This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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2
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3answers
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Reference Request: Parameter Dependent Center Manifold Theorem for ODEs

Suppose we have an $n$-dimensional first order ODE of the form $\frac{dx}{dt}= f_{\mu}(x)$ with $\mu \in \mathbb{R}^k$ a parameter and with an equilibrium at $x=0$ $(f_{\mu}(0) =0)$. For fixed $\mu$ ...
1
vote
0answers
20 views

Are there infinite self-locating strings in the decimal expansion of $\pi$?

I came across the following interesting objects. Self-locating strings within $\pi$: numbers $n$ such that the string $n$ is at position $n$ (after the decimal point) in decimal digits of $\pi$. The ...
0
votes
0answers
17 views

Video Lectures on Multi variable Calculus in n dimensions

I am teaching myself multi variable calculus from various resources on the internet. I have completed the famous 18.02 offered at MIT. This is an introductory course on Multi variable Calculus and is ...
-1
votes
2answers
60 views

What are the books that I should study for college? [closed]

Baccalaureate exam approached Real Analysis (limits, differentiation and integration), Abstract Algebra, Functional Algebra, Linear Algebra, Combinatorics, Complex numbers, Vector Geometry, Analytical ...
1
vote
0answers
37 views

Reference for Coefficient Extraction of Multiple Sum

In a post here, the final answer is obtained by coefficent extraction of the quadruple sum. ...
3
votes
1answer
88 views

Book recommendation and online learning resources for mathematics as a hobby.

The career and field of study I am choosing is not related to maths much.But I am still interested and I love maths.But I just know math up to highschool level.So I wish to learn more,say as a hobby ...
0
votes
1answer
23 views

Reference request for Prof. Wilhelm Blashke's DG book

Please suggest how can one get a copy of the book mentioned here: TA Guess_on_W Blashke Is there an English translation of Professor Blashke's book on Differential Geometry? (Einführung in die ...
3
votes
2answers
32 views

Isometric embedding of $\ell ^ 1$ in $\ell ^\infty$ in finite dimensions

It is well known that $\mathbb{R}^n$ with $\ell ^1$ norm can be embedded into $\mathbb{R}^k$ with $\ell^\infty$ norm for some $k\in \mathbb{N}.$ But I guess, this is not true in complex case that is ...
1
vote
0answers
19 views

Transformations between Pell[-like] equations

I’m looking for [non-trivial] transformations that take a Pell-like equation $$ u^2-dv^2=w $$ and turn it into another Pell-like equation $$ x^2-my^2 = z. $$ Best-case scenario, one could always use ...
0
votes
0answers
27 views

On Kapranov's “On the derived categories of coherent sheaves on some homogeneous spaces”

As a graduate master student I am reading Kapranov's paper "On the derived categories of coherent sheaves on some homogeneous spaces" (1988). One problem is that the paper assume lot of notations and ...
0
votes
2answers
36 views

Probability of a number in the real line

I have read that the probability to pick a rational number in the real line is null. My problem is: If $S$ is a dense set in the real line, what is the probability to pick an element of $S$? There ...
1
vote
3answers
29 views

Algorithm: Intersection of two conics

I am looking for a detailed description of an algorithm for the classical problem of computing the intersection of two conic curves. The curves are given by two equations of the form: $$ a x^2 + b ...
0
votes
1answer
27 views

Reference for finite sum of $k^{\alpha}$

in this answer, we are given a great formula for $\sum\limits_{k=1}^n k^{\alpha}$ for all real alpha. For a paper I'm writing, I need a reference for a textbook or a paper which contains this result ...
1
vote
1answer
75 views

Reference for zero sum problems?

I am looking for books/ references which deal with the analysis of zero sum problems and weighted zero sum problems. I have found some articles on the internet, but they seem insufficient. Any ...
2
votes
2answers
55 views

References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
1
vote
1answer
54 views

reference request: Category theory

I am sure that a similar question has been asked before, but I make my ideal textbook and situation more specific. I would like a textbook on category theory designed for someone who knows basically ...
2
votes
1answer
30 views

General property of Fitting series

Let $G$ be a finite solvable group. $F(G)$ (Fitting subgroup) is defined to largest normal nilpotent group contained in $G$. Then $F_2(G)$ is defined to be inverse image of $F(G/F(G))$. i.e ...
3
votes
1answer
65 views

How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ ...
1
vote
0answers
43 views

“Closure” of a polynomial ring by fraction field

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular noetherian $k$-algebra, $K$ the fraction field of $A$ and $\bar{K}$ an algebraic closure of $K$. Does there exist a ...
0
votes
1answer
22 views

reference request: $C^k(\overline\Omega)$ as restriction of $C^{k}$ functions on $\Omega$

Let $\Omega\subset\mathbb{R}^d$ be an open set. $C^k(\Omega)$ is defined as the space of functions $f:\Omega\to\mathbb{R}$ such that $\partial^nf$ is continuous for $0\leq|n|\leq k$. There are ...
0
votes
0answers
37 views

Is there any way to retain Russell's original proof of induction in Appendix B of PM 1925?

Recently I was reading this question again and the following question occurred to me, Can there be some new interpretation of the system of PM $1925$ so that Russell's proof of $^\ast89.16$ is not ...
1
vote
0answers
61 views

A very detailed book for calculus 1-3.

Is there a very good book covering the whole calculus in detail, explaining all topics in calculus 1-3 for self-learning? I'm in geometry I, so I will start calculus in two years, and finish in five ...
0
votes
0answers
19 views

Examples of applications of the Theorems of Pappus and Ménélaüs.

I'm going to present an exposition about applications of the Theorems of Pappus and Ménélaüs. I need some simple examples of these two theorems. Any links, please?
7
votes
1answer
56 views

On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR

Convention : All rings are commutative with unity unless stated otherwise. By ideals we will mean to include $\{0\}$ and $R$ also. Let us call an integer $n>1$ a "principal number" if any ring ...
0
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2answers
27 views

On those integers $n>1$ such that there exists a commutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer; we call $n$ a "ring number" if there exists a commutative ring $R$, with identity, having exactly $n$ ideals (including $\{0\}$ and $R$); now since for every $n>1$, ...
0
votes
0answers
17 views

Can you identify this stochastic process?

So I run into this problem the other day and I cannot even think of the keywords I need to use to look it up. For the discrete random variable $X$ we have: $P_{\Delta X(t)} = F\big(X(t-1), ...
0
votes
1answer
26 views

How to use monotone convergence theorem to show that $\int \sum |f_n| = \sum \int |f_n|$

In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state $$\int \sum |f_n| = \sum \int |f_n|$$ without proof. Can ...
0
votes
0answers
15 views

Condition for global cascade

Assume a unidirectional, unweighted network generated according to a degree distribution. Each node is given a value between 0 and 1 called threshold $\phi$. We topple some nodes, the neighbours will ...
1
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0answers
34 views

Does this theorem hold for real part of a toric variety? + Reference request - Real toric varieties.

Let $X$ be a complex toric variety and let $X_\mathbb R$ be its real part, that is the $X_\mathbb R$ consists of all the real valued points of $X$. I would like to learn a little more about the ...
0
votes
1answer
41 views

What book about algebraic combinatorics is it?

Recently I found a fragment of a book about algebraic combinatorics on the internet coincidentally. And I found it's really an excellent resource of learning polynomial method, about Combinatorial ...
0
votes
0answers
16 views

Spigot algorithms for transcendental numbers

I'm trying to write a program that will compute digits of transcendental numbers using a spigot algorithm. While researching, I found the BBP Formula, and a Compendium of BBP-Type Formulas, alas, I ...
0
votes
0answers
6 views

CG as an orthogonal projection

I have heard that the Conjugate Gradient method can be viewed as an orthogonal projection onto the Krylov subspace $K(A,r_0)$, but I can't find a reference that deal with it in this way. Could you ...
1
vote
1answer
53 views

Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
0
votes
0answers
21 views

Is this orthogonal distance a common pseudometric?

Define $d: V \times W \to \mathbb{R}$ such that $$d(v,w) = \sup_{z \perp w} \frac{\langle z, v \rangle}{\|v\|\|z\|}.$$ Is this a pseudometric that anyone has utilized in the literature? Does it have a ...
1
vote
3answers
66 views

2nd degree differential equation

Can someone please tell me how to solve this differential equation? $${d^2y\over dx^2} +y=\tan(x)$$ I am a beginner in ODE and have absolutely no idea how to proceed. Can you also site a reference ...
0
votes
0answers
21 views

measure-theoretic probability, (sets of) null events and non-zero probability

Assuming a well-defined probability space $ (\Bbb{R},\mathscr{B},\Bbb{P}_X) $, where $\mathscr{B}$ is the Borel $\sigma$-field, and for a random variable $X$ having a continuous probability density ...
2
votes
1answer
43 views

Help finding paper from the 1920's

I have not been able to find a copy of this paper anywhere! B. Knaster еt C. Kuratowski: Sur quеlquеs propriétés topologiquеs dеs fonctions dеrivéеs. Rеnd. dеl Сirc. Math. di Palеrmo, 59 (1925), ...
2
votes
1answer
60 views

Reference Request: Differential Geometry Book [closed]

What is a good self study book in Differential Geometry. Keep in mind I won't have the advantage of being able to ask a professor any questions.
10
votes
3answers
420 views

Extension of real analytic function to a complex analytic function

I just learned that real analytic functions (by real analytic, I mean functions $f: \mathbb{R} \to \mathbb{R}$ which admit a local Taylor series expansion around any point $p \in \mathbb{R}$) cannot ...
-2
votes
3answers
42 views

ordinary differential equation project suggestion [closed]

My professor asked to write a project on ODE just to experience on how to write projects. It need not be a research project. Being in second rate school from third world country, we never did those ...
0
votes
0answers
35 views

How can I correctly catalog this partition problem?

Studying the partition problems, I tried to do an special version to apply it to a kind of model of "orbits and energy levels" (explained below), but I am having problems to properly catalog this. ...
0
votes
1answer
27 views

Literature on generating functions for networks

Are you aware of any material the presents all (or most, or many) the properties and applications of generating functions in the context of graphs? For example I am aware of 'Generating ...
0
votes
1answer
14 views

Are there any connected graphs with constant link which are not vertex transitive?

By constant link, I mean for any vertices $v,w$ of a graph $G$, the subgraph of $G$ induced by the neighborhood of $v$ is isomorphic to the subgraph induced by the neighborhood of $w$. $C_n + C_m$ ...
8
votes
1answer
84 views

Why is complex analysis so nice? And how is it connected/motivating for algebraic topology?

This is very much a soft question, but after seeing Cauchy's integral formula in lecture today I was really struck by how neat complex analysis is. I don't understand how all of these amazing analytic ...
5
votes
1answer
69 views

Wallpaper groups for the hyperbolic plane

I would be grateful if someone could direct me to a reference that classifies the equivalent of the wallpaper groups (and the frieze groups and the point groups, if possible) for the hyperbolic plane, ...
2
votes
1answer
35 views

Graph of a $G_\delta$-function

Let $f:\mathbb R \to \mathbb R$ be a function. It is well known that if $f$ is continuous ($f^{-1} [A]$ is closed whenever $A$ is closed), then its graph is closed in $\mathbb R ^2$. Here is an ...
2
votes
1answer
56 views

Any Video Lectures Of An MIT, Harvard, Stanford, UC Berkeley, Yale, or Princeton Analysis Course Based On Baby Rudin?

I'm learning analysis from the book Principles of Mathematical Analysis by Walter Rudin, third edition. This book, popularly known as Baby Rudin, is being used for analysis courses at such elite ...
4
votes
1answer
80 views

Well-ordering of sets of cardinal numbers

Proposition For every cardinal number $m$ there is a definite next larger cardinal number. This proposition is proved on page 136 of "Proofs from the Book" using the fact that any set of ordinal ...
0
votes
1answer
34 views

Reference request: alternative proof for every open set in $\mathbb{R}^n$ can be expressed as countable disjoint union of open boxes

A "box" is a cartesian product of intervals of the type $[a,b]$ I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite ...
1
vote
1answer
27 views

Generators of the congruence subgroup $\Gamma (5) \subset SL(2,\mathbb Z)$

Recall that $SL(2,\mathbb Z)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}: \det(A)=1; a,b,c,d \in \mathbb Z \right\} $ and $\Gamma(5)=\left\{A=\begin{pmatrix} a&b \\ c&d ...