inf\limits \{ ~\sum_{k=0}^{\infty} \text{vol}(Q_{k}):~\{Q_{k}\}_{k} \text{ are rectangles such that } E ...


1

The group of unitary matrices is compact, and for any $\epsilon>0$, there are distinct integers $m$ and $n$ such that $\| U^m- U^n\| < \epsilon$. So $\|U^{m-n}-I\| <\epsilon$. This seems so simple that a reference would not be required.


1

First, I think you need to fix your theorem: $\bar X = \frac{\sum X_i}{n}\sim \mathcal{N}(\mu,\frac{\sigma}{\sqrt{n}})$. You need to subtract the mean and divide by the standard deviation, so that $\sqrt{n}\frac{\bar X-\mu}{\sigma} \sim \mathcal{N}(0,1)$ That aside, the answer will depend on the underlying distribution of the population. Lets say all ...


0

Synge and Schild's "Tensor Calculus" has the old, component-heavy, "physics-style" discussion of tensors, and it has bits devoted especially to GR. I would recommend you supplement it with a more modern text like Bishop and Goldberg's "Calculus on Manifolds" or "Geometry, Topology and Physics" by M. Nakahara because thinking of tensors exclusively in ...


0

I highly recommend this online course: https://www.coursera.org/course/maththink It is starting again in the next few days, and I found it very helpful in getting to understand the concept of a proof, and what the various types of common proof are. I'm not sure if the course will end in time for you, but fingers crossed it does.


0

It is definitely for someone who already knows the subject and is looking for a different perspective. It is not advanced and it is not introductory, more a supplement. It is also sloppy and very hard to follow for someone who does not know the subject. The praises are from people who know the subject and like the presentation and a few things not easily ...


0

I would suggest 2 references: "Quantum Groups", Kassel "Algebraic Operads", Loday & Vallette The book by Kassel is very well written and contains an expository part on algebras / coalgebras with modules / comodules as well. The second reference is an advanced book on algebraic structures called operads. Chapter I ("Algebras, coalgebras, homology") ...


1

When $K=\mathbb{R}$, this is Taylor's theorem. Perhaps it is more recognizable if we write it as $P(x+h) = P(x) + h\cdot Q(x,h)$. The fact that $Q$ is a polynomial follows from the fact that all but finitely many of the derivatives of $P$ are zero. Note that since $Q$ is arbitrary, the only real content of the statement (even for general $K$) is that $h$ ...


1

One interpretation is that he is talking about cyclic branched coverings of knots. This is definitely talked about in Rolfsen's "Knots and Links" but is probably also described in any other book about knot theory, Lickorish could well be a better option. In particular, there is a lot of interest in double branched coverings due to the Montesinos trick, which ...


3

(Unfortunately I don't remember much of the details, and I don't have time to look them up now, so this will be very sketchy, but perhaps it will be of some use until someone else gives a better answer...) There are indeed restrictions that $\mu$ must satisfy in order for the space of polynomials to be dense in $L^2(\mu)$. It has to do with limit ...


-2

See capters 8 and 9 of Commutative algebra by N.S. GOPALAKRISHNAN (a good note with more details)



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