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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14
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3answers
466 views

Has someone seen a discussion of the (divergent) summation of $\sum\limits_{k=0}^\infty (-1)^k (k!)^2 $?

In extending my studies of the Eulerian matrix and its suitability for a matrix-based divergent summation procedure I'm trying to proceed to sums of the form $$ S = \sum_{k=0}^\infty (-1)^k (k!)^2 $$ ...
1
vote
0answers
19 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
vote
0answers
36 views

Reference for Coefficient Extraction of Multiple Sum

In a post here, the final answer is obtained by coefficent extraction of the quadruple sum. ...
29
votes
1answer
3k views

GRE past papers

As it is required for most students who wish to do a Ph.D in maths in the US to sit the GRE subject specific mathematics exam, I hope this question will be of interest to the mathematical community ...
1
vote
2answers
3k views

What is a third proportional?

I searched online, couldn't find anything clear. If I had two numbers, $a,b$, what is their third proportional? Apparently it can be either $c$ such that $a/b=b/c$ or $b/a=a/c$, but obviously these ...
3
votes
2answers
31 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 ...
3
votes
6answers
3k views

Best practice book for calculus [closed]

I tried with every inch in me to not ask a question such as this but I just couldn't resist asking this. What is the best Calculus practice book? I tried looking around but couldn't find a ...
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 ...
8
votes
3answers
14k views

Abstract Algebra book with exercise solutions recommendations.

I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books. I am trying to improve my understanding by doing exercises. However none of the books I am ...
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 ...
3
votes
1answer
171 views

I need a textbook! Information theory and probability

I have posted some questions: Probability result - 3 discrete random variables Markov chain - a notation I don't understand Random variables identities - how to make a formal proof. These ...
3
votes
3answers
316 views

Introduction to Information Theory

I'm studying bioengineering but in conversations and reading I've found that a great background in information theory as it applies to probability, statistics, random process, causation and inference ...
1
vote
0answers
28 views

Alpha representation of probability distribution

Probability vector is an $n$-dimensional vector $p=(p_1,...,\ p_n)$ that the sum of whose components equals one, i.e. $p_1+...+p_n=1$. If we take the square root of each component of probability, we ...
6
votes
0answers
75 views

Self-studying Information Geometry

I was recently exposed to the topic of Information Geometry by a friend of mine, and was looking for a good book to begin self-studying this topic. Any suggestions? Also, what subject matter would ...
0
votes
1answer
70 views

Books on graphical models

I looked at Koller & Friedman's Probabilistic Graphical Models, but their use of non-standard notation is prompting me to see if there is anything else out there. I'd like to find an introductory ...
0
votes
1answer
60 views

Introductory (online) texts on Bayesian Network.

I would like to ask for some recommendation of introductory online texts on Bayesian Network. What I am searching for is some accessible and instructive text not necessarily covering the subject in ...
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 ...
6
votes
1answer
135 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
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 $n>1$, ...
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 ...
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 ...
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
0answers
40 views

Toposes in algebraic geometry

I know the definition of topos and read an introduction to algebraic geometry. I heard that topos is used for algebraic geometry and want to know detail. Where can I read about this?
8
votes
4answers
1k views

Where can one find (freely, online) mathematical articles with a fighting chance to be understood by high school students and undergraduates?

I am an undergraduate non-math major. I just finished my university's engineering calculus series, looking forward to linear algebra in the coming semester. To be frank, I always despised math because ...
26
votes
4answers
5k views

Primes of the form $n^2+1$ - hard?

I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book ...
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 ...
48
votes
16answers
38k views

Best book for topology?

I am a graduate student of math right now but I was not able to get a topology subject in my undergrad... I just would like to know if you guys know the best one..
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 ...
9
votes
2answers
196 views

Books about harmonic numbers

I'm looking for books about harmonic numbers, where I could find proofs of results about them. For example a proof for the fact, that the generating function of the generalized harmonic numbers is $$ ...
15
votes
0answers
108 views

Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
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?
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
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 ...
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
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 ...
5
votes
2answers
171 views

Is there any wide sense but introductory Book series/websites for mathematics literature? (just to be familiar with)

Is there any Book/Book Series/Website which illustrate advanced mathematics but in concise and basic form. I just want to be familiar with the literature and I don't want a deep exploration of the ...
0
votes
1answer
71 views

A kind of planar figures

Studying issues related to the planar shapes I've found some attribute, useful for my investigations: Any segment with origin in mass center and end point on figure's boundary is contained within ...
2
votes
1answer
55 views

Hodge numbers of a cartesian product of copies of $\mathbb{C}P^1$

I wonder if some works have been done in the context of cohomology space of projective complex manifolds. Specifically I want to study the Hodge diagrams of $\mathbb{C}P^1\times\mathbb{C}P^1$ and ...
3
votes
1answer
111 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
4
votes
1answer
144 views

Fractional powers of positive self-adjoint operators

Consider two positive unbounded operators $A$ and $B$ densely defined on a Hilbert space $H$ self-adjoint on a domain $\mathcal{D}(A) = \mathcal{D}(B) = H_1$. By the spectral theorem, we can define ...
1
vote
3answers
77 views

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such ...
3
votes
1answer
102 views

Is there a garden of derivatives?

I've found a book called A Garden of Integrals, in which the author shows the evolution of the concept of Integral. I follow AnalysisFact on Twitter, some days ago, they posted the following: The ...
0
votes
1answer
61 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
0
votes
1answer
78 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
1
vote
1answer
93 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
3
votes
0answers
51 views

How many ternary functionally complete connectives are there?

Today I was reading up once more on some of the nice results regarding functional completeness, notably Post's celebrated classification theorem with the 5 classes that need to be avoided. (See this ...