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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7 views

Distributions of components to distribution of vector

Suppose that I have independent variables $x_1,\ldots,x_n$ with tractable (not necessarily identical) distributions. I'm interested in the distribution of $\boldsymbol{x}=(x_1,\ldots,x_n)'$ and, if ...
0
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0answers
8 views

Counting the numnber of (labelled and unlabelled) rooted trees on $n$ vertices with height $h$

As far as I know, the number of labelled rooted trees on $n$ vertices is $n^{n-1}$. Is there a known result for counting the number of (labelled and unlabelled) rooted trees on $n$ vertices having ...
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0answers
54 views
+50

Coordinate-free notation for tensor contraction?

I am not sure if I can prevent this question from being too vague or with too large an overlap with other similar math.SE questions, but I will do my best... A standard linear operation in tensor ...
1
vote
1answer
26 views

Reference for the $\Bbb A^1_k$-rigidity of abelian $k$-varieties

Is there a reference that shows, for a field $k$, that abelian $k$-varieties are $\Bbb A^1_k$-rigid? A smooth variety $X$ over $k$ is $\Bbb A^1_k$-rigid if and only if the canonical map $$ ...
5
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3answers
1k views

Is there an English translation of Jordan's “Cours D'analyse”

I am trying to find an English translation of Camille Jordan's work "Cours D'analyse". Only the French edition is on Amazon, so since this is a somewhat specialized topic, I thought perhaps someone in ...
5
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0answers
41 views

When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
0
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1answer
37 views

Reference for: simple closed curves generate the fundamental group

In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not ...
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0answers
40 views

Is there any linear algebra textbook presented using logical symbols?

I'm currently going through a book called Linear Algebra Done Right by Axler, and to be honest, his book seems to be very loose with what things he defines. For instance , the symbol 0 could be mean a ...
1
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1answer
18 views

Universal UHF algebra

I am reading on quasidiagonal $C^*$-algebras. There the phrase "the universal UHF algebra" appears. I know what UHF algebras are, but I don't know what the universal UHF algebra is. I would be glad ...
1
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0answers
8 views

How to make a probabilistic sense of the semigroup of a positive operator

Consider the operator $\mathcal{L}$ acting on the function $f:\{0,1\}\mapsto \mathbb{R}$ defined as following: $$\mathcal{L}f(x)=f(1-x)-f(x)$$ This is the infinitesimal generator of a continuous time ...
3
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1answer
27 views

One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...
2
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1answer
68 views

An example of frame operator.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in ...
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0answers
9 views

Ask for reference convergence of implicit euler method for initial value problem with dissipative source term

I am considering the convergence of implicit euler method for solving the following initial value problem: \begin{cases} u'(t)=f(t,u(t)),t\in[0,T]\\ u(0)=u_0\in \mathbb{R}, \end{cases} where ...
2
votes
0answers
29 views

Name of dominated convergence for sums

Having a sequence $(a_n(j))_{n}$ where every element of the sequence also depends on $j\in\mathbb{N}$. If $\sum_{n=1}^\infty \sup_{j\in\mathbb{N}} |a_n(j)| < \infty$, then the following (assuming ...
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0answers
20 views

Reference Request: Monologues on Lie Groups/Algebras and Differential Geometry

I find that before really diving into a subject, I prefer to get a general idea of it. For instance, before studying ergodic theory through a standard textbook I enjoyed Paul Halmos' lecture notes on ...
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votes
3answers
293 views

Where has this common generalization of nets and filters been written down?

It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters. They are of course essentially equivalent, ...
0
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2answers
289 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
2
votes
1answer
46 views

double integrals on quantum calculus

I need references or book recommendations to find properties of double integrals on quantum calculus. Especially i need analogue of Fubini's theorem on q-calculus.
2
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2answers
914 views

What does it mean by piecewise smooth boundary?

I will be highly obliged if anyone can give me any reference where i can get the definition of domain (in $\mathbb{R^n}$) with piecewise smooth boundary. My question is when a domain in ...
0
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0answers
22 views

Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where ...
0
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0answers
15 views

What are the good references on tame hereditary algebras?

I have Thomas Brustle's Typical Examples of Tame Algebras, but I still do not have a systemic understanding of what tubes are and what regular roots of a tame hereditary algebra are. I'm also looking ...
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0answers
18 views

Should I ask about math books? I.e. scan of index page or references page? [on hold]

I mean it's not harmful to anyone if I ask about certain page of certain book? I need page 263 from this book ...
6
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1answer
235 views

A book on advanced math for a “novice” mathematician, but “mature” thinker

I enjoy giving interesting problems to my peers who have not seen a lot of mathematics. It is a constant conversation I have with many friends who would interchange "mathematical thinking" with ...
3
votes
1answer
27 views

“Shape” of solutions of 2nd order homogeneous ODEs

Consider a second order homogeneous ODE: $$P(x)y''+Q(x)y'+R(x)y=0.$$ If $P,Q,R$ are constant functions, then we know that the general solution has the form $$y=c_1e^{r_1x}+c_2e^{r_2x},$$ ...
3
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0answers
60 views

Reference request for Grothendieck's work on “Integration with values in a topological group”

Recently I was reading the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found, An anecdote survives about Grothendieck's arrival in Nancy: ...
2
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1answer
210 views

Tutorials for Sprague-Grundy Theorem/Nimbers?

Help needed in understanding S-Grundy Number , any good tutorial. I am trying to solve Mathalon Problem 146 S-Grundy Game (dead link).
4
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0answers
50 views

Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
2
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0answers
26 views

How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
4
votes
1answer
44 views

Finite Almost Simple Groups

I want to study finite almost simple groups but I am not sure which would be the best texts to look at. Can someone please refer me to some books that teach the theory of finite almost simple groups?
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0answers
59 views

The free cocompletion of a complete locally small category is complete

$\DeclareMathOperator{\colim}{colim}\newcommand{\cat}{\mathbf}\DeclareMathOperator{\Nat}{Nat}$I'm looking for a reference that talks about the free cocompletion $\hat{\cat C}$ of a (large) locally ...
4
votes
1answer
1k views

Preferable Order of Mathematics Study

I was just wondering if someone would be kind enough to tell me in what order (I know that there is no real "best order") one would most profitably study these subjects/books: (edited to conform with ...
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0answers
53 views

Find a polynomial such that this proposed root finding algorithm fails.

Is this polynomial root finding algorithm below known, and under what conditions for the choice of polynomial coefficients does it find at least one root? Description of the algorithm: Consider the ...
2
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1answer
55 views

Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...
2
votes
1answer
39 views

Is there a projective morphism from the quadric surface to the projective plane with degree 1?

Is there a projective morphism from the quadric surface $\mathbb{P}^1\times\mathbb{P}^1$ to the projective plane $\mathbb{P}^2$, with degree $1$?
48
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4answers
68k views

Eigenvectors of real symmetric matrices are orthogonal

Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = ...
2
votes
0answers
29 views

What came first: pythagoras number or pythagorean fields? [migrated]

Which concept was first introduced: the pythagoras number of a field or pythagorean fields? I have not found anything on this matter, but my gut feeling says the latter. One can more directly link the ...
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2answers
233 views

series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

I found an interesting series $$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( ...
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0answers
33 views

Reference Quest: Measure Theoretic and Functional Analytic Intro to Stochastic Processes

Does anyone have any recommendations for a good book which introduces and cleanly and rigorously explains the measure theory and functional analysis implicit in and relevant to stochastic processes, ...
4
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2answers
183 views

Method to solve $xx'-x=f(t)$

I would like to resolve this differential equation: $xx'-x=f(t)$ any suggestions (or any online texts on similar differential equation) please? Thanks.
0
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0answers
12 views

An empirical correspondence in algebra

In the article Simplicity of Jordan superalgebras and relations with Lie structures by C. Martinez, the author states: "What is know about Jordan superalgebras with non-semisimple even part? Here the ...
1
vote
1answer
32 views

Is there a list of recommended problems to do in each chapter of Spivak's Calculus anywhere?

I've recently been self-studying Spivak's Calculus, and since I don't have the time to do every problem from every chapter at a and finish at reasonable rate, I've looked for a course syllabus or ...
2
votes
0answers
11 views

Reference request: Relationship between Entropy and Lyapunov Exponent

If $\lambda$ is the largest positive Lyapunov exponent of a piecewise linear dynamical chaotic discrete in time map, then is there a relationship between the entropy and its $\lambda$. I remember ...
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vote
2answers
44 views

Proving Euler's spiral is an isometric embedding with bounded image

$\newcommand{\al}{\alpha}$ I am trying to prove Euler's spiral is an isometric embedding of $\mathbb{R}$ into $\mathbb{R}^2$ with bounded image. Here is the definition of the spiral: $(*) \, ...
3
votes
1answer
409 views

Demystifying the asymptotic expression for the partition function

A partition of an integer $n$ is a way of writing $n$ as a sum of integers. The partition function $p(n)$ counts the number of distinct partitions of $n$. In 1918, Hardy and Ramanujan proved the ...
0
votes
1answer
254 views

Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
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0answers
45 views

Intuitive, short explanation of differential forms and exterior calculus

Are there any introductory lecture notes on differential forms and exterior calculus, preferably aimed at physics students studying General Relativity and Black holes? I have some familiarity with GR ...
2
votes
1answer
71 views

Classification theorem of the coverings of a given space

I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as Classification of covering spaces ...
2
votes
0answers
84 views

Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E*(du^2+dv^2)$ and where $E>0$ is an harmonic function? I know that this metric is a special kind ...
0
votes
0answers
38 views

Book reference for theory of differential equations (not Coddington's book)

I'm looking for references to study theory of ordinary differential equations. I'm looking for a similar book to Coddington's book, theory of ordinary differential equations but not this one, because ...
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0answers
13 views

References to: If $C\subset\mathbb{R}^n$ is convex and $0\notin C$ then there exists $v\in C$ such that $C$ is in the closed halfspace $H_v$.

For each $v\in\mathbb{R}^n$, we define the notation $H_v=\{u\in\mathbb{R}^n:\langle u,v\rangle\geq0\}$, where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in $\mathbb{R}^n$. Recently, ...