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
47 views

Reference Request: Differential Geometry Book [closed]

What is a good self study book in Differential Geometry. Keep in mind I won't have the advantage of being able to ask a professor any questions.
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4answers
2k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
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0answers
6 views

CG as an orthogonal projection

I have heard that the Conjugate Gradient method can be viewed as an orthogonal projection onto the Krylov subspace $K(A,r_0)$, but I can't find a reference that deal with it in this way. Could you ...
0
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0answers
26 views

How can I correctly catalog this partition problem?

Studying the partition problems, I tried to do an special version to apply it to a kind of model of "orbits and energy levels" (explained below), but I am having problems to properly catalog this. ...
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1answer
41 views

Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
1
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1answer
103 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
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0answers
21 views

Is this orthogonal distance a common pseudometric?

Define $d: V \times W \to \mathbb{R}$ such that $$d(v,w) = \sup_{z \perp w} \frac{\langle z, v \rangle}{\|v\|\|z\|}.$$ Is this a pseudometric that anyone has utilized in the literature? Does it have a ...
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1answer
75 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
0
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0answers
18 views

measure-theoretic probability, (sets of) null events and non-zero probability

Assuming a well-defined probability space $ (\Bbb{R},\mathscr{B},\Bbb{P}_X) $, where $\mathscr{B}$ is the Borel $\sigma$-field, and for a random variable $X$ having a continuous probability density ...
2
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1answer
42 views

Help finding paper from the 1920's

I have not been able to find a copy of this paper anywhere! B. Knaster еt C. Kuratowski: Sur quеlquеs propriétés topologiquеs dеs fonctions dеrivéеs. Rеnd. dеl Сirc. Math. di Palеrmo, 59 (1925), ...
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3answers
40 views

ordinary differential equation project suggestion [closed]

My professor asked to write a project on ODE just to experience on how to write projects. It need not be a research project. Being in second rate school from third world country, we never did those ...
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0answers
10 views

Exponential matrix using Laplace transform - reference request [closed]

I am looking for a book that covers the concept of exponential matrix in detail using the Laplace transform plz
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3answers
399 views

Extension of real analytic function to a complex analytic function

I just learned that real analytic functions (by real analytic, I mean functions $f: \mathbb{R} \to \mathbb{R}$ which admit a local Taylor series expansion around any point $p \in \mathbb{R}$) cannot ...
2
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1answer
75 views

Linear ordinary differential equations and their evolution operators for measurable operators

Consider the following homogeneous IVP: $$\begin{cases} \dot{u}(t)+A(t)u(t)=0 \\ u(0)=u_0 \end{cases}$$ for $u:[0,1]\to \mathbb{R}^n$ (some interval to some finite dimensional Hilbert space, let's ...
1
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4answers
1k views

n-th roots of unity form a cyclic group in a field of characteristic p if gcd(n,p) = 1

Let $n$ be a positive integer, and let $\mathbb F$ be a field of positive characteristic $p$ with $\gcd(n,p) = 1$. Where can I find some proofs that the group of all $n$-th roots of unity (in an ...
2
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0answers
59 views

Tight bounds for harmonic measure

I recently came across a question concerning harmonic measure here, and was wondering if there is a good reference summarizing different methods of estimating harmonic measure? Specifically, I would ...
0
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2answers
113 views

The Adjunction $\_\times A\dashv (\_ )^A$ for Preorders: The Deduction Theorem.

The following is from Turi's Category Theory Lecture Notes. Definition 11.11 Let $A$ be an object of a category $\mathbb{C}$ with binary products. The right adjoint of $\_\times ...
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1answer
60 views

Wallpaper groups for the hyperbolic plane

I would be grateful if someone could direct me to a reference that classifies the equivalent of the wallpaper groups (and the frieze groups and the point groups, if possible) for the hyperbolic plane, ...
16
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1answer
2k views

Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel ...
3
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1answer
111 views

Dual of holomorphic functions (with the $L^1$ topology)

Let $\Omega$ be a connected domain of the complex plane, and let $E$ be the vector space of integrable holomorphic functions on $\Omega$. Then it can be checked that $E$ is a closed subspace of ...
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3answers
2k views

What is the origin of the expression “Yoneda Lemma”?

Thank you very much in advance for telling where the expression “Yoneda Lemma” comes from. EDIT 1. On page -14 of Reprints in Theory and Applications of Categories, No. 3, 2003. Abelian Categories, ...
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6answers
10k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
27
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6answers
840 views

Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...
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1answer
82 views

Luna-Vust theory for embeddings of homogenous spaces

I'm interested in the theory of Luna and Vust of embeddings of homogenous spaces like presented in D. Luna, Th. Vust: Plongements d'espaces homogènes, Comment. Math. Helvetici 58 (1983) 186-245. ...
7
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1answer
77 views

Why is complex analysis so nice? And how is it connected/motivating for algebraic topology?

This is very much a soft question, but after seeing Cauchy's integral formula in lecture today I was really struck by how neat complex analysis is. I don't understand how all of these amazing analytic ...
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1answer
12 views

Are there any connected graphs with constant link which are not vertex transitive?

By constant link, I mean for any vertices $v,w$ of a graph $G$, the subgraph of $G$ induced by the neighborhood of $v$ is isomorphic to the subgraph induced by the neighborhood of $w$. $C_n + C_m$ ...
2
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1answer
34 views

Graph of a $G_\delta$-function

Let $f:\mathbb R \to \mathbb R$ be a function. It is well known that if $f$ is continuous ($f^{-1} [A]$ is closed whenever $A$ is closed), then its graph is closed in $\mathbb R ^2$. Here is an ...
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1answer
40 views

Reference needed with derivation of the equations for the cubic and quartic and proof of impossibility of quintic equation

My background is an undergrad year long course in algebra which got me through to some basic finite field and field extension stuff (most of fraliegh or Ian Stewart's book are familiar to me if that ...
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2answers
51 views

Probability spaces over graphs: which area has focus on them?

Suppose a simple graph $G$. Now consider probability space $G(v;p)$ where $0\leq p\leq 1$ and $v$ vertices. I want to calculate globally-determined properties of $G(v;p)$ such as connectivity and ...
1
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1answer
19 views

PreLie Operads and Free PreLie Algebras

Is there any relationship between Free Pre-Lie Algebras and Free Algebras over the Operad Pre-Lie (like in KOSZUL DUALITY FOR OPERADS in pg 13 )? If there is, can someone indicate a reference for ...
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0answers
24 views

Please recommend books for Python and what is scope of Pyhton ??? [closed]

I am looking for good books of Python but I am not able to find good one. Please suggest me some.
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1answer
43 views

Any Video Lectures Of An MIT, Harvard, Stanford, UC Berkeley, Yale, or Princeton Analysis Course Based On Baby Rudin?

I'm learning analysis from the book Principles of Mathematical Analysis by Walter Rudin, third edition. This book, popularly known as Baby Rudin, is being used for analysis courses at such elite ...
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1answer
51 views

Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
0
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1answer
33 views

Reference request: alternative proof for every open set in $\mathbb{R}^n$ can be expressed as countable disjoint union of open boxes

A "box" is a cartesian product of intervals of the type $[a,b]$ I am using Terence Tao's introduction to measure theory and on page 24 a proof of title statement is given, however, it is quite ...
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1answer
24 views

Generators of the congruence subgroup $\Gamma (5) \subset SL(2,\mathbb Z)$

Recall that $SL(2,\mathbb Z)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}: \det(A)=1; a,b,c,d \in \mathbb Z \right\} $ and $\Gamma(5)=\left\{A=\begin{pmatrix} a&b \\ c&d ...
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0answers
9 views

How to formulate and analyze systems of stochastic differential equations?

I'm having trouble finding reference material on how to deal with systems of stochastic differential equations. Specifically, I'm interested in ecological models. For example, consider the standard ...
1
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1answer
55 views

Can I ask about copyright here? [closed]

I don't know it is okay if I post about copyright here. If there is any problem, please leave a comment. I will delete this post. Thank you. I am a graduate student studying engineering and ...
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0answers
31 views

An identity for $q-$Fibonacci numbers if $q$ is a root of unity.

In his proof of the Rogers-Ramanujan identities I. Schur introduced two $q-$analogues of the Fibonacci numbers ${F_n}({q})$ and ${G_n}({q})$, which satisfy ${F_0}({q})=0$, ${F_1}({q})=1$ and ...
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1answer
34 views

Bernstein Inequality (wiki correction?!)

I am having trouble with one of the statements made on this wikipedia page, in particular the second Bernstein Inequality on: https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory) ...
0
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1answer
61 views

Short exact sequences from the Euler sequence.

I was reading an article in which the author said that the sequence $\require{AMScd}$ \begin{CD} 0 @>>>\Omega ^1_{\mathbb{P}^n} @>>> \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus n} ...
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0answers
30 views

Good problems to do while reading Hardy's book on divergent series?

I am reading Hardy's text on divergent series and to my great dissapointment it has no exercises. I wonder if anybody among you know of some suitable references with problems to read simultaneously ...
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1answer
20 views

Good book on combinatorics for beginners in statistical mechanics

Im studying stat mech and i want to have a better understanding on counting microstates. What book in combinatorics do you guys recommend for beginners like me? Thanks in advance
2
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1answer
68 views

Does anyone know a book on sketching surfaces?

Is anyone aware of books or sources dedicated to sketching surfaces? Sort of like Forst's An Elementary Treatise on Curve Tracing, but on surfaces; or a book that has a fair few chapters dedicated to ...
2
votes
2answers
892 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 ...
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1answer
40 views

Good introductory book in geometric probability

I recently came across the proof of the Buffon theorem and I was fascinated by geometric probability. Could someone indicate me a good introductory book? Maybe with many exercises?
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1answer
40 views

Does equivalence of algebraic categories imply bi-interpratibility of their theories?

By an algebraic theory $\mathcal{T}$ I mean any category with finite products such that the objects are given by all finite powers of some object $X$. Let $Alg\mathcal{T}$ be the concrete category of ...
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1answer
36 views

rigorous statistics book recommendations

I am learning statistical inference by myself, I have skim through a few books like Casella Hoggs and I find it omitted lots of details, for example, they didn't introduce the conditional expectation, ...
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1answer
64 views

Errata for Mathematics: Its Content, Methods and Meaning?

I'm new here, so I hope this is the right place to post this! I am currently reading through the Dover edition of the textbook Mathematics: Its Content, Methods and Meaning, by Aleksandrov, ...
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0answers
51 views

Finding a problem book in algebraic topology

I simply need book with problems solved with greatest explanation possible. I know about Hatcher and have a great lecturer, so I do not need theory. I need problems solved in detail.
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0answers
10 views

reference for convex function results

Here is a simple property of a concave function from $\mathbb{R}$ to $\mathbb{R}$, Given $x,x'\in \mathbb{R}$ with $x'>x$, if $\exists \kappa'\in (0,1)$ such that ...