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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3
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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
42 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
18 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
votes
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
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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
vote
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
52 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
42 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), ...
1
vote
1answer
56 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 ...
1
vote
0answers
10 views

How to formulate and analyze systems of stochastic differential equations?

I'm having trouble finding reference material on how to deal with systems of stochastic differential equations. Specifically, I'm interested in ecological models. For example, consider the standard ...
1
vote
1answer
60 views

Can I ask about copyright here? [closed]

I don't know it is okay if I post about copyright here. If there is any problem, please leave a comment. I will delete this post. Thank you. I am a graduate student studying engineering and ...
1
vote
0answers
33 views

An identity for $q-$Fibonacci numbers if $q$ is a root of unity.

In his proof of the Rogers-Ramanujan identities I. Schur introduced two $q-$analogues of the Fibonacci numbers ${F_n}({q})$ and ${G_n}({q})$, which satisfy ${F_0}({q})=0$, ${F_1}({q})=1$ and ...
1
vote
1answer
21 views

PreLie Operads and Free PreLie Algebras

Is there any relationship between Free Pre-Lie Algebras and Free Algebras over the Operad Pre-Lie (like in KOSZUL DUALITY FOR OPERADS in pg 13 )? If there is, can someone indicate a reference for ...
1
vote
1answer
41 views

Bernstein Inequality (wiki correction?!)

I am having trouble with one of the statements made on this wikipedia page, in particular the second Bernstein Inequality on: https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory) ...
0
votes
0answers
32 views

Good problems to do while reading Hardy's book on divergent series?

I am reading Hardy's text on divergent series and to my great dissapointment it has no exercises. I wonder if anybody among you know of some suitable references with problems to read simultaneously ...
0
votes
1answer
62 views

Short exact sequences from the Euler sequence.

I was reading an article in which the author said that the sequence $\require{AMScd}$ \begin{CD} 0 @>>>\Omega ^1_{\mathbb{P}^n} @>>> \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus n} ...
0
votes
1answer
41 views

Reference needed with derivation of the equations for the cubic and quartic and proof of impossibility of quintic equation

My background is an undergrad year long course in algebra which got me through to some basic finite field and field extension stuff (most of fraliegh or Ian Stewart's book are familiar to me if that ...
5
votes
1answer
73 views

Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
1
vote
0answers
53 views

Finding a problem book in algebraic topology

I simply need book with problems solved with greatest explanation possible. I know about Hatcher and have a great lecturer, so I do not need theory. I need problems solved in detail.
0
votes
0answers
11 views

reference for convex function results

Here is a simple property of a concave function from $\mathbb{R}$ to $\mathbb{R}$, Given $x,x'\in \mathbb{R}$ with $x'>x$, if $\exists \kappa'\in (0,1)$ such that ...
2
votes
1answer
71 views

Does anyone know a book on sketching surfaces?

Is anyone aware of books or sources dedicated to sketching surfaces? Sort of like Forst's An Elementary Treatise on Curve Tracing, but on surfaces; or a book that has a fair few chapters dedicated to ...
1
vote
0answers
20 views

Looking for concrete examples of semigroups, with explicit computations done to help make the theory more understandable.

I would appreciate if someone can help me find a resource which has a bunch of worked out examples showing how to find, for example, the smallest congruence containing a relation on the semi groups, ...
1
vote
0answers
33 views

I'm looking for a good book on FOL and set theory.

I finally decided to really learn some axiomatic set theory, at least the basics. I've studied a bit of FOL, but a review would be nice. In short, I'm looking for a book that focuses on $\sf ZFC$ or ...
3
votes
1answer
23 views

Sum greater than 1; minimization for non-strict inequalities

I want to show that if $x_k>0$ for $k=1,2,...,n$ and $\sum_{k=1}^n x_k=1$, then $\sum \frac{x_k^2}{y_k}\ge 1$ for any $y_1, y_2,...,y_n>0$ so that $\sum_{k=1}^n y_k=1$. I tried solving the ...
2
votes
1answer
75 views

Errata for Mathematics: Its Content, Methods and Meaning?

I'm new here, so I hope this is the right place to post this! I am currently reading through the Dover edition of the textbook Mathematics: Its Content, Methods and Meaning, by Aleksandrov, ...
0
votes
0answers
34 views

How useful is Prentice Hall Algebra 2?

I am currently a sophmore in highschool, and I wish to continue learning mathematics over the summer. My school laptop has a copy of Algebra 2 published by Prentice Hall. I have so far been unable to ...