Questions tagged [reference-request]
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.
19,750
questions
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Continuity on the transform $f\mapsto f^{-1}$ in Wiener algebra
Intro. Norm in a Wiener algebra on the unit circle defined as
$$
\left\|\sum_{n\in \mathbb{Z}}a_n e^{inz}\right\| = \sum_{n\in\mathbb{Z}}|a_n|.
$$
The Wiener $1/f$ theorem states that the function $F$...
- 540
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0
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30
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When does equality hold in the Minkowski's inequality $| x - y |^a = | x - z |^a + | y - z |^a?$
More precisely, for which points $x = (x_1, x_2), y = (y_1, y_2)$ and $z = (z_1, z_2)$ in the plane does hold
$$
| x - y |^a = | x - z |^a + | y - z |^a
$$
where $a \in (0,1)$?
- 5,359
1
vote
0
answers
25
views
Sampling Spin Configurations in Ising Models
Considering an instance of the Ising model, with N number of spins, I am sampling from the Boltzmann probability distribution: $$\mu(s)=\frac{e^{-E(s)/T}}{Z}$$
using Monte Carlo Markov chains (MCMC). $...
- 21
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votes
5
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698
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What are examples of Halmos's claim that a single small concrete special case can capture every instance of a concept of great generality?
Paul Halmos states:
It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
What are examples of ...
- 1,975
3
votes
1
answer
59
views
Are there "measurable" properties?
Consider two measurable spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$, consisting of sets $X$ and $Y$ and some $\sigma$-algebras $\mathcal{A}$ and $\mathcal{B}$ defined on each of them respectively.
...
- 3,144
1
vote
1
answer
37
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Classification of torsion-free nilpotent groups of class 2
Some background
Let $G$ be a torsion-free nilpotent group of class $2$ and rank $2$ (i.e., generated by two elements). Then, $G$ has to be isomorphic to the Heisenberg group.
This is relatively easy ...
- 1,945
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votes
1
answer
49
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Name of a particular probability distribution
Suppose the probability mass function $\{p_n\}_{n \geq 0}$ takes the form
\begin{equation}\label{eq:p*n_example}
p_n = \left\{
\begin{array}{ll}
n\,p_0\,r^n, & \quad \text{if}~ n \geq 1,\\...
- 2,382
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votes
0
answers
9
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Request for Literature Recommendations on Isotonic Mappings
I am looking for literature recommendations on classical and generalized isotonic mappings of sets. Specifically, I am interested in any scientific and technical literature on this topic that is ...
- 1
1
vote
0
answers
21
views
Free $S^1$-action on Seifert manifolds
It is certain that not every fixed point-free action is free.
I saw in an earlier post (https://mathoverflow.net/questions/77715/s1-action-in-three-dimensions) in MO regarding non-trivial $ S^1$-...
- 11
1
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0
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27
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The Summer Before University: Self-Studying for Proofwriting skill vs. Comprehensiveness
Apologies for the soft question. I have been puzzling over what to spend my time studying. I'm a high school senior, who only recently (during the pandemic) became interested in mathematics. I've ...
- 823
-4
votes
0
answers
64
views
Is there such a theorem in general topology
In general (point set) topology, I believe that an isolated point set must be nowhere dense. But after searching online, I can not find such a theorem, So I wonder if there is a theorem on this fact ...
- 16.2k
2
votes
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answers
23
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Interpolation between $L^p(\mathbb R^n)$ and $\operatorname{BMO}(\mathbb R^n)$
Consider a measurable, real function $f$ defined on $\mathbb R^n$ which belongs to $L^p(\mathbb R^n)\cap \operatorname{BMO}(\mathbb R^n)$, for some $1\leq p<\infty$. An interpolation inequality ...
- 2,457
7
votes
1
answer
529
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Did Paul Halmos state “The heart of mathematics consists of concrete examples and concrete problems"?
Paul Halmos is quoted as stating “The heart of mathematics consists of concrete examples and concrete problems.”
Google shows me many people quoting him as stating this, but no authoritative reference ...
- 1,975
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answers
35
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Period of Lehmer sequences
In his thesis (1930), D. Lehmer did not provide the general formula for the period of his sequences. And it does not appear in HC. Williams book about E. Lucas work. And, since Lehmer sequences are ...
- 303
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13
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Reference Request: Hausdorff dimension of special Cantor sets
Currently I want to refresh my knowledge about Hausdorff dimensions and I looking for some references of the following kind:
Typical examples of calculating the dimensions are known to me but I am ...
0
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0
answers
28
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Reference request: manipulations, AM/GM, norms and inequalities
Consider the following questions:
Continuity of this two variable function with inequalities
Continuity of a multivariable function: doubts on how to reason
Another question on the procedure of ...
- 1,985
-1
votes
0
answers
33
views
Soviet/Russian Textbooks for Discrete Mathematics? [closed]
I have been on the hunt for some Soviet/Russian textbooks to supplement my university coursework. I found them to be able to fill in the gaps that my university coursework disregards. I have been ...
- 1
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votes
0
answers
32
views
finite dimensional skew-symmetric representation
I was reading the article "Flat Nonunimodular Lorentzian Lie Algebras", and I found that they are using this result as a known result. However, I have no idea about the reference or the ...
- 1
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0
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57
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+50
Wanted reference/list of elliptic integral of the second kind $\operatorname E(x,m)$ parameter transformations.
$\def\F\{\operatorname F}\def\E{\operatorname E}$
Elliptic F$(x,m)$ appears in elliptic E$(x,m) =\int_0^x\sqrt{1-m\sin^2(x)}dx$ parameter $m$ transformations. Luckily the DLMF has reciprocal/imaginary ...
- 7,669
1
vote
1
answer
116
views
An unusual equivalent form of Riemann hypothesis
Let $G(x)=\sum_{k\leq x}\frac{\mu(k)}{k}$, where $\mu$ is the Mobius function. From this question and its answer, its mention the Riemann hypothesis is equivalent to $G(x)=O(x^{-\frac{1}{2}+\epsilon})$...
- 1,145
-2
votes
0
answers
24
views
Optimizing a cost function where the feasible set is given by a distribution
I was wondering if the following optimization problem has somewhat been studied somewhere.
Suppose you have a closed set $\Omega \subset \mathbb{R}^n$ and a cost function $f$ (say a positive convex ...
- 5,166
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votes
0
answers
39
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Reference request: Lie group theory with emphasis on Riemannian geometry
I was hoping there is a textbook focused on Lie groups that has particular focus on the Riemannian geometry side of Lie groups: Left-invariant, bi-invariant metrics, relationship between the Lie and ...
- 2,607
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votes
0
answers
39
views
Understanding $AB$ as a transform of matrix $A$ by matrix $B$
Given matrix $A$ and vector $x$, we can view $A$ as a linear transform, so that $Ax$ means $x$ transformed by $A$. The powerful insight is that $A$ transforms basis vector $e_j$ to $A_{:j}$ (the $j^{...
- 1,975
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vote
0
answers
75
views
A result similar to the Mean Value Theorem
During my search for a solution to the challenging math problem, I came across the following problem by mathematician Vasile Cîrtoaje.
I planned to use the intermediate value theorem, but I didn't ...
- 165
1
vote
1
answer
40
views
Composition of function and delta function
This wikipedia article https://en.wikipedia.org/wiki/Probability_density_function has the formula
${\displaystyle f_{Y}(y)=\int _{\mathbb {R} ^{n}}f_{X}(\mathbf {x} )\delta {\big (}y-V(\mathbf {x} ){\...
- 314
3
votes
1
answer
110
views
Ji Chen's lemma
https://artofproblemsolving.com/community/u797276h2556237p27284705
In the solution of the VMF member, Ji Chen's lemma was mentioned. I asked VMF but have not received an answer yet. Could you provide ...
- 165
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votes
0
answers
31
views
Defining Fourier Transfrom using Banach Algebra from $L^1(\mathbb R)^n \to L^1(\mathbb R)^n$
Context
A way to define the Fourier transform on $L^1$ is by finding the sets of all homomorphism $\left\{ \phi \right\}$ from $L^1(\mathbb R) \to L^1(\mathbb R)$ where in the domain the product is ...
- 5,166
2
votes
0
answers
62
views
Searching for Martin Gardner / Dr. Matrix reference
In a recent interview on Robinson's Podcast, Joel David Hamkins described an infinite game he called "The chocolatier and the glutton" (also at this link). The 2 player alternate-move game ...
2
votes
0
answers
50
views
Texts to initiate in the theory of homotopy
I want to begin a study of homotopy theory and its applications, I have come across the introductory texts by Arkowitz and Brayton Gray. Particularly, I have found the text of Arkowitz a little more ...
- 738
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43
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hello everyone, l am new in the platform, l am looking for a book where l can read on proof of Prime Decomposition as Difference of Power of Integers
Need: reading materials, I want to try prove these questions:
$n^k − m^k$, for $n, m ∈ N$ cannot be a prime number if $k$ is not a prime
number.
$n^k − m^2k$, for $n, m ∈ N$ cannot be a prime number ...
3
votes
1
answer
112
views
Category theory textbook for Banach algebras and Banach spaces
Do you know of a textbook at the graduate level about the common categories of functional analysis (such as the 2 categories of Banach spaces, the 2 categories of Banach algebras, the category of ...
- 713
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vote
0
answers
78
views
Induced action of cohomology
Let us consider a $G$-space $M$ with $M$ topological space and $G$ topological group and let $H^\bullet(M,\mathbb{Z})$ be the singular cohomology with integer coefficients. I would like to ask if the ...
- 29
1
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1
answer
27
views
Database of lattices and lattice properties
Are there any websites that are databases of lattices?
I'm also interested in databases distributed as libraries in a programming language or similar.
I'm studying a little bit of lattice theory on ...
- 9,246
3
votes
1
answer
89
views
Which "book(s)" complements "Combinatorial Problems and Exercises by László Lovász"? [closed]
I am trying to prepare for my entrance exam/interview to an Integrated-Master's and PhD programme in Mathematics in my country.
Currently, I am an undergraduate student.
I have been told that "...
- 31
1
vote
0
answers
99
views
Some references on Abstract Algebra [closed]
About two years ago, I read several pages of a book about set theory by chance, and I felt a sense of familiarity with an universe that I had deemed unfriendly since I was a youngster. Because of the ...
- 4,386
2
votes
0
answers
27
views
Textbooks focused around generalisation of Infinite series
I'm not very into integral to be honest. But I love theorems related to generalisation of Infinity series such as Abel-Plana formula, Poisson summation or analytic continuation of series derived by ...
- 354
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votes
0
answers
41
views
Deformation retract of $\Delta$ complex to Klein bottle
The same question has been asked before, for instance here
Hatcher exercise 2.1.2 deformation retract of $\Delta$-complex to Klein bottle by edge identifications
and here
Show that the $\Delta$-...
- 10.1k
8
votes
1
answer
212
views
Adjoining a function to a ring: what is this called?
There's a kind of construction of an extension of a ring, and that I've seen used e.g. here (although that may not be the best example) which is essentially adding a function into the ring, and taking ...
- 2,634
2
votes
0
answers
37
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Reference on the sum of absolute differences between $n$ samples from a random variable
Let $X_1, X_2,\ldots X_n$ be $n$ samples taken of a random variable with a given distribution (so in particular it is i.i.d.). Is there literature on or a name for the random variable defined by $$Y = ...
0
votes
1
answer
50
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The solution $X$ of the SDE $\mathrm d X_t = f(t, X_t) \mathrm d t + g(t, X_t) \mathrm d B_t$ is a Markov process
I'm reading Section 5.4 Markov property from these notes, i.e.,
$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\,\mathrm d}$Let $B$ be a standard Brownian motion and $(\mathcal F_t, t \ge 0)$ its ...
- 13.5k
4
votes
2
answers
40
views
Reference for the "Quotient Variance"?
I have recently encountered the expression
$$\frac{\mathbb{E}[X^2]}{\mathbb{E}[X]^2}$$
in my research. This can be thought of as an analogue of the variance $\mathbb{E}[X^2] - \mathbb{E}[X]^2$ where ...
- 6,834
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vote
0
answers
16
views
Reference request: dimension of convex hull of n random points
Let $X$ be an i.i.d. random sample of $n$ points in $\mathbb{R}^d$ from a probability distribution that is absolutely continuous with respect to the $d$-dimensional Lebesgue measure. Assume $n>d$. ...
- 167
2
votes
1
answer
82
views
Reference about equivalent form of the Riemann hypothesis
I saw a statement about the Riemann hypothesis in Wikipedia, stating the following:
$\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\frac{1}{\zeta(s)}$ holds for $Re(s)>\frac{1}{2}$ is equivalent to the ...
- 1,145
3
votes
1
answer
60
views
Jacobian of a 1-form on a manifold
Let $X$ be a smooth vector field on a manifold $M$. The Jacobian of $X$ at a critical point $x^*$ is the linear map
$$X'(x^*): T_{x^*}M \rightarrow T_{x^*}M$$
where $T_{x^*}M$ is the tangent space to $...
- 384
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votes
0
answers
86
views
Nonstandard Complex Analysis?
I recently discovered Nonstandard Analysis and am slowly working my way through Kelsier's textbook and Foundations companion. However while I have found plenty of stuff about real nonstandard ...
0
votes
0
answers
16
views
Factoring Sparse or Lacunary Polynomials of Single Variable
I want to learn about the factorization of lacunary polynomials of the form $f(x) = a_{0} + a_{i_{1}} x^{i_{1}} + \ldots + a_{i_{k}}x^{i_{k}}$, where $i_{1} < i_{2} < \ldots < i_{k}$ and $i_{...
- 1,585
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votes
0
answers
50
views
Best online lecture notes on probability [closed]
Are there any short expository articles (say 1~50 pages) on probability, combinatorics, optimization, or mathematical statistics? I am looking for something in the style of Keith Conrad or Andy Putman:...
- 378
2
votes
0
answers
43
views
Connections between Modular forms and Hyperbolic geometry
I am taking a course each on Modular forms and Hyperbolic geometry currently and I have begun to like the nice connections that exists between them. I am still a beginner in both these subjects and ...
2
votes
0
answers
28
views
Reference request: Tensoring with Barratt-Eccles is $\Sigma_* $-cofibrant replacement
In some different places (e.g. in Berger & Fresse, Combinatorial operad actions on cochains), it is stated as a classical result that the Barratt-Eccles operad $\mathcal{E}$ (in chain complexes) ...
- 31
2
votes
1
answer
36
views
Reference request: infinitary Ramsey theory
I was reading about the (various) infinite versions of Ramsey's theorem, and stumbled across a text containing a proof of the Bolzano-Weierstrass Theorem using it:
Unfortunately, I forgot where I ...
- 1,299