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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17
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3answers
1k views

Why does what I've written fail to define truth?

I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I did that, it seemed to me as though one could ...
1
vote
0answers
47 views

Slick Definition of the Category of Cartesian Closed Categories

I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ...
1
vote
0answers
34 views

Book 2 of Visual Complex Functions

I am having a lot of fun in reading Visual Complex Functions by prof Wegert. (it is a very interesting read and well-recommended by me). Inside it, he regularly let things be and postpone until part ...
1
vote
1answer
41 views

Generating all prime powers $\leq N$

Some very good algorithms exist to generate all primes $p$ up to some bound $N$, like the sieve of Erastothenes and the sieve of Atkin. However, suppose I want to generate a (sorted) list of all prime ...
2
votes
1answer
27 views

All riemannian isometries between open subsets of $\mathbb{R}^n$ are affine

I heard that there is a theorem of Liouville (Something like "Liouville's rigidity theorem") which states the following: Every Riemannian isometry between open subset of $\mathbb{R}^n$ is affine. ...
2
votes
1answer
740 views

Expected state of a Markov chain

Let's start with a slightly trivial Markov chain defined as follows: the beginning state is called $1$ and the set of states is $\mathbb{N}$. At each step, when the current state is $n$, the ...
2
votes
1answer
51 views

Pentagonal tiling

I am currently working on a research project in my last year of high school. For this paper we are discussing Eschers tesselations, both in the euclidian and the non-euclidian plane. At the moment I ...
6
votes
3answers
474 views

What does “the average continuous function is nowhere monotonic” mean?

I plan on asking my professor what he meant by "average continuous function," but as it is possible that this is a concept as vague as the statement, I was hoping to get some interesting ...
0
votes
0answers
19 views

A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 ...
2
votes
1answer
22 views

Building a hidden markov model with an absorbing state.

I'm working on trying to implement a hidden markov model to model the affect of a specific protein that can cut an RNA when the ribosome is translating the RNA slowly. Some brief background: The ...
0
votes
0answers
29 views

Is Engelking and Sieklucki's “Topology: A Geometric Approach” a Good Introduction to Algebraic Topology?

I only found this book incidentally while looking at Engelking's more well-known "General Topology". I posted a link here. ...
4
votes
1answer
45 views

Using calculus results for functions of operators

I am interested in the conditions required for functions of operators to be manipulated as if it were a real valued function with a real domain. In an applied maths text I am using the following is ...
0
votes
0answers
27 views

Representable bifunctors

Is there a notion of representability for functors in the form $F:C^{op} \times C \to Set$? Can anyone please give me a reference? Thanks.
4
votes
0answers
54 views

Hilbert's Inequality - improved???

Assume for convenience that $a_n\ge0$ (this also clarifies why certain inequalities below are in fact stronger than certain other inequalities below). Of the various inequalities Hilbert proved, I'm ...
0
votes
2answers
238 views

Machine learning: beginner study material.

Can anyone suggest to me some beginner study material for Machine learning applications in fields of 1) Financial forecasting and 2) Online advertisement? Thanks in advance!
5
votes
1answer
113 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
0
votes
0answers
40 views

Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
0
votes
1answer
23 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
0
votes
1answer
16 views

Proving This Theorem on Independence

I'm trying to find a good resource for proving the following theorem, stated in Shreve's "Stochastic Calculus for Finance II," p. 73: Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let ...
3
votes
1answer
43 views

irreducible components of subscheme

Let $f : X \to Y$ be a closed immersion of (noetherian) schemes. Is there any "general" result on $f$ out there ensuring that $X$ has the same number of irreducible components as $Y$ ?
1
vote
1answer
30 views

Asymptotic analysis references

I'm self studying asymptotic analysis with Bruijn (1981) - Asymptotic Methods in Analysis Bleistein and Handelsman (1986) - Asymptotic Expansions of Integrals but the texts are terse, without too ...
0
votes
0answers
35 views

Is there a name for these inequalities? Where can I look them up?

Consider the operators $A,B,C$ on Hilbert space $\mathcal H$: Show that: $$ \left \vert \left \vert AB \right \vert \right \vert \le \left \vert \left \vert A\right \vert \right \vert \left \vert ...
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) ...
5
votes
1answer
1k views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
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 ...
1
vote
1answer
17 views

When does $A\mathbf{v} = \lambda B\mathbf{v}$ admit a basis of solutions?

Let $A, B \in \mathbb{C}^{n \times n}$ be Hermitian matrices, and consider the so-called generalized eigenvalue problem $$A\mathbf{v} = \lambda B\mathbf{v}$$ where $\lambda \in \mathbb{C}$ is called a ...
1
vote
0answers
50 views

Self-extensions of a skyscraper sheaf

Let $V$ be a smooth variety over a field $k$. For a point $x \in V$ we denote the skyscraper sheaf of length 1 by $$ k(x) = \mathcal{O}_x/m_x. $$ Then by taking the Koszul resolution of $k(x)$ one ...
3
votes
0answers
38 views

Matrix with roots of unity entries

For a given prime p, i am interested in the norms of matrices which have root of unity entries, i.e., $M_{k,l} \in \{1, \zeta, \dots, \zeta^{p-1}\}$ where $\zeta = \exp{(2\pi I/p)}$. Are there any ...
0
votes
0answers
45 views

reference request for $L^p(\partial\Omega)$ in real analysis textbooks

Let $\Omega$ be a bounded open set in $\mathbb{R}^d$. Would anybody come up with a real analysis textbook which contains detailed introductory treatment of the space $L^p(\partial\Omega)$?
2
votes
1answer
54 views

bounded generation and groups with infinitely many ends

Following section 7.1 in Peterson-Thom's paper here, we say a countable group $G$ is boundedly generated by the subgroups $G_1, \cdots, G_n$, if there exists an integer $k\in\mathbb{N}$, such that ...
6
votes
4answers
727 views

Top 10 math mnemonics

If you study undergraduate medicine, mnemonics are almost indispensable - there is so much factual material to learn. I was never given any mnemonics in my time as a maths undegraduate. But Robert ...
3
votes
2answers
66 views

Self-study mathematics subject sequence and recommended books

I am a Physics student but I finally found that I've entered the wrong department that I am in fact much more interested in mathematics. I want to self-learn mathematics. I am now reading Artin ...
0
votes
0answers
22 views

Classical Complex Analysis by Mario O. Gonzalez

My friend found a copy of the book Classical Complex Analysis by Mario O. Gonzalez, and we found that this book was being sold online for upwards of $800. I am really curious what would drive the ...
9
votes
1answer
112 views

Independent Transcendental Numbers

I've been thinking about numbers which have not yet been proven nor disproven to be transcendental, such as $e + \pi,\, \pi - e,\, \frac{\pi}{e},\, \gamma,\,\zeta(3),$ etc. Some of these numbers ...
0
votes
2answers
75 views

Soft Question: Weblinks to pages with explanation on quadratics.

I recently placed a question based on quadratics and received a few valuable answers. One of them was a comment in an answer with a link in it which I found useful. But unfortunately the webpage (of ...
0
votes
0answers
20 views

Relation between the eigenvalues of $\Delta$ and counting lattice points

I was reading a paper with the following information: "Let $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ be the flat torus, let $\varphi$ be the eigenfunctions and $\lambda$ the eigenvalues of the ...
5
votes
2answers
78 views

Book about intuition behind Lebesgue measure

I recently completed a course in Real analysis covering Lebesgue and Borel measure, Fourier series, $L^p$ spaces and such. I can solve problems in these topics but am afraid that I do not truly ...
0
votes
2answers
32 views

Reference recommendation in dynamical system

I need a good book that is self-study in dynamical system. I have the book "Geometric Theory of Dynamical Systems" by Jacob Palis but it is difficult and is not a self-study book. I need a book that ...
1
vote
1answer
60 views

Learning Abel-Ruffini

I took an introductory abstract algebra course at my university and was fascinated by the content. I would love to learn more and go into greater depth with groups, rings, and fields, but ...
0
votes
0answers
16 views

Euler/Pfaff transformations for generalized hypergeometric functions $_pF_{p+1}$ functions

For hypergeometric function $_2F_1(a_1,a_2;b_1;z)$ there exists Euler/Pfaff transformations: $$_2F_1(a_1,a_2;b_1;z)=((1-z)^{b_1-a_1-a_2})_2F_1(b_1-a_1,b_1-a_2;b_1;z),\quad \text{Euler ...
0
votes
0answers
9 views

Near $z=\infty$ solutions for generalized hypergeometric functions $_pF_{p+1}(z)$

For differential equation that is satisfied by the hypergeometric function $_2F_1(a_1,a_2;b_1;z)$, around $z=\infty$, if $a_1-a_2$ is not an integer, one has two independent solutions ...
0
votes
1answer
42 views

Which rules of inference does Suppes use?

I'm reading Axiomatic Set Theory by Suppes, and I'm having a bit of trouble understanding which rules of inference (logical system) he is using, here's an example (capital letters are used for sets): ...
3
votes
1answer
70 views

groups with infinitely many ends are not boundedly generated?

Recall that a group $G$ is boundedly generated if it can be written as a finite product of cyclic subgroups. And there are a lot of examples of groups that are (not) boundedly generated. I am ...
15
votes
4answers
856 views

Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose ...
7
votes
3answers
404 views

Stochastic geometry, point processes online lecture

Does any of you know where to find online lecture/podcast introducing stochastic geometry and/or point processes? Thank you! Riccardo
1
vote
0answers
48 views

A proof of Cayley-Hamilton using the Algebraic Closure of a Field

As most are surely aware, there are lots of proofs of the famous Cayley Hamilton Theorem. I was told by a friend of a proof which is claimed to be rather direct and short. Its strategy goes as ...
2
votes
0answers
49 views

Defining perpendicular lines in the 3D space

Is there a universal agreement about the definition of "perpendicularity" between two stright lines in the 3 dimensional euclidean space? Do they need to meet or it is enough to have perpendicular ...
0
votes
0answers
23 views

Basic Asymptotic Theory book

I would love if someone can recommend me a book where Basic Asymptotic Theory is thoroughly covered and explained with some examples. I'm currently reading Econometric Analysis of Cross Section and ...
0
votes
0answers
14 views

Some books about plane Affine geometry?

I want to learn some introductive notions about next topics: Plane affine geometry ( Axioms, Theorems, Models), The group of affine transformations of a vector space, Affine invariants. I need a book ...
7
votes
3answers
181 views

Arithmetic rules for big O notation, little o notation and so on…

There are many asymptotic notations like the big O notation: big Omega notation, little o notation, ... Thus there are many arithmetic rules for them. For example Donald Knuth states in Concrete ...