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6

It's called the prime constant. When you enter that number into WolframAlpha and you see the $\mathcal{P} = 0.41468250985111166$, notice that in the bottom right-hand corner of that cell it says "$\mathcal{P}$ is the prime constant", which links to the Wolfram Mathworld page explaining what it is.


5

Mertens' Theorem says: $$\lim_{n \rightarrow \infty} \ \frac{1}{\log p_n} \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p_k}}} = e^{\gamma}.$$ Euler's product formula for the $\zeta$ function and his evaluation of $\zeta(2) = \pi^2/6$ says that $$\zeta(2) = \lim_{n \rightarrow \infty} \ \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p^2_k}}} ...


3

André Weil has written about the origin of the Weil conjectures: "In Chicago, in 1947, I felt bored and depressed, and not knowing what to do, I began reading Gauss's two memoirs on biquadratic residues [from 1828 and 1832], which I had never read before. (...) This led me in turn to conjectures about varieties over finite fields."


3

It seems like Stephen Abbott's Understanding Analysis is just what you are looking for. From page vii of the Preface: Each chapter begins with the discussion of some motivating examples and open questions. The tone in these discussions is intentionally informal, and full use is made of familiar functions and results from calculus. The idea is to freely ...


3

This would be the corona of $K_n$ and $K_1$, usually denoted $K_n \circ K_1$. The original definition was by Harary and Frucht in 1970 in their paper "On the Corona of Two Graphs" See this question for an application of it to more general graphs: Eccentricity in corona product


3

This can be found in the proof of Theorem 5.2 of the following paper: Handel, David. "On products in the cohomology of the dihedral groups." Tohoku Mathematical Journal, Second Series 45, no. 1 (1993): 13-42. In particular, for $m$ even and $n>0$, we have: $$ H^n(D_m;\mathbb{Z}) \;=\; \begin{cases} (\mathbb{Z}/2)^{(n-1)/2} & \text{if }n\equiv 1\pmod ...


2

Numerical Optimization by Nocedal and Wright is quite a classic in the field. It also contains a few appendices introducing an overview of the necessary background.


2

Here are three (somewhat standard) references: Jurgen Jost, "Riemannian Geometry and Geometric Analysis." Peter Li, "Lecture Notes on Geometric Analysis." Thierry Aubin, "Nonlinear Analysis on Manifolds. Monge-Ampere Equations." Jost's book is on its sixth edition. Aubin's book has a first and second edition, although my understanding is that the first ...


2

I am not sure if this exactly what you want, but a homotopical perspective on homology is given in the book partially titled Nonabelian Algebraic Topology, EMS Tracts, vol 15 (2011) (pdf available there). The main results do not assume singular homology, but nevertheless give results such as the Relative Hurewicz Theorem, (!), and results on second ...


2

This is not true. In fact, Kolmogorov constructed (1923) an example of a $L^1$ function whose Fourier series diverges almost everywhere (later improved to everywhere divergent). Om the other hand, if $f\in L^p$ for some $p>1$, it's a deep theorem by Carleson (the $p=2$ case) and Hunt ($p>1$) that the Fourier series of $f$ converges pointwise almost ...


2

Yes, there are past examples, and it is likely still possible. Here are some recent examples that I've heard of: Newton's polygons (which Newton described in letters he wrote in the seventeenth century) have evolved into Newton's polytopes, which have interesting applications in algebraic geometry. The Bernstein–Kushnirenko theorem (a result from the ...


1

I've been talking with the speaker since I asked this question and today he gave me an old copy of a French algebra book which has the same scheme the geometry books quoted by him. "Éléments D'Algèbre" by M. Bourdon, from 1907. A very interesting book indeed, exactly as he said: a property or problem is presented and worked out in a very straight and clear ...


1

It seems the following. A space $X$ is almost $\omega_1$-Lindelöf [Par][Mat, p. 92], if every open cover of cardinality at most $\omega_1$ has a countable subfamily whose union is dense in $X$. It is easy to check that a space $X$ has Property A iff $X$ is almost $\omega_1$-Lindelöf. See the diagrams at [Mat, p.92] and at [DRRT, p.94] about the relations of ...


1

What's going on is $d$ is supposed to mean "an infinitesimal change in". To make it clearer what's happening, imagine we had the following situation. The quantities $a$, $b$, and $c$ all depend on time. Let's write $a(t)$ to mean the value of $a$ at time $t$. We also write $b(t)$ and $c(t)$. Furthermore, you know that they solve those two equations: ...


1

Let $f : A \to X$ be a based map of based spaces. The homotopy pushout $X \coprod_A \text{pt}$ is called the homotopy cofiber, cofiber, or mapping cone of $f$; I'll denote it by $X/A$. Iterating this construction produces the cofiber sequence or Puppe sequence $$A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma X/A \to \dots$$ which is in some sense the ...


1

"Channel codes - Classical and Modern" by William E. Ryan and Shu Lin should be a great place to learn about both linear codes and LDPC codes in particular as the book devotes a huge part to the latter topic with a self-contained introduction to the former.


1

The biggest number that is not too big to be the limit is one way to describe the limit superior, $\limsup_{x\to a}f(x)$, and the smallest number that is not too small to be the limit is one way to describe the limit inferior, $\liminf_{x\to a}f(x)$. Any number $l\in[\liminf,\limsup]$ is a number that is neither too big nor too small to be the limit. And if ...


1

The metric you described is the standard metric on the projective space: in the real case it can be visualized as the angle between lines (thinking of the elements as lines). It arises as the quotient of the spherical metric on $S^n$ by the group of isometries $\{x\mapsto \alpha x, \ |\alpha|=1\}$ where $\alpha$ belongs to the ground field, $\mathbb{R}$ or ...


1

For $M=\mathbb R^n$ take a smooth function with compact support $\phi\in C_0^\infty(M)$. Then $$ \phi = (-\Delta+\lambda I)^{-1}((-\Delta+\lambda I)\phi) $$ obviously. Now consider translations $\phi_k(x):=\phi(x+kv)$ for $v\in \mathbb R^n$, $k\in \mathbb N$. Take the norm of $v$ large enough such that the supports of different $\phi_k$'s are empty. The ...


1

Donald L. Cohn, Measure Theory (Birkhäuser 1980) First Edition Theorem 8.3.6, page 275 (2nd Edition, p. 259) alternate proof: Exercise 5, page 277. Outline: (a) Every Borel subset of a Polish space is Borel isomorphic to a Borel subset of $\{0,1\}^{\mathbf N}$. (b) Each uncountable Borel subset of a Polish space has a Borel subset that is Borel ...


1

That random graph is denoted by $\mathbb{G}_{1-out}$. A common generalization is $\mathbb{G}_{k-out}$, where in step $j$, we choose $k$ vertices out of $V\setminus\{v_j\}$ and add the $k$ edges $\{v_j, \cdot\}$. Then at the end, delete multiple edges. One place that you can read about this models is Alan Frieze and Michal Karoński's new book on Random ...


1

I learned it from Mathematical Modeling by M. Meerschaert. The problems allow for interesting questions that go beyond his suggested exercises, so it's a great source of problems. Also, he writes problems that give you an excuse to learn things like Maple or R Regarding what Calculus to review for this text, you should learn about Newton's Method, the ...


1

A very good Introductory book with detailed explanation http://www.amazon.co.uk/Linear-Nonlinear-Optimization-Igor-Griva/dp/0898716616 The tip I would give is just like all the other maths please practice and write detailed solution. In Optimization solutions can be very very long but i highly recommend solving them. Also one often forgets the algorithm ...


1

Let continue your calculus : $$\frac{1}{2\pi}\int ^{\infty}_{-\infty}\frac {e^{iut}}{iu+1}du$$ $$\frac{1}{2\pi}\int ^{\infty}_{-\infty}\frac {(\cos(ut)+i \sin(ut))(-iu+1)}{u^2+1}du$$ $$\frac{1}{2\pi}\int ^{\infty}_{-\infty}\frac {\cos(ut)+u \sin(ut)}{u^2+1}du + \frac{i}{2\pi}\int ^{\infty}_{-\infty}\frac {-u\cos(ut)+ \sin(ut)}{u^2+1}du$$ $\int ...


1

Much longer comment turned answer: Howard Eves' An Introduction to the History of Mathematics seems like a perfect fit. The chapters that would be of particular use are chapters 11 through 14 (relevant sections included on the side): Chapter 11: The Calculus and Related Concepts [11.9 Newton; 11.10 Leibniz] Chapter 12: The Eighteenth Century and the ...


1

Suppose that $x=\langle x^{(j)}:j\in J\rangle\in (E^I)^J$, where each $x^{(j)}=\langle x_i^{(j)}:i\in I\rangle\in E^I$. Similarly, let $y=\langle y^{(j)}:j\in J\rangle\in (E^I)^J$, where each $y^{(j)}=\langle y_i^{(j)}:i\in I\rangle\in E^I$. Then we want to define $\mathfrak{W}$ so that $$\begin{align*} ...


1

One can also prove the theorem using Nevanlinna theory. See here, for example.


1

This is maybe not a full answer, but too long for a comment. Any two Morse functions are homotopic through the obvious homotopy $t\mapsto t\,f+(1-t)g$. This homotopy will not be through Morse functions. This is easy to see for a closed manifold as the number of critical points cannot change through such a homotopy. The homotopies can be done through ...


1

I think Electromagnetic Theory and Computation: A Topological Approach by Gross and Kotiuga might be just what you're looking for. However, it does assume that you know some general and algebraic topology to start with. I would recommend that you read John Lee's Topological Manifolds first. The text covers what you would expect in a typical topology book, ...


1

First consider the case that $\sigma$ is a simple function, i.e. $$\sigma(s) =\sum_{j=1}^n 1_{[t_{j-1},t_j)} \xi_j \tag{1}$$ where $(\xi_j)_{j=1,\ldots,n}$ are random variables independent from the Brownian motion $W$. Without loss of generality, we may assume that $t_k = t$ for some $k \in \{1,\ldots,n\}$ (otherwise we add the point to the partition). ...



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