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1

This actually has nothing to do with zeta values, it is pure symmetry - a special case of $$e_n=Z(P_n)(p_1,p_2,\cdots,p_n) \tag{1}$$ where $e_k$ and $p_k$ are the elementary and power-sum symmetric polynomials, in infinitely many variables $x_1,x_2,\cdots$, respectively. For zeta values we simply evaluate at $x_k:=k^{-s}$. The recursive identity you wrote ...


0

Of course, knowing a symmetry makes it possible to reduce an ODE system to a system with less components. You may wish to look into, say, Chapter 2 of the book Applications of Lie groups to differential equations by Peter Olver, for details.


0

This is meant as a question directed at @studiosus and anybody who's familiar with his example. M. Dyer's theorem seems to state, as a special case, that Theorem Suppose $X,Y$ are finite, at most 2 dimensional, connected CW complexes with isomorphic fundamental groups (so called $[G,2]$-complexes, where $\pi_1(X)\simeq G\simeq\pi_1(Y)$). If $X$ and ...


1

Easy. There are exactly five subfields of ${\mathbb Q}(\sqrt{2},\sqrt{3})$ : $K_1={\mathbb Q},K_2={\mathbb Q}(\sqrt{2}), K_3={\mathbb Q}(\sqrt{3}), K_4={\mathbb Q}(\sqrt{6}), K_5={\mathbb Q}(\sqrt{2},\sqrt{3})$ and ${\mathbb Q}(\gamma)$ is one of them. It follows that $\gamma$ is a primtive element iff $\gamma\not\in K_2,\gamma\not\in K_3,$ and ...


0

You can try Modern Analysis and Topology (Universitext). Here's the table of contents: part 1 part 2 part 3 part 4 part 5 part 6 part 7 part 8


1

As usrtt1 suggests, Introduction to Topology: Pure and Applied by Adams and Franzosa is a great book introducing topology and many of its applications, although it stops short of material based on homology. An alternative might be Basener's Topology and Its Applications, based on its table of contents, but I do not have the latter book. There are many ...


5

It was a problem for me too when I started learning conformal differential geometry in 2008. As I prefer to learn from openly available sources even though our University has an excellent library within a minute of walk, some of my references will be links to such online resources. My first encounter with the proof of the conformal transformation of the ...


0

I would recommend Introduction to Topology by Gamelin and Greene for a few reasons: Covers the Point-Set Topology that will be very helpful to know when studying Real Analysis. The authors do an excellent job of covering applications of metric space topology to Analysis. The authors provide solutions or at least guidance on a large set of the problems in ...


1

I learned a lot of this material (real analysis, and some of the basic ideas of point-set topology) from Elements of real analysis by Bartle.


3

My favorite analysis book is that by Pugh. It's similar to Rudin, but readable and with tons of fantastic problems. So far, I really like Runde for topology. It's rigorous, and short enough that you'd want to read it front to back. It has what you need if you want to continue on to more advanced topics, and is enough even if you don't! Another nice book to ...


2

As Adam mentions, my paper with Benedikt Löwe on the The modal logic of forcing is concerned with the modality $\square\varphi$, which means that $\varphi$ holds in all forcing extensions, and the corresponding modality $\Diamond\varphi$, which means that $\varphi$ holds in some forcing extension. It is remarkable, in my opinion, that these set-theoretic ...


3

I consider Folland's “Real Analysis: Modern Techniques and Their Applications” as the best textbook ever written, on any subject. I warn you, though, it is extremely dense: If you want to be thorough and check everything for yourself (as many really easy results are just mentioned in passing without proof and there are a lot of exercises, some quite easy, ...


1

I have heard good things about Abbott's "Understanding Analysis", though I have only glanced at it myself. For self-learning topology I used (many years ago, but a situation like yours) George Simmons' "Introduction to Topology and Modern Analysis" and recommend it highly.


0

W. Walter, Ordinary Differential Equations, Springer (1998) See section 9, page 89 -- 97 .


6

Analyis: Spivak - "Calculus" or Abbott's "Understanding Analysis." Might also want to pick up Gelbaum's "Counterexamples in Analysis" Topology: Munkres - "Topology", as well as Steen's "Counterexamples in Topology" to go with it.


1

The book I would recommend for an introductory course to real analysis is Real Analysis by Bartle and Sherbert. I found it perfect for a first course in real analysis. As for topology, the book I prefer is Topology by J. Munkres. Another book that I would recommend for real analysis is Mathematical Analysis by T. Apostol.


10

You might try "Principles of Mathematical Analysis" by Walter Rudin.


1

I like Doug West's book called Introduction to Graph Theory. It's a breadth book, covering the basics including cycles, paths, trees, matchings, covers, planarity, and coloring. There are algorithms covered like Dijkstra, Kruskal, Ford-Fulkerson, Bipartite Matching, Huffman Encodings, and the Hungarian algorithm. There is also a lot of relevant theory you ...


0

An answer I found on Quora: It looks like ILA is a more introductory book, where LAaIA assumes that the reader is already familiar with the basics of matrices and vectors. ILA also seems to have some material introducing the abstract view of linear algebra, whereas LAaIA looks like it's mostly focusing on material that's relevant for engineering ...


3

I would recommend practicing bounding things with inequalities, so when you encounter a problem that requires the use of the triangle inequality (it comes up fairly frequently), you can find a bound fast. Many of the definitions used in analysis deal with inequalities as well, such as convergence of a sequence, limit of a function, continuity, etc... Also ...


0

I would say $n$-ary trees in which each internal node can be labeled with an operator from $O$ and all leaf nodes can be operators or symbols. If the number of symbols is finite, then $S$ will be finite too, since the maximum size of $O$ is finite. If you take a tree that has operator leaves, then it is an operator, and if all its leaves are filled with ...


1

Jim Hefferon has a freely available book Linear Algebra that discusses various applications as well as giving a solid theoretical base.


0

Strang revisited with a valid free download: https://archive.org/details/flooved1323 "Computational Sciences and Engineering - Applied Linear Algebra"


0

Hey I hope you're still interested in reading this article!


2

There is an innovative course Coding the Matrix offered by Philip Klein which consists of a book and a course offered on Coursera and other places. It even has a Twitter account for keeping updated. The reviews are controversial, see also here and here, but it looks as an interesting challenge to try. It is designed, according to the author's website, as a ...


3

Find the (real) value of $a$ such that the curves \begin{eqnarray} y &=& a^x \\ y &=& \log_a (x) \end{eqnarray} intersect exactly once. Find also the $x$ and $y$ values where they intersect. Note that's the logarithm of base $a$ in the second curve. I think this is a pretty tough problem. It doesn't involve advanced calculus, but you need to ...


3

Linear Algebra and its Applications- Gilbert Strang seems to be very recommended.


1

One resource I haven't seen mentioned is the trusty Table of Integrals, Series, and Products by Gradshteyn & Ryhzik. While known primarily as an integral table, this book has a ton of other material including good coverage of special functions. Further, I second (or third, etc) the suggestion of the Handbook of Mathematical Functions by Abramowitz & ...


1

Linear Algebra and its Applications by David C. Lay is a simple book containing many references to real-world problems, including computer science.


1

I think it is best known as the fair division problem. Here's wikipedia to get started http://en.wikipedia.org/wiki/Fair_division However, I haven't seen the exact protocol you specify. It seems similar to the economic concept of directed search. http://www.nber.org/papers/w17746


7

This limit has been floating around since the 1990s (if not longer), though I haven't seen it "in the wild" since 1996: If $a$ is real, evaluate $$ \lim_{x\to\infty} e^{e^{e^{[x + e^{-(a + x + e^{x} + e^{e^{x}})}]}}} - e^{e^{e^{x}}}. $$


4

Here's another one to see how you understand calculus. $f(x)\geqslant 0, \forall x \geqslant 0$ $f(x)\leqslant c\int_0^xf(t)dt, \forall x\geqslant 0 ,\exists c>0$ Prove that $f(x)$ is identically zero. Solution : Let $F(x)=\int_0^xf(t)dt $ $$f(x)-cF(x)\leqslant 0$$ $$f(x)e^{-cx}-ce^{-cx}F(x)\leqslant0$$ $$(F(x)e^{-cx})'\leqslant 0$$ ...


0

Try : $$ \int e^{-2x}\left[(e^x)^2 +(x^x)^2\right]\ln x dx$$


2

This will test all of your analytical and mathematical skills. $f(x)$ is a differentiable function and $g(x)$ is a double differentiable function such that $|f(x)|\leqslant 1$ and $f'(x)=g(x)$. If $$f(0)^2+g(0)^2=9$$ then prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.


3

This was on this years intervarsity paper in Ireland. It's kinda fun. $\large{\int\limits_0^4 \frac{dx}{4+2^x}}$


0

If you haven't seen it before, then this should put your integration skills to the test: $$\int \! \sec^3(x) \, \mathrm{d}x$$


1

I personally prefer online resources: DLMF Wolfram Functions They are much more convenient to use as compared to books. Otherwise: Abramowitz-Stegun and three-volume set by Prudnikov-Brychkov-Marychev (for more special stuff). Finally, for the proofs, I would recommend Whittaker-Watson and Bateman-Erdelyi.


2

As OP stated he was a physics student in a comment, there are a lot of books called (something similar) to "Mathematical Methods for Physicists", which have relevant references, details and techniques for physicists. I'd probably start with the first two. Arfken, "Mathematical Methods for Physicists" Boas, "Mathematical Methods in the Physical Sciences " ...


2

The Bateman Manuscript Project is a good source, as is the NIST Digital Library of Mathematical Functions.


0

Have you looked through anything by V. I. Arnold? The book " Geometrical Methods in the Theory of Differential Equations" might be of interest to you. Also, John Guckenheimer would be another reference for studying problems in bifurcation theory. Home Page: http://www.math.cornell.edu/~gucken/


1

You may consider looking at Oldham, Myland, and Spanier's book An Atlas of Functions which can be purchased used quite cheaply on Amazon. I would dare suggest it is a beautiful book, with very nice graphs on glossy pages. It is large and it is organized by classes of functions. There aren't many proofs in the book, so I'm not sure if that would deter you ...


3

...it turns out there are non-trivial high dimensional smooth knots $S^n\subset S^{n+2}$ such that $\pi_1$ of the knot complement $X$ is infinite cyclic. The inclusion of the meridian $S^1 \subset X$ is a homology isomorphism, a $\pi_1$ isomorphism but is never a weak equivalence since by a result of Levine if it were then the knot would be ...


1

I hate to be that guy, but in order to properly understand Quantum Mechanics, you need to have a solid understanding of Newtonian Physics. Since you're already on Coursera, try and slow down a bit and take one or two introductory physics courses before moving on to quantum. For the first few weeks (or likely the entire first course), you won't be learning ...


0

In wikipedia there is explantion with a good number of figures. Wouldn't this be enough?


3

Another way to make a curve with many branches through a point is to start with $\mathbb A^1$ over $\mathbb C$ (say) and then glue together the points $t = 0, 1, \ldots, n$ (for some the value of $n$). That is, we let $A$ be the $\mathbb C$-subalgebra of $\mathbb C[t]$ consisting of polynomials $f$ such that $f(0) = f(1) = \ldots = f(n).$ This a finitely ...


5

I am going to plug my undergraduate professor's book again, but it is honestly the best book I know to prepare oneself for the math involved in QM. (I should know, as I experienced his course as a Math/Physics double major.) The book is Applied Analysis by the Hilbert Space Method by Samuel S. Holland. It is now available in paperback and relatively ...


4

I think that Higher Maths for Beginners – Zeldovich, Yaglom and Elements of Applied Mathematics books have the math you need for QM. They're written by Zeldovich, a co-father of Soviet nuclear bomb project. The latter book has a related volume called Elements of math physics. Noninteracting particles, unfortunately it's not translated to English. The ...


4

Consider the curve singularity at the origin of the image $C$ of the map $z \mapsto (z^e, z^{e + 1} \ldots, z^{e + n})$ where $e$ is a large integer (say larger than $n + 1$). Any polynomial equation for $C$ must have vanishing constant and linear terms. Hence the Zariski tangent space of $C$ at $0$ has dimension $n + 1$.


7

It depends on what type of QM course you want to take. Courses in QM for engineers, undergraduate physics majors, graduate students in physics, and graduate students in mathematics are all pretty different. I will assume you're seeking an "undergraduate physics major" understanding of QM. I have two recommendations: In my opinion the best book for ...


3

It depends a lot on the point of view you want to take on QM. For an experimental physicist's point of view, you will need real/complex analysis, linear algebra and probability theory. If you want the theoretical physicist's/mathematician's point of view, then add functional analysis (maybe with a focus on $C^*$-algebras/algebras of operators) and ...



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