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6

If a function $m$ satisfies your condition, then $m(x) = 1$ for all $x \ne 0$. Just observe the following inequality: $$2^{m(x)}||x|| = ||2x|| = ||x + x|| \le ||x|| + ||x|| = 2||x||$$


4

Gödel did epoch-making work in a number of fields: In pure logic, he was the first to prove the completeness of a system of the predicate calculus. In what we might call the proof theory of formal systems, he proved the incompleteness [different sense!] of any formal system strong enough to encode a certain amount of arithmetic. (This required developing ...


3

We have $$\frac{a_n}{\pi/2} = \prod_{k = n+1}^\infty \frac{(2k-1)(2k+1)}{(2k)^2} = \prod_{k = n+1}^\infty \biggl(1 - \frac{1}{4k^2}\biggr).$$ To estimate products, it is often convenient to take logarithms. Here we can get the easy upper bound $$\log \prod_{k = n+1}^\infty \biggl(1 - \frac{1}{4k^2}\biggr) = \sum_{k = n+1}^\infty \log \biggl( 1 - ...


3

Euler's "Introductio in analysin infinitorum" (1748), though a bit dated and not up to date with modern notation and standards of rigor. It covers an enormous swath of the area, much more than what is customary today. Spivak or no Spivak.


3

http://www.amazon.com/gp/offer-listing/0936428066/?tag=wwwcampusboocom667-20&condition=used http://product.half.ebay.com/Between-Nilpotent-and-Solvable-by-Henry-G-Bray-John-F-Humphreys-David-Johnson-Paul-Venzke-and-W-E-Deskins-1982-Hardcover/717850&item=345068498941&tg=videtails These are some sites I found after searching, hope this helps.


2

Unlike integration, differentiation is a very unstable operation. It is very hard to make assumptions on $\{f_n\}_n$ so that $\{f'_n\}_n$ converges. For instance, let $f_n(x)= \frac{\sin (nx)}{n}$: $\{f_n\}_n$ converges to zero uniformly, but the derivatives $f'_n$ are oscillating. The only "elementary" theorem about differentiation of sequences of ...


2

Define $M(N)=\sum_{n\leq N}\mu(n)$. Then $M(N)\ll N^{1/2+\varepsilon}$ is equivalent to the Riemann Hypothesis (Aleksandar Ivic, The Riemann Zeta-Function, page 47). Define $S(N)=\sum_{n\leq N}\frac{\mu(n)}{n}$. We will prove $M(N)\ll N^{1/2+\varepsilon}$ if and only if $S(N)\ll N^{-1/2+\varepsilon}$. Proof: Suppose $M(N)\ll N^{1/2+\varepsilon}$. ...


2

From what I've been told, Brownian Motion and Stochastic Calculus by Karatzas and Shreve is the gold standard. Continuous Martingales and Brownian Motion by Revuz and Yor is also a great reference. What you have listed as background knowledge is sufficient.


2

I would go with ACoPS because this book teaches you how to solve hard problems, and gives you ideas and techniques that are quite new that you can use them again or refine them so you can solve a broader class of questions.


1

Try the book Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.


1

Reading more mathematics is not going to help you understand more mathematics unless you read with pencil and paper next to you and try to fill the gaps in the proofs that you do not understand or find hard to follow. You don't get better at swimming by watching people swim, you must swim. To get more fluent in mathematics, you must do it as well.


1

The term "logarithmic convex hull" is in use for this object, because it can be obtained by convexifying the image of a domain under the logarithm map, and then coming back. It comes up in complex analysis in several variables, due to the following fact: a domain $D\subset\mathbb{C}^n$ is a region of convergence of some power series (centered at $0$) if ...


1

UPDATE 2: I managed to prove rigorously that the conjecture I had formulated in the first update holds: The expressions for $\pi_j$ in both formulae are off by a factor of $t$. I also figured out the most plausible reason for this mistake. Consider the partition of the state space $\{S_0,\ldots, S_{t-1}\}$ mentioned above and let $q_{ij}\equiv p_{ij}^{(t)}$ ...


1

Yes, "your" statement was very popular in classic Italian texbooks. Now I do not have them on my desk, but I believe that your theorem appears more or less explicitly in Prodi's book "Analisi matematica" and in the old treatise by Luigi Amerio. In Rudin's book it is not stated because Rudin approaches every topic from a rather abstract point of view. Limits ...


1

Perhaps it would be wiser to use the following expansion $$E(f+h)=E(f)+dE(h)+o(h),$$ from what you have calculated, we see that $$\int \nabla f\cdot\nabla h+\frac{1}{2}\nabla h\cdot\nabla h=E(f+h)-E(f)=dE(h)+o(h).$$ Since the the last term on the LHS is $o(h)$, we obtain $dE(h)=\int\nabla f\cdot\nabla h$. Note that this is the Frechet derivative at the ...


1

I found this exposition of the Smallest Eigenvalues of a Graph Laplacian by Shriphani Palakodety to be readable and informative. The article begins with a discussion of eigenvectors for the smallest eigenvalue, which in the case of the graph Laplacian happens to be zero. The number of eigenvectors for this eigenvalue gives the connected components of the ...


1

You could try a Monte Carlo approach. Basically, you can simulate a large number of strong solutions and then evaluate the sample mean and variance of the specific instant of interest. Depending on the structure of the diffusion coefficient, it is possible to perform exact simulation. In this case, no approximation error will be propagated to your ...


1

I am adding the answer I received by email from Doctor Tadashi Tokieda, he is the director of studies in mathematics at Trinity Hall, University of Cambridge, some bio here and here, I was following his Topology and Geometry open lectures at Youtube for the AIMS, so I dared to send him an email yesterday (I did not expect an answer, just tried) and received ...


1

As pointed out in my earlier comment, we need to assume $s\in (0,1)$. Out of habit, I'm going using the convention $\mathbb{T}\cong[0,1)$. You can modified everything to your own convention. The expression $$[f]_{W^{2,s}(\mathbb{T})}=\left(\iint_{\mathbb{T}\times\mathbb{T}}\dfrac{\left|f(x)-f(y)\right|^{2}}{\left|x-y\right|^{1+2s}}dxdy\right)^{1/2}$$ ...


1

"generatingfunctionology" by Herbert S. Wilf and "A=B" by Marko Petkovsek, Herbert Wilf and Doron Zeilberger. These are available as free downloads at https://www.math.upenn.edu/~wilf/DownldGF.html and https://www.math.upenn.edu/~wilf/AeqB.html


1

You first prove that the right-shift operator $R$ is a linear operator on (doubly infinite) sequences. Then you define addition and scalar multiplication on operators so that you can given a sequence $f$ obeys a recurrence relation rewrite the recurrence relation as $(P(R))(f) = 0$ for some polynomial $P$, which is the characteristic polynomial. Now $P(R)$ ...


1

Let's deduce $(1.3)$. Let $(x_1,…,x_n)$ be a positive integral solution of $$\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}\leq1.$$Now put $y_i=x_i-1\geq0$ for all $i$. We claim that $(y_1,…,y_n)$ is a non-negative integral solution of $$\frac{y_1}{a_1(1-a)}+\cdots+\frac{y_n}{a_n(1-a)}\leq1.$$ In fact, ...


1

You need to tell us something more. Are you interested in very formal book with all the technicalities or rather something which mostly emphasises ideas and results. Is your background in pure mathematics or rather applications like physics etc. If you would choose second answer to both questions I strongly recommend Frankel, but it's actually a huge book. ...



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