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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1answer
29 views

Ind- and pro-objects, reference request

Can someone point me to a good exposition of ind- and pro-objects, the intuition behind, and how one "in practice" works with them (i.e. prove things)? The nlab page is nice (especially for the ...
1
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1answer
56 views

reference for classifying groups of order $p^2q^2$

In a previous question I asked about the number and structure of groups of order $p^2q^2$ where $p,q$ are primes and with the help of Prof. Derek Holt I understand it now (see here non-abelian groups ...
2
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0answers
32 views

$A^{\frac 32}$ for $A\geq0$ self-adjoint as an integral of the Resolvent

Let $A\geq0$ be a bounded self-adjoint operator on a Hilbert space. I would like to show that $$A^{\frac 32} =c \int_0^\infty A^2 (y+A)^{-1}y^{-\frac 12}\text{d}y,$$ where $c>0$ is an appropriate ...
3
votes
1answer
54 views

What is the difference between high dimensional and low dimensional chaos?

Often I read of high and low dimensional chaos. But, I don't know what is their difference. I have thought the following answer. Let us consider a time series $\{x_i\}_{i\in\mathbb N}$. According to ...
4
votes
1answer
120 views

What is a number theory book I can read in bed?

I am looking for a good book that is very easy going but not a "pop science" account i.e. something that goes through theory that would be on a basic undergraduate course for someone who finds the ...
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0answers
33 views

The approximation formula $\left|\alpha -\frac{p}{q}\right| \le \frac{1}{\sqrt{5}q^2}$

I have seen a result about the approximation of irrational numbers and want to find its proof. Suppose $\alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=...
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0answers
39 views

A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
3
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2answers
95 views

The graph of the function $f(x)= \left\{ \frac{1}{2 x} \right\}- \frac{1}{2}\left\{ \frac{1}{x} \right\} $ for $0<x<1$

Let for reals $$\{x\}=\text{Frac}(x)$$ the fractional part function, take for example the more common definition, the first (there is a different definition as you see in this MathWorld's Page, ...
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1answer
39 views

How do we evaluate this Dirichlet L-series

In this answer, David Speyer, whose answer is magnificent, states that "The sum $\sum \chi_3(n)/n$ is only slightly less well known; it is $\pi/(3 \sqrt{3})$.", where $\chi_3(n)$ is the character ...
1
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0answers
13 views

Looking for an entry level discussion on convex analysis

I have been studying for a qualifier and every so often I come across questions such as: Let $f_n:[a,b] \to \mathbb{R}$ be convex functions and suppose that $f(x) = \lim_{n \to \infty} f_n(x)$ exists ...
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0answers
28 views

PDE with Stochastic Coefficients

Does anyone have reference suggestions for pde's with stochastic coefficients? I've seen many papers on more advanced problems, but it would be great to have a reference discussing the basic theory ...
3
votes
1answer
90 views

Formalizing splitting into cases

Let $x$ denote a fixed but arbitrary real, and suppose we're trying to solve an equation like $$(x^2-1)^2 = 1.$$ The 'high school' approach is to just shuffle the functions on one side onto the other ...
4
votes
2answers
68 views

For which classes of matrix can the matrix exponential be easily computed?

We have diagonal matrices $A = \mbox{diag} (\lambda_1, \ldots, \lambda_n)$ for which matrix exponential has simple form $e^A = \mbox{diag} (e^{\lambda_1}, \ldots, e^{\lambda_n})$, and it can be ...
-1
votes
0answers
42 views

Killing fields and the Laplacian on $S^{2n + 1}$

If $X_{ij} = (x_i\partial_j - x_j\partial_i|_{S^n}$, where $(x_1,..,x_{n + 1}) \in \mathbb{R}^{n + 1}$, then it is known that the Laplacian $\Delta$ on $S^n$ is given by $\Delta = \sum_{i \neq j} X_{...
2
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2answers
69 views

Evaluate $\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$ for fixed integers $n,k\geq 1$

My question is the following Question. Can you compute some of the following $$c_{n,k}=\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$$ where $n\geq 1$ is a fixed integer and $k\geq 1$ is ...
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0answers
9 views

Input Error estimation

I was wondering what are the methods used to detect the input's error when having the output's error of a model. I thoroughly searched on google, but I failed to find a well explined method, or ...
2
votes
1answer
67 views

Name of the theorem: If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$?

Note: I am not asking for a proof of this theorem or any other theorem or help with a mathematical problem. This question is a reference request. I use the following well-known and somewhat-easy-to-...
4
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0answers
66 views

Reference about $\sigma$-linked posets and related notions

In this link, the following list appears: Some chain conditions [of posets], listed from easiest to satisfy to hardest to satisfy: ccc powerfully ccc productively ccc $\sigma$-...
4
votes
1answer
50 views

A topology on the natural numbers

Is there a name of or a reference to the following topology on $X=\mathbb N$: $A\subseteq X$ is closed if and only if $n\in A\wedge m|n\implies m\in A$?
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0answers
22 views

Green functions

do you know some litterature about green functions for the heat equation ? in particular for the non-linear equation : $\frac{\partial u(x,y,t)}{\partial t}-\frac{\partial^2 \left[ f(u(x,y,t))u(x,y,t)...
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1answer
47 views

Are there any examples of consistent proper axiomatic extensions of classical logic?

By a proper axiomatic extension, I mean a logic with the same set of well formed formulas as classical logic, but with the set of theorems of the logic a proper superset of the theorems of classical ...
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0answers
20 views

Rappaports algorithm for the convex hull of multiple circles.

I would like to use D. Rappaports convex hull algorithm for discs in a JavaScript application, which I am developing. http://www.sciencedirect.com/science/article/pii/092577219290015K I found ...
1
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1answer
66 views

Books about the foundations of (calculus) functions?

I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe ...
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0answers
24 views

On a stronger property than being an Armendariz ring

A ring $R$ is said to be Armendariz if $f(x), g(x) \in R[x]$ are such that $f(x)g(x) = 0$, where $f(x) = a_nx^n + \dots a_0, g(x) = b_mx^m + \dots + b_0$, then $a_ib_j=0$ for all $i,j$. In other ...
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1answer
38 views

Reference request: $W^{2,2}$ estimates of elliptic PDE with measurable coefficients

I have some questions on solvability of the following elliptic PDE: in $\mathbb{R}^2$, for $f\in L^2$, $$a^{ij} u_{x^i x^j} +b^i u_{x^i} + c u = f.$$ Here $\{a^{ij}(x)\}_{i,j=1,2}$ is symmetric ...
1
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1answer
43 views

self teach algorithms [closed]

What are some good resources to self teach the subject of Algorithms for someone with background in mathematics? That is, does there exists a more theoretical and abstract approach versus practical ...
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0answers
15 views

Reference for the equivalence of categories between the categories of affine group schemes and commutative hopf algebras.

Where can I find the proof or a discussion that the category of affine group schemes is equivalent to the category of commutative hopf algebras? Thanks.
0
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0answers
17 views

A “contraction” on tensor product spaces

Let $X$ be a topological vector space. Let $X'$ denote its continuous dual. Consider the (algebraic) tensor product $\mathbb{X} := X \otimes X'$. For simple tensors $x \otimes x' \in \mathbb{X}$ set $...
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1answer
46 views

Courant-Hilbert's Book: Weyl's asymptotic law for eigenvalues - Planar domains

In the book Strauss W.A. Partial Differential Equations - an Introduction (Wiley, 2008, 1st Ed.) page $311$, there is a comment Now an arbitrary plane domain $D$ can be approximated by unions of ...
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2answers
33 views

Curvature of curves on surfaces

Are there ways to know the curvature of a curve $\gamma$ that lives on a surface $\mathcal{S}$starting from the gaussian curvature of $\mathcal{S}$? In general, is it possible bound the curvature of ...
4
votes
1answer
89 views

Applications of PDEs in many variables

One reason that solving systems of partial differential equations is so important is the many applications of PDEs in science and engineering (eg. the heat equation, the wave equation, etc.). Often ...
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0answers
42 views

In which quadrant of the circle does the angle of $90^\circ$ lie?

By definition and with an authoritative reference, in which quadrant or quadrants does $90^\circ$ lie? (There are non-authoritative references which answer the question, and a related question which ...
0
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1answer
12 views

Decidability of quantifier-free formulae in Peano- and True Arithmetic

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...
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1answer
20 views

Differential forms vector space over function field?

Let $V$ be a vector space and $V^*$ be its dual space. Then I know that $V^*$ is considered a vector space because we can scale the basis covectors by real numbers and add them together and all of ...
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0answers
26 views

Legendre's Conjecture Theme (Part II)

This is a continuation of this question. My main question is that, in the previous question we were mainly concerned about the sign of, $$f_{2}(n)=\pi\left((n+1)^2\right)+\pi\left(n^2\right)-2\pi\left(...
0
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1answer
20 views

On terms “Orientation” & “Oriented” in different mathematical areas?

The goal of this question is to help to deal with different meanings of the words such as "orientation" and "oriented" in different mathematical areas. Are different oriented concepts somehow ...
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0answers
8 views

Are Oriented Graphs Related to Oriented Matroids?

My professor said that oriented matroids make it easier to investigate things such as connectivity. Recall that an oriented graph is a digraph without multiple edges or loops. Now Are oriented graphs ...
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12answers
405 views

The Big List of the Mathematical Songs [closed]

I recently watched a music video in Youtube about a finite simple group of order two! (See also this link). What are other examples of songs about mathematicians and mathematical objects? Please ...
0
votes
2answers
39 views

Special Relativity-Book

I would a good book to study the Special Relativity. In my course the professor has treated the following topics: $(1)$ Lagrangian and hamiltonian dynamic of a charged particle; $(2)$ Relaticistic ...
12
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2answers
242 views

modern calculus or analysis text that emphasizes Landau notation?

Is there a comprehensive calculus or analysis textbook or problem book, written in the last twenty years, that emphasizes the use of Landau notation (big and little oh), especially for making ...
1
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0answers
13 views

Area of Operations Research on graph theory and reliability engineering? [closed]

I am confused by the jargon in Operations Research (OR) when it is the same as in Graph theory such as component but it can mean just a vertex. So I am confused to the extent that reliability ...
2
votes
0answers
64 views

Legendre's Conjecture Theme (Part I)

Main Question Recently I have been thinking about the Legendre's Conjecture. I noticed that a proof of the conjecture can be obtained if we can prove any one of the following, Conjecture 1. For ...
2
votes
0answers
36 views

Introduction to p-adic vector spaces

I'm interested in learning about vector spaces over $\mathbb{C}_p$ and $\mathbb{Q}_p$. Most textbooks on p-adic numbers (Koblitz, Schikhof) focus on analysis and number theory. Is there any ...
3
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0answers
37 views

Image of the norm map in imaginary quadratic fields

Let $K=\mathbb{Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$. I want to know the image of the norm map $$ N^K_{\mathbb{Q}}:\mathcal{O}_K\to\mathbb{Z} $$ and the values of $N^...
4
votes
1answer
31 views

Ultrafilter on $[0,1]$ consisting of closed sets

Today we learned about filters and ultrafilters in the General Topology course. I am trying to play around with these definitions. I wish to ask a question that I am unsure about. Let us say, we have ...
0
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0answers
48 views

Books on inference for stochastic analysts

I realize that book recommendations to learn statistical inference is a hackneyed topic but I have something more specific in mind. I work on diffusions and would like to quickly and effectively learn ...
3
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2answers
85 views

Maclane/Birkhoff's “Algebra” as a first book on the subject?

Would the more knowledgeable and well-versed members of this community be so helpful as to give their opinion on using Birkhoff & MacLane's famous "Algebra" for a first course in Abstract Algebra? ...
3
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2answers
97 views

Spivak's Calculus?

I have seen many users here asking questions about problems in what they call "Spivak's Calculus Book". I have never seen the book, and information online is scarce. From what I've gathered, it is ...
0
votes
1answer
35 views

A good reference for irreducible and noetherian spaces

I am looking for a comperhensive reference for irreducible and noetherian topological spaces. Also, a reference for prime spectrum of a commutative ring.
0
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2answers
83 views

Reference for Algebraic Topology

I know undergraduate algebra (groups, rings, fields, Galois etc), undergraduate differential geometry, undergraduate real/complex analysis and now I feel as though to get to the next level, i should ...