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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.

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What formulas of differential geometry am I missing?

I was reading a book on Riemannian analysis and the author assumes some formulas of differential geometry, which may be basic but I have a lack of knowledge on those. Specifically: $\int \delta (a(x)...
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0answers
14 views

Does an empirical distribution converge to the underlying distribution?

Let $\mu_n$ be an empirical distribution of $n$-iid points from the underlying distribution $\mu$. In 1D, it is well-known by Kolmogorov's theorem, Glivenko–Cantelli theorem that for any $x$, let $...
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0answers
27 views

What is symmetry in physics in the mathematical sense, using (Lie) groups

In physics, we sometimes say that, for example, a certain classical system has a certain symmetry, which is given by some group. I don't feel like I understand this well enough. Are there some good ...
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0answers
8 views

Scalable size of feasible linear constraints generator in matlab

I need to generate linear constraints in Matlab with the hope that I can scale the size. I can easily generate constraints of the following form $$ i/n \leq x_i \leq n/i, \ \forall i\in[n].$$ Here, ...
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0answers
15 views

Number of $S_n$-orbits in $P^k(\{1,\dots,n\})$

Let $n$ and $k$ be integers with $n\ge1$, $k\ge0$, and let $a(n,k)$ be the number of orbits of the symmetric group $S_n$ on the $k$-th iterated power set $$ P^k(\{1,\dots,n\}) $$ of the set $\{1,\...
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0answers
61 views

How well-studied is origami field theory?

It's well known that angle trisection cannot be done with straightedge and compass alone, as Theorem 1. If $z \in \mathbb C$ is constructible with straightedge and compass from $\mathbb Q$, then $$...
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1answer
11 views

Existence of classical Dirichlet series from Lebesgue integrable inverse Mellin transform

Let $f(s)$ be meromorphic in $\mathbb{C}$. Let the following inverse Mellin transform be Lebesgue integrable for all real positive $x$ at some complex point $s$ with some real $c$: $\frac{1}{2\pi i} \...
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0answers
32 views

Partial sums of binomial series

I am working with the following function for integers $n,m$, defined in $[0,1]$. $$f_{n,m}(x)= \sum_{i=0}^m \binom{n+i}{i}x^i$$ I know that this can be written as a hypergeometric function, using ...
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0answers
16 views

Suggestion of article or book for convolution

I need a good book or article to learn about convolution.I have a course in Neural Networks and we have to make calculations by hand.
1
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1answer
47 views

Hardy-Littlewood Inequality for Sobolev spaces

After making the mistake of applying Hardy-Littlewood-Sobolev(H-L-S) for the infinity case, I was wondering if it is possible to bound it by a Sobolev norm. Fix dimension to be $3$. H-L-S says that ...
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0answers
22 views

Expressing power law decay in terms of exponentials

I'm trying to find out how power law decays can be represented, or approximated, by exponential functions. Any papers or textbook suggestions would be particularly helpful. But in particular, on the ...
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0answers
32 views

Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

$\newcommand{\Spec}{\operatorname{Spec}}$ It is a well-known fact that a smooth projective variety over $\mathbb Q$ has good reduction almost everywhere, i.e. everywhere apart from finitely many ...
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2answers
84 views

Is there any elementary solution for this problem on colored interval?

The problem is as following. Assume $m,n$ are two coprime odd numbers, consider the interval $[0,mn]$. We cut the interval by $m,2m,\ldots,(n-1)m$ and $n, 2n,\ldots, (m-1)n$ into $m+n-1$ pieces of ...
2
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0answers
30 views

Support for Fulton reading (chap 3 and 5) [on hold]

I am studying on my own the book of the Fulton (algebraic curves). My goal is to get to chapter 5. I happen to encounter difficulties in two chapters, 3 and 5. So I would like to know if there is ...
3
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0answers
43 views

The axiomatic minimum required to have unique solutions to the Schrödinger equation

Let us consider the free non-relativistic Schrödinger equation $$i\partial_t \psi =-\frac{1}{2}\partial_x^2 \psi=:H\psi.$$ Adapting Fritz John's pathological solution to the heat equation, I find that ...
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0answers
25 views

Partition of Maximum

Suppose $n\in \mathbb{N}$, let $P_n$ be set of $n$-tuples from $\{0,a,b\}$ with at least one and at most two zero components. That is, standard members of $P_n$ are of the form $$x=(x_1,\dots,x_{i-1},...
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1answer
49 views

Good textbook for questions/answers?

Does anyone know of any good textbooks which contain a lot of exercises and solutions? A lot have exercises, and that's useful but I really would prefer having solutions if I'm going to be doing 100+...
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0answers
54 views

Alternative sources for a topos theory description of zeroth order logic

I have recently been reading Robert Goldblatt's fantastic book Topoi: The Categorial Analysis of Logic. Through chapters 6-8, Goldblatt produces a topos theoretic approach to zeroth order logic, where ...
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0answers
27 views
+50

Gale's Theorem Analog for smaller caps

Let $S^k$ denote the unit hypersphere in $\mathbb{R} ^{k+1}$. For a point $a\in S^{k}$ let $H_{\epsilon}(a):= {\{ x | \langle x,a \rangle > \epsilon}\}$. If $\epsilon =0$, then $H_{0}(a)$ is simply ...
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0answers
53 views

Machine learning book with robust linear algebra approach

I am looking for machine learning book - neural network, deep learning etc etc - that use linear algebra in a robust manner. I found satisfactory the old book of Simon Haykin : Neural Networks : A ...
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0answers
64 views

Seeking references to scholarly comparisons of $\mathbb{N}_1=\{1,2,\dots\}$ versus $\mathbb{N}_0=\{0,1,2,\dots\}$.

Since this question has been twice down-voted without explanation, I wish to explain my purpose in asking it. The document Personal review notes on the topic of algebra is one of 49 such files all of ...
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1answer
46 views

List of basic integrals? [on hold]

can anyone recommend a list of the most essential integrals in problem solving, that is the most common ones a student should know. I know there are tables of them but I want to try to memorise the ...
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0answers
14 views

Comprehensive list of power identities

Is there any comprehensive list of power identites like Worpitzky Identity, MacMillan double binomial sum (see eq. 12), Identity in Strirling numbers of the second kind and so on ?
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0answers
11 views

Semi-locally uniform convergence

I am looking for a topology on a set of functions which is somehow between uniform convergence and locally uniform convergence. For simplicity reasons I will explain my idea for the space of functions ...
0
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1answer
26 views

Reference Request for Ordinary Differential Equation problem book

I am looking for a good problem book in Ordinary Differential Equations at Graduate's level. Can someone suggest me the book for problem practice. To be precise our study revolves around the analysis ...
4
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1answer
49 views

Have similar theories like knot theory been developed in higher dimensions?

Well, my question is kind of basic but I hope it would be taken seriously by the community. Also, I'm very new to this topic and I want to study knot theory in future. Knot theory is the study of ...
0
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0answers
34 views

Book about Prime numbers [duplicate]

Is there any book that tells theorems about prime numbers with proofs. I know undergraduate calculus. I need a book that explains everything in detail.
1
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1answer
19 views

Do filters correspond to the collection of their ultrafilter extensions?

Let $X$ be a set, let $\mathscr{F}$ be the collection of filters on $X$, and let $\mathscr U$ be the collection of ultrafilters on $X$. Define $\pi: \mathscr F \to 2^{\mathscr U}$ by $$\pi(F) = \{U \...
3
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0answers
45 views
+50

Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that, evaluated at $x=e^t$, it minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\...
6
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1answer
66 views

Generalization of the fundamental theorem of duality

The "fundamental theorem of duality" states: If $X$ is a real linear space and $f, f_1,...,f_n$ are linear functionals on $X$, then $f$ lies in the span of $f_1,...,f_n$ (i.e. $f = \sum_{i=1}^n \...
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0answers
11 views

Completeness relation for Jacobi Polynomials

I was wondering if there exists a completeness relation for Jacobi Polynomials, $P^{\alpha,\beta}_{n}(x)$ as in the case of Hermite polynomials, $H_{n}(x)$ such that $$ \sum^{\infty}_{n=0} \psi_n(x) \...
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0answers
27 views

Alternative to Marcus's Number Fields with complete proofs

I originally learned algebraic number theory from Marcus's Number Fields book and think it is a wonderful book. Unfortunately, almost every proof leaves at least one unproved statement to the reader. ...
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0answers
26 views

Distance satisfying only positive definiteness

Given a set $X$ and a map $d:X\times X \rightarrow [0,\infty)$. Assume that given $x, y \in X$: $d(x,x) = 0$ $d(x,y) = 0 \Rightarrow x = y$ $d(y,x) = 0 \Rightarrow x = y$ Hence, $d$ is a metric ...
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0answers
21 views

Integro-differential equation with convolution

Given a $\mathcal{C^\infty}$ matrix-valued function $f$ from $\mathbb{R}^+$ to $\mathbb{R}^{n,n}$, I'd like to solve the following integro-differential equation: $$\ddot x(t) + \int_0^t f(\tau) \dot ...
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0answers
23 views

How to prove every holomorphic vector bundle on $\mathbb{C}P^n$ is an algebraic vector bundle

It is well-known that every holomorphic vector bundle on $\mathbb{C}P^n$ is an algebraic vector bundle, as a part of GAGA principle. Where can I find a reference for a detailed proof of the above ...
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1answer
43 views

Real numbers in theater [closed]

I am an Italian math student; on next Wednesday, the 20th of February, I will try to qualify for the following: http://famelab-italy.it foo . Have you any suggestions on how to make interesting the ...
0
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0answers
17 views

Modelling Random Variables with Specified PDF and Correlation

I am trying to develop a radar simulation system that is able to generate random processes whose elements are taken from a specified probability density function and have also have a specified ...
3
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0answers
32 views

Probability on Hilbert spaces

I have been looking to generalize a result concerning random variables with values in $\mathbb{R}^d$ to random variable with values in function spaces (in particular a space of smooth functions). I ...
2
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1answer
110 views

Is $10^{2^{21}}+1$ known to be composite?

I looked at the generalized Fermat-prime-numbers. According to factordb, the case $$10^{2^{21}}+1$$ is unknown. Neither a factor is displayed nor $C$ for "composite". Hence my question : Is $$10^{2^...
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1answer
41 views

Each ordered semigroup is cancellative: reference?

It is easy enough to show that $a+b < a+c\Rightarrow b < c$ holds in totally ordered semigroups. Indeed this must be very well known. Can anyone please provide a reference for this result? A ...
2
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1answer
47 views

Isotonic regression with a linear constraint

I'm trying to find a direct approach to solving (for some fixed vector $y$): $$ \begin{aligned} \min & \; \|x - y \|^2 \\ \mbox{s.t. } & \alpha^\top x \leq 0 \\ & x_i \...
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0answers
20 views

Definition of the surface measure in some Book

I am studying PDEs and in some place I found an integral integrated by the surface measure on M=($C^k$ - Hypersurface of $R^n$). 1)Is there any reference to see how are defined this Measure on M? 2)...
0
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1answer
36 views

Non-standard examples of smash products and joins

While working with joins and smash products (wedge sums) of topological spaces, I noticed that all examples I know (and find in textbooks) are either involving discrete spaces or spheres, e.g. that $S^...
1
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1answer
24 views

Asymptotic independence on the tail field is equivalent to a probability measure's being trivial

Let $(\Omega, \mathcal B, P)$ be a probability space and $\mathcal A = \bigcap_n \mathcal{A}_n$, where $\mathcal{B} \supset \mathcal{A}_1 \supset ... \supset \mathcal{A}_n \supset ...$ is a decreasing ...
2
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1answer
21 views

Differential on tensor product of $A_\infty$ modules

Let $A$ be an $A_\infty$-algebra over a commutative ring $k$, and $M$ is a left $A$-module and $N$ is a right $A$ module. By modules I mean $A_\infty$-modules. Then we can define $A_\infty$ tensor ...
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0answers
23 views

Questions about studying abstract harmonic analysis

Motivation I got interested in abstract harmonic analysis when I was reading representation theory of groups. In chapter 4 of J.-P. Serre's classic text Linear Representations of Finite Groups, the ...
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1answer
21 views

Vector which corresponds to angle between subspaces

The wikipedia article Angles between flats discusses the principal angles between two subspaces of $\mathbb{R}^n$. It states, "if the largest angle is $π / 2$, there is at least one vector in one ...
0
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2answers
47 views

Good book on Partial Differential Equations

I know that this question was already asked. But I am looking for a book on partial differential equations that covers the basics . And specifically explains reduction to canonical forms with examples....
0
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1answer
63 views

Formula for coefficients of interpolation polynomial, general case?

(Question) Formula for coefficients of interpolation polynomial, general case? Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)$. Suppose ...
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1answer
104 views
+50

Numerical Analysis and Differential equations book recommendations focusing on the given topics.

I am looking for an introductory book on Numerical Analysis and Differential Equations. I have done my B.Sc. in Math and I'm preparing for M.Sc entrance exams. The syllabus for the exam contains the ...