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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5
votes
1answer
105 views

Construction of Monotone function which is differentiable on the given set

Given a set $A \subset \mathbb{R}$ of measure $0$, is it possible to construct a monotone function whose set of non differentiable points is $A$ ?
4
votes
2answers
56 views

recommend math books [on hold]

So i completed an year ago my schooling and i am pretty good at maths well at my level and i am very interested in maths and want to learn as much maths as possible and i like stuff like number ...
6
votes
6answers
422 views

Proving that $E=mc^2$ [closed]

What are the axioms of special relativity? Is there a book or paper that introduces the theory of special relativity in a rigorous manner, and proves that $E=mc^2$ after appropriate definitions?
26
votes
5answers
1k views

Proving the existence of a proof without actually giving a proof

In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. ...
0
votes
0answers
12 views

Learning outcomes of reading textbooks [on hold]

So I've densely mined this site for the scholarly materials I need most. And have prudently written many of them down, so I can sort them if I see fit. But, are the books always the preferred method ...
31
votes
4answers
838 views
+500

How to get intuition in topology concerning the definitions?

Most topology texts go on directly to give definition of topology, then they give some examples and that's it, like they directly tell you right Let $X$ be a set and let $τ$ be a family of subsets ...
1
vote
1answer
52 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
0
votes
0answers
10 views

Source for Space-Time Fourier transform theory

I need to do research on Space-Time Fourier transforms (specifically applications within EM theory). Since the resources for learning this digitally seem to be limited (by my very advanced Google ...
5
votes
1answer
896 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 ...
3
votes
3answers
45 views

A good companion to Axler's “Linear Algebra Done Right”?

Seeing as Axler is very reluctant to talk about determinants and generally avoids computations and playing around with algebra, I'd like to get a book that will serve as a companion to Axler's ...
0
votes
0answers
20 views

Pell's Equation sources

I am researching about Pell's equation and wanted to ask what the best resources are for it? So far I have Stopple's book and Hardy's book.
0
votes
0answers
35 views

Reference request for a very particular problem solving skill

I want to start with an apology for a very verbose description of my question but if there is a way to cut it down, please let know and I will do so right away. I have been trying to get better at ...
2
votes
1answer
127 views

Regularity of heat kernel

I'm trying to find some references dealing with regularity and properties of the heat/Gaussian kernel $$ G_t\left(x,y\right) = \frac{1}{\sqrt{2\pi t}}\, e^{-\left.\left(x-y\right)^2\right/2t}, ...
4
votes
1answer
40 views

References for information theoretic statistical tools

Strange statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence with respect to these quantities are necessitated in my ...
1
vote
2answers
24 views

reference request Schur Zassenhaus Theorem

I am looking for a reference for the Schur Zassenhaus Theorem, saying that any normal Hall subgroup admits a complement. An on-line search show that it is supposed to be in "The theory of groups" by ...
1
vote
1answer
29 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\theta \colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi ...
3
votes
0answers
81 views

Name for categories with a certain property on coproducts

Is there a name for categories with the following property: The category has zero morphisms, coproducts, and for each family $(X_i)_{i \in I}$ of objects the natural map $$\hom(Y,\bigoplus_{i \in I} ...
7
votes
0answers
50 views

Are weakly étale ring homomorphisms of finite presentation étale?

Following [Stacks, 092A], say a ring homomorphism $A \to B$ is weakly étale if both $A \to B$ and $B \otimes_A B \to B$ are flat. Question. Are weakly étale ring ...
2
votes
1answer
42 views
+50

Seeking more information regarding the “hybriation function.”

Definition 0. Given a pair of finite sets $Y$ and $X$, write $Y_X$ for the set of all collections $\mathcal{K}$ of functions $f : Y \leftarrow X$ that are closed under "hybridization", by which I ...
3
votes
1answer
60 views

Rank 2 vector bundle

$E$ is a rank $2$ vector bundle. Why is $E\simeq E^*\otimes \det E$? Any generalization (arbitrary rank, $E$ non locally free etc.)?
1
vote
0answers
81 views

Request a paper by Gelfand and Ponomarev

I am looking for the following paper by Gelfand & Ponomarev: I. M. Gelfand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a ...
14
votes
4answers
1k views

Introduction to Filters in Topology

Question: What are some good resources for a student who has taken algebraic and point-set topology and who wishes to learn how filters and ultrafilters are applied in topology? Motivation: I've seen ...
0
votes
1answer
29 views

Partitioning a convex object without cutting existing convex subsets

$C$ is a convex object in the plane. $D_1,\dots,D_n$ are pairwise-disjoint convex subsets of $C$ such that $D_1\cup\dots\cup D_n \subsetneq C$, like this: Is it always possible to partition $C$ to ...
-2
votes
1answer
47 views

How good is Naive Set Theory by Halmos? [on hold]

I happened to run into this book in an old shop and got it for like half a dollar. Has anyone read this book? Is it worth the time? (Please don't respond things like "every math book is worth the ...
0
votes
0answers
13 views

Material on Koszul complex

I have tried a lot of books to understand Koszul complex but eventually failed. I have tried Eisenbud's book, Weibel's homological algebra, but I just can not how to construct Koszul complex by tensor ...
2
votes
2answers
76 views

Reference for entropy of the binomial distribution?

The Wikipedia page Binomial distribution says that the entropy of the Binomial(n,p) is $\frac{1}{2}\log_2\left(2\pi e n p (1-p)\right) + O\left(\frac{1}{n}\right)$. What is a reference (paper or ...
0
votes
0answers
11 views

Infinite series, continued fractions, nested radicals and such - is there a general recursive algorithm theory?

The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time ...
3
votes
1answer
46 views

On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
2
votes
0answers
63 views
+50

Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
2
votes
0answers
15 views

A sufficient condition for factorization in a complete local ring

I think something like the following statement is true, but I don't recall a reference. Suppose $f(x,y)\in k[[x,y]]$ is power series with no constant or linear terms. Then, if the quadratic terms ...
1
vote
1answer
125 views

Clarification if a disconnected function has a derivative at defined points.

I know so far for a derivative to exist. -The point should not exist as a discontinuity -It should not have a vertical tangent -There should be no sharp corner/ cusp at the point ...
1
vote
0answers
42 views

Estimates for the Dedekind number $M(9)$

The Dedekind number $M(n)$ is the number of antichains in the partial order of subsets of $\{1,\dotsc,n\}$. It is only known for $0 \leq n \leq 8$. Question. What are some known upper and lower ...
1
vote
3answers
110 views

Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
0
votes
1answer
44 views

Using derivatives to get some trigonometric identities

Is there a way of using derivatives to get some trigonometric identities in a straight-forward fashion? I use to forget them, so that would help me a lot... For example, since when we get the ...
0
votes
0answers
21 views

Reference request for numerical invariants of modules which are not finitely generated

Suppose that $R$ is an integral domain with subring $S$ and that both rings are finitely generated $k$-algebras ($k$ an algebraically closed field). $R$ is integral over $S$ if and only if $R$ is ...
0
votes
1answer
47 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
1
vote
2answers
46 views

DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
2
votes
1answer
54 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
3
votes
1answer
72 views

Lie groups pre-requisites and reference

What are the minimum pre-requisites in analysis (differential geometry) required to study Lie-groups? And for that material, what are some good references? I have done basic courses in Metric spaces, ...
10
votes
0answers
77 views

Category Theory Zoo

There are a few very useful websites when it comes to either finding a specific object with certain properties (and maybe lacking other properties) or finding out which properties a certain object ...
20
votes
4answers
2k views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
0
votes
0answers
27 views

“Peak lemma” and explicit monotone subsequence

Looking at the proof of Bolzano–Weierstrass theorem, it found an interesting lemma (called the peak lemma here) : Every sequence $(x_n)_{n\in \mathbb{N}}\in \mathbb{R}^\mathbb{N}$ has a monotone ...
0
votes
0answers
31 views

On even almost perfect numbers other than the powers of two, as compared to odd perfect numbers given in Eulerian form

(Note: I have edited this question to conform to the further details added in the cross-post to MO.) Antalan and Tagle (in a 2004 preprint titled Revisiting forms of almost perfect numbers) show ...
0
votes
1answer
18 views

Asymptotic bounds on sum of primes

Let $p_i$ denote the $i$th prime number, and let $p_k\#$ denote the $k$th primorial, $p_k\# \overset{\textrm{def}}= \prod_{i \le k} p_i$. I am interested in asymptotic upper bounds for the ...
1
vote
1answer
27 views

Is the braid category biclosed and bicomplete?

Let $\mathcal{B}$ be the braid category, as in Categories for the Working Mathematician §XI.4 p.262 (objects are natural numbers and morphisms are the braids $n\to n$). Then this can be given a ...
0
votes
0answers
8 views

Regarding Smirnov domains

Suppose $G$ is a Smirnov domain that contains infinity in the plane (we can think of it as the exterior to a closed Jordan curve) and $\phi$ is a conformal mapping from $\mathbb{D}$ onto $G$. Can we ...
0
votes
0answers
21 views

Best resources for learning about regular and context free languages

I would like to train myself when it comes to finding out if a language is regular or context free. I would be grateful for pointing what are the best places/books for training.
3
votes
1answer
1k views

Intuition for gradient descent with Nesterov momentum

A clear article on Nesterov’s Accelerated Gradient Descent (S. Bubeck, April 2013) says The intuition behind the algorithm is quite difficult to grasp, and unfortunately the analysis will not be ...
1
vote
1answer
680 views

video lectures on Lie algebra

Is there any video lecture on first course on Lie algebra available online? , by the first course I mean, The complete book of Introduction of Lie algebra and its representation theory by James ...
6
votes
1answer
50 views

Survey of varieties of non-standard analysis?

Is there a reliable, reasonably up-to-date, survey article doing a "compare and contrast" on varieties of non-standard analysis?