Questions tagged [reference-works]

Reference works include encyclopedias, dictionaries, books of tables (which may include numerical tables, tables of integrals, series, and products, tables of Fourier transforms, tables of finite groups, tables of combinatorial designs, etc.).

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limit pochhammer symbol $\lim_{n \rightarrow \infty} \left(\frac{1}{\sqrt{n}},\frac{1}{\sqrt{n}}\right)_n= 1$

I am looking for a proof of the following relation: $$\lim_{n \rightarrow \infty} \left(\frac{1}{\sqrt{n}},\frac{1}{\sqrt{n}} \right)_n= \lim_{n \rightarrow \infty} \prod_{i=1}^n \left( 1 - \left(\...
Grandes Jorasses's user avatar
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Who firstly gave the following definition of cartesian product?

If $\mathfrak X$ is a collection indexed over a set $I$ then it is usual name the set $$ \tag{1}\label{def: cartesian product}\prod_{i\in I}\mathfrak X:=\left\{x\in\left(\bigcup_{i\in I}X_i\right)^I:x(...
Antonio Maria Di Mauro's user avatar
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Reference for Zassenhaus bound of roots of polynomial

I remember author of some textbook alluding to Zassenhaus and Knuth's bound of the zeros of a complex polynomial.Unfortunately after even after days of search I could not find some reference or ...
AgnostMystic's user avatar
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Are there any known lower bounds on $xP(X>x)$ when $E(X) =\infty$?

I have encountered following problem while I was working on something. For what follows, let $X$ be a non-negative random variable. If we know $E(X^p)\leq \infty$, then it is well known that $x^pP(X&...
Tiramisu's user avatar
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Third cohomology of a group

I'm familiar with two primary ways to discuss the n$^{th}$ cohomology of groups with coefficients in an abelian group $A$: (1) Through the exploration of n-fold extensions. (2) By examining the map $...
MANI's user avatar
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160 views

Reference on the cumulant generating function (basic properties)

The cumulant generating function $K(t)$ of a random variable is defined as $$K(t) = \log \mathbb{E} [e^{Xt}]$$ for any $t$ such that the exponential moment is finite. In Wikipedia, it is said that: ...
Goulifet's user avatar
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Double cone with countably many cones instead of two

Is there a notion of a singularity of a real "algebraic variety" which looks locally like a double cone but with a countable infinite number of cones which meet in the same point? A short ...
Jfischer's user avatar
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Derivative of the minimiser of convex optimization problem with respect to a parameter

I consider a bivariate function $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f(x,\cdot)$ is strictly convex for any $x$. The strict convexity implies that $$ y^*(x) = \arg \min_{y \in \mathbb{...
Goulifet's user avatar
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Binomial identity reference request

Math Overflow answer https://mathoverflow.net/a/297916/113033 references the binomial identity \begin{equation} \sum_{t} \binom{r}{t} \frac{(-1)^t}{r+t+1} \binom{r+t+1}{j} =\begin{cases} ...
Petro Kolosov's user avatar
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Most relevant literature on topology and modern geometry.

I’ve been collecting and reading some articles and publications of history’s most influential mathematicians from different sources, so i’ve got a clear historical picture of their work, their ...
Simón Flavio Ibañez's user avatar
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Looking for bibliography

I was going through : Prove a convex and concave function can have at most 2 solutions Despite the provided answers are correct and flawless, I was wondering if there was some literature about since I ...
dodo's user avatar
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Is there an index of mathematical objects by notation?

I am looking for something like an encyclopaedia of mathematics, but with the main difference from regular references of such kind being that one could use it to search for mathematical objects by ...
Allawonder's user avatar
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Number theory book by Borevich and Shafarevich

I just saw a book of Number Theory by Borevich and Shafarevich. It is currently not in print. On the other hand, there are many volumes of Number Theory (I, II, III, IV etc) by Shafarevich by Springer,...
Maths Rahul's user avatar
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How many ways can we seat 100 people around 20 different circular tables in such a way that there are five people per table?

I saw the accepted answer posted by Brian M. Scott( (https://math.stackexchange.com/users/12042/brian-m-scott), Seating Multiple People at Multiple Tables, URL (version: 2013-04-25): https://math....
Elizabeth Huffman's user avatar
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Diagonal union in ZFC: a picture

Is it possible to draw-create a picture even by hand, the diagonal union concept in set theory ? It is denoted by $\triangledown\cal A$ and defined to be $\{z ∈ Z : \text{for some }x ∈ z, z ∈ A_x\}$ ...
user122424's user avatar
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Resources for learning computations using Adam Spectral Sequences.

I am trying to learn computations of homotopy groups of spheres using Adam and Adam Novikov Spectral sequences. Are there any resources that help me in learning this as quick as possible? I want to ...
Emptymind's user avatar
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5 votes
2 answers
254 views

Intersection of tangent of parabola with directix and tangent at vertex

I saw this question from Advanced Porblems in Coordinate Geometry by Vikas Gupta for JEE Advanced pertaining to Conic Section. If the line $x + y −1 = 0$ is a tangent to a parabola with focus $(1, 2)$...
Elizabeth Huffman's user avatar
2 votes
3 answers
118 views

Understanding reference from book.

At page 34 of the book The Theory of Matrices (Chelsea Publishing Company), of Cyrus Colton MacDuffee, there is a reference, but I can't understand it. It says: ...
M159's user avatar
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Could someone explain this technique to solve $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$?

Question: Find a fast calculation for $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$ I saw the above question in an another thread (linked below) but I'm just a beginner to calculus. I would appreciate ...
Elizabeth Huffman's user avatar
3 votes
2 answers
97 views

Understanding a proof that, if $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$, then the minimum value of $|a-b|$, over distinct reals $a$ and $b$, is $2$.

I saw this epic question in Advanced Problems in Mathematics by Vikas Gupta: If $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then find the minimum value of $|a-b|$, where $a$ and $b$ are distinct real numbers....
Elizabeth Huffman's user avatar
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1 answer
88 views

How does log come into play?

$$\begin{align*}I & = 2\int_{0}^{\frac{1}{\sqrt{2}}} \dfrac{\sin^{-1} x}{x}\ dx - \int_{0}^{1} \dfrac{\tan^{-1} x}{x} \ dx \\ & = 2\int_{0}^{\frac{\pi}{4}} \dfrac{\theta \cos \theta}{\sin \...
Elizabeth Huffman's user avatar
1 vote
3 answers
129 views

Why did he write $f_\min=3/2$?

I don't have much reputation to comment everywhere as as I'm new here. Could some explain me what and why this line is being written in this anser to "Find minimum value of $f(b)$ where $f(b)$ ...
Elizabeth Huffman's user avatar
2 votes
2 answers
132 views

How do I solve exercise 2.2.4 Blackburn/de Rijke/Venema’s “Modal Logic”

Here’s a transcript of the original exercise. (There‘s even a hint given by the authors in the textbook as you can see. But precisely this hint confuses me). 2.2.4 Consider the binary until operator $...
user avatar
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1 answer
48 views

Are there useful websites that shows directly the results from various sources?

One of the biggest problems in mathematical research is that it takes time, and for me, an enormous amount of time is wasted searching for results, and by results, I mostly mean references but not the ...
RanWang's user avatar
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0 answers
62 views

Exponential sums reference

Looking for reference on exponential sums, in particular Jacobi, Gauss, Kloosterman and Ramanujan sums. The books mentioned in https://mathoverflow.net/questions/65429/exponential-sums-for-beginner ...
Ximenez's user avatar
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Reference For the Question Written Below

I have had the question below on my term test a few weeks ago; however, I do not know, and I cannot find what textbook this has come from. If could you let me know where that came from, it would be ...
VirgOpta's user avatar
2 votes
1 answer
95 views

How to solve this integral equation $\int_{-\infty}^{\infty} e^{xy} g(x) dx=0$? [closed]

How do we solve integral equations like $$\displaystyle \int_{-\infty}^{\infty} e^{xy} g(x) dx=0?$$ Is there ia non-trivial solution? This is an arbitrary equation, however I am looking for any ...
stephan's user avatar
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4 votes
1 answer
148 views

This Meijer G function identity on Wikipedia is **wrong**. Can it be corrected?

While thinking about this question about the Meijer G function, I encountered the following identity listed at https://en.wikipedia.org/wiki/Meijer_G-function#Basic_properties_of_the_G-function (the ...
K.defaoite's user avatar
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3 votes
2 answers
158 views

Proof by contradiction in Game Theory: is this paper correct?

I have doubts whether this paper is sound and correct in its very basic setting. It discusses the relationship between $\varphi\vdash\psi$ and $\varphi\Rightarrow\psi$ and denies it in general, i.e. ...
user122424's user avatar
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1 vote
0 answers
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Need help finding G. Pick's 1917 article "Über affine Geometrie iv: Differentialinvarianten der Flächen gegenüber affinen Transformationen"

I'm studying affine geometry and I need help finding this article "Über affine Geometrie iv: Differentialinvarianten der Flächen gegenüber affinen Transformationen" written by G. Pick in ...
User0212's user avatar
4 votes
1 answer
71 views

Is there a comprehensive list of complexity theoretic reductions from and to prime number factorization?

I am interested in the complexity theoretic equivalences of prime number factorization. I am especially interested to learn wether there are some not initially obvious reductions. Im sure there is a ...
ckrk's user avatar
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-1 votes
1 answer
76 views

Can someone provide a brief list of when, and when not to, use each finite difference approximation for derivative?

So far, I've received one possible scenario, and may be the only one, for using the forward difference in approximating the derivative rather than its central difference counterpart. And that is when ...
user5896534's user avatar
1 vote
0 answers
238 views

English translation of Opera Omnia, Euler

I want to read Euler’s works, and yes, I am aware that I can read them online, on Euler Archive. But, I was looking for physical books in which Euler’s works are compiled. When searching for the ...
Srinidhi kabra's user avatar
3 votes
1 answer
258 views

What computer methods are used to quickly calculate the $\zeta$-function (if any)?

So I can think of how I could compute $\zeta(\sigma + it)$ in principle. We can take $\zeta(\sigma + it)$ for $\sigma>1$ by the usual $\sum_{n} n^{-(\sigma + it)}$ summation. We can use the ...
Maximal Ideal's user avatar
1 vote
1 answer
297 views

Generalized mean value property for the Poisson equation

It is well known that solutions to the Laplace equation in a region $\Omega\subseteq\mathbb{R}^n$, $\nabla^2u=0$ satisfy the mean value property, namely for all $x\in\Omega$, and for all $r>0$, $$ ...
Joshhh's user avatar
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1 vote
1 answer
184 views

Mathematical Journal on Logic and Set Theory

I'm looking for free to access mathematical journal about logic and set theory. I found some earlier but all of them need payment/subscription. Is there anyone that know some reference? I'm a little ...
Mikaela Gentica's user avatar
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0 answers
51 views

Work of Omar Khayyam on cubic equations

I am trying to see the work of Omar Khayyam on cubic equations. It seems that he solved some types of cubic equations. However, I am not getting proper references for it. Can anybody suggest a ...
Maths Rahul's user avatar
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1 vote
0 answers
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Missing a page in a journal paper, is it normal?

My question comes from the paper Propagation of chaos for Burgers' equation published at Annals of the IHP Theoretical Physics, Tome 39 (1983) no. 1, pp. 85-97. However, the page 95 is clearly missing....
Fei Cao's user avatar
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-3 votes
1 answer
362 views

Pseudonyms in mathematics and groups of mathematicians similar to the Bourbaki group.

In the thread at the end of the post the asker asks for pseudonyms for famous mathematicians or (secretive) groups of famous mathematicians. The famous "Bourbaki book series" is an example ...
hm2020's user avatar
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4 votes
1 answer
297 views

Are there applications of $\mathrm{\int \frac {dx}{tan^{-1}(x)}}$?

This will be a follow up question to my unexpectedly popular question: Is there an exact solution for $\large\int \frac{dx}{\tan^{-1}(x)}$? which is also nicely related to: Indefinite Integral $\...
Тyma Gaidash's user avatar
1 vote
2 answers
108 views

Proof that there's only one function that satisifes the properties of a trace. [duplicate]

I've been reading Mathematics for Machine Learning, by Deisenroth, Aldo Faisal and Cheng Soon Ong, and it contains the following lines about the trace of square matrices that I need help with: The ...
Gaurav Chandan's user avatar
3 votes
1 answer
118 views

Is there any English translation of the Gergonne paper "Variétés. Essai de dialectique rationnelle" ("Varieties. Essay about rational dialectic")?

Is there any English translation of this Gergonne paper? "Variétés. Essai de dialectique rationnelle", Annales de Mathématiques pures et appliquées, tome 7 (1816-1817), p. 189-228. ("...
Aris Makrides's user avatar
1 vote
2 answers
976 views

English translation of Riemann's complete works

Browsing in the library I came across with the mathematical (and some philosophical) papers of Riemann, collected by Weber and Dedekind in the original German (although published by Dover). But ...
DanielKatzner's user avatar
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0 answers
52 views

Where to find known bounds on expressions?

Example. In a problem I was working on, I had an expression of the form $(1-e^{-\alpha n})^{e^{\beta n }}$. I wanted to find an upper bound $f(\alpha, \beta, n)$ on this that makes it easier to see ...
user56834's user avatar
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0 votes
1 answer
394 views

"NONCOMMUTATIVE Algebra with a view towards Algebraic Geometry"?

Is there a noncommutative algebra book that is similar to Eisenbud's "Commutative Algebra with a view towards Algebraic Geometry" in the sense that fundamental and geometrically motivated ...
Dat Minh Ha's user avatar
-1 votes
2 answers
161 views

How To Learn Mathematics Basics [closed]

I wanna learn mathematics from zero to advanced which books should I read and which websites should I visit?
Issam Mohamed Hammadi's user avatar
1 vote
1 answer
280 views

Lecture notes of complex analysis

I just want to go through the topics of complex analysis, in order to get an overview of the subject. Does there exists any good lecture notes for complex analysis which covers all the topics quickly? ...
roydiptajit's user avatar
7 votes
0 answers
343 views

Is there an equivalent of differential geometry for infinite dimensional spaces?

Differential and (Semi/Pseudo-)Riemannian geometry provide a framework for doing calculus on finite dimensional manifolds and have applications in physics (general relativity) and dynamical systems ...
CasaBonita's user avatar
2 votes
0 answers
84 views

Is there a bibliography of Lebesgue’s publications (with translations)?

I’m reading in the history of measure and integration and can’t find a complete list of Lebesgue’s publications, which I understand comprise his PhD thesis, some 50 papers, and two monographs. (...
symplectomorphic's user avatar
1 vote
1 answer
103 views

Mathematical analysis book with a specific result of continuity.

Let $I^{n}\overset{_\mathrm{def}}{=}\left\{x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}|\;\;a_i\leq x_{i} \leq b_{i}\,;\; i=1, \ldots, n\right\}$ be an $n$-dimensional closed interval and $I$...
Elias Costa's user avatar
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