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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0
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1answer
76 views

Practise with Smooth Functions and Manifolds

I'm trying to get an intuition for smooth manifolds, and in particular the smoothness of transition functions. I haven't done that much calculus on $\mathbb{R}^n$ before, and would like to practice ...
1
vote
1answer
108 views

Sampling and probability generating functions - reference wanted

Suppose I have a huge (effectively infinite) population of widgets. The number of widgets that are broken is given by a random variable $X$, whose probability generating function is $p(z) = E(z^X)$. ...
86
votes
7answers
9k views

Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
3
votes
1answer
500 views

Longest Odd/Even Sequence in Composite Patterns

NOTE I have completely reworded this because I made a complete hash of it the first time, it got worse as I added to it. I apologize to anyone who might have been confused, and hope that this will be ...
1
vote
1answer
233 views

Short First Course in linear algebra text

I am currently reading a bit of real analysis from Dr Pugh's book and I intend to study abstract algebra from Dummit and Foote in a short time. However, I wish to explore linear algebra in the ...
3
votes
1answer
203 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...
2
votes
1answer
354 views

reference recommendation on mathematical logics

It is a bit of strange that I haven't be trained on such a topic. It wouldn't influence my daily work but I would like to study some materials on my own. So, dear fellows, you got any classic ref in ...
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2answers
798 views

Prerequisites for understanding the Hodge conjecture

The Hodge conjecture is a major open mathematical problem that states that on a complex manifold $X$ and its respective Hodge classes, defined as $Hdg^k(X)= H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X)$ that ...
0
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1answer
58 views

What is known about moduls $M = F^n$ over a ring $R$ where $F = R/I$ is a field

If $R$ is a ring and $I$ is an ideal of $R$, then $F = R/I$ is a homomorphic image of $R$, i.e. there is a homomorphism $f: R \rightarrow F$. If you let $M = F^n$, and define $(\cdot): (R,M) \...
4
votes
1answer
752 views

Conformal parametrization of an ellipse

I am looking to a formula for the conformal map from the unit disc in the interior of an ellipse centered in $0$ and with semiaxes $a,b>0$. I know that depends on elliptic function, but I didn't ...
0
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3answers
596 views

Algorithm to find all vertices exactly $k$ steps away in an undirected graph

This question may be better served at cs.SE, but I am not very familiar with CS lingo, so I'm hoping the maths community would be able to answer it as well... I have an undirected graph, and I am ...
1
vote
1answer
55 views

Need help to understand some definitions

I found following two definitions in a Ben Israel's book whose title is Generalized inverses: Theory and applications: For any $A, B \in \mathbb{C}^{m\times n}$, define $R (A, B) = \{Y = A X B \in ...
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4answers
6k views

A good introductory discrete mathematics book.

I am professor of mathematics and I am scheduled to teach a newly designated discrete math class for computer science majors. The prerequisite for the class is only Calc 1 and Id like a book that ...
5
votes
0answers
150 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
1
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2answers
74 views

products of logarithms

(When one says several things in a question, then several things may get answered and others neglected. Hence this posting overlaps with one of my earlier ones, but (I hope) this one will be short, ...
1
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0answers
108 views

Existence and Unicity Weak, Strong, Pathwise, In Law, etc… for SDE's

I feel always confused with weak, strong, pathwise unicity and or existence for Stochastic Differential Equations. It is mainly my own fault and I should do something about it. But it would be much ...
1
vote
2answers
84 views

How should I understand probability distribution defined as $P^X(A)=\frac{1}{3}\delta_0(A)+\frac{2}{3}P_2(A)$?

I get the following question from Zastawniak's Probability Through Problems: Assume that the distribution function of a random variable $X$ on a probability space $(\Omega,{\mathcal A},P)$ is ...
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3answers
918 views

Project in computer science and mathematics.

My background I'm a third year student. I study mathematics combined with computer science (with focus on modeling, simulations and visualization). In order to get my degree, I have to make a ...
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5answers
6k views

Simpler definition of manifold

I'm new to topology but I must understand how it works to progress in my research. First, can anybody point me to a document that introduces topology in a "gentle" way. What pre-requisites do I need ...
4
votes
1answer
584 views

Good textbooks for lattice and coding theory

I am looking for good textbooks for lattice and coding theory. Lattice and coding theory are very interesting on their own, but I have application of the theory to K3 surfaces & modular forms (and ...
1
vote
2answers
729 views

Ways to solve a 2D Laplace equation

I'm looking for a survey of methods to solve Laplace equation in two dimensions. Is there a book describing them with hints regarding their applicability for various cases? I mean analytical methods,...
3
votes
0answers
84 views

What is a good resource for functional derivatives and functional determinants?

What is a good resource for functional derivatives, functional determinants, etc.? What is the branch of mathematics dealing with those things? It is not in my functional analysis book. What is a ...
5
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0answers
181 views

Some rare binomial identities

Long ago , I once saw a nontrivial appealing binomial type of identity that I never saw again. It was something along the line of $\Sigma$$\binom{a(x)}{b(y)}$= where $a$ and $b$ where polynomials not ...
6
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2answers
646 views

Resources to learn the meaning of any math symbol

There is lots of symbols and may be operators like || in this expression $$ 3^k||n$$ that I would like to be able to quickly find the meaning of. I tried Wolfram|Alpha but I think it expects the ...
1
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1answer
55 views

Why is it that $\mathbb{C}[\{m_{ij}\}]^{G} \subseteq \mathbb{C}[\textrm{tr}(m),\ldots, \det(m)]$?

Let $G= GL(n,\mathbb{C})$ act on the set of all matrices $M_n$ by conjugation, i.e., for $g\in G$ and $m \in M_n$, $g\circ m = gmg^{-1}$. Then if $m=(m_{ij})$, then the $G$-invariant ring $\mathbb{...
1
vote
1answer
276 views

Good substitutes for Ross's book on Probability Models

I was wondering if there are any FREE good alternatives to Sheldon Ross's Probability Models which are more succinct? Are there any free online resources (websites/PDFs/course notes) which cover more ...
2
votes
3answers
141 views

Polynomials as concrete structures

Motivation The structuralist point of view on mathematical objects has two aspects: On the one side, a mathematical object is seen as a concrete structure of abstract dots, e.g. a graph. On the ...
1
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1answer
68 views

Generalization of groups of the form (R-{0}, regular multiplication)

Result: Fix $a \in \mathbb{R}$. Then $(\mathbb{R} \backslash \{a\}, *)$ is a group, where our group operation is defined by $x*y = (x-a)(y-a) + a$. One consequence of this is the standard fact that ...
2
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0answers
98 views

Where read recent developments in Math presented in a way that is easy to understand?

I posted a similar question a few months ago, but I don't think people understood my question. I often find articles such as this which discuss exciting developments in science. I have a great ...
15
votes
1answer
339 views

Do lay persons really believe that all of mathematics is known?

I once found a professor of medicine to be under the impression that everything in mathematics is already known. It's probably commonplace to hear that the masses labor under that misconception. In ...
27
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1answer
3k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
9
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0answers
125 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
3
votes
1answer
375 views

Cyclic Algebras over local fields- reference request

I am looking for an introductory text to the subject of Cyclic Algebras, and in particular ones defined over a local field. A cyclic Algebra, to the best of by understanding is defined as follows: ...
3
votes
0answers
145 views

Has the notion of a unique factorization category been defined and studied?

I am using a this notion in a paper that I am writing. The notion is that $(\mathcal{C},\otimes, I)$ (which we will just call $\mathcal{C}$) is a symmetric tensor category, with a unit $I$. Then a ...
30
votes
4answers
7k views

What are the required backgrounds of Robin Hartshorne's Algebraic Geometry book?

It seems that Robin Hartshorne's Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about the modern AG. But is it possible for someone who is out of the ...
1
vote
0answers
80 views

Good resources for learning Probability [duplicate]

Possible Duplicate: probability textbooks I recently started taking Probabilistic Graphical Models on coursera, and 2 weeks after starting the course I am beginning to believe that I am not ...
0
votes
1answer
58 views

Polynomial behavior on hyperbolic plane

More a reference request / more information. I was reading some websites about hyperbolic geometry and got to thinking about how would polynomials $(x^2-2)$ behave in such a geometry. So, I need ...
4
votes
0answers
115 views

Connection between modular forms and line bundles

I need a good reference about the connection between modular forms and line bundles. I found only Milne's note that treats briefly this argument. I've already checked, but without finding anything, ...
2
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0answers
44 views

Smallest dimensional irreps of semi-simple Lie algebras

I'm wondering if there is a reference that lists the first couple smallest dimensional irreducible representations of each semi-simple Lie algebra. I know these can be found using the Weyl dimension ...
6
votes
0answers
271 views

de Rham Cohomology of Non-Flat Bundle

Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$. If $E$ ...
4
votes
1answer
144 views

Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?

Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$. An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space. What are the necessary and ...
3
votes
1answer
391 views

Fluid mechanics resources for pure mathematicians?

I'm currently taking a course on fluid mechanics, and I'm finding it very difficult to become motivated and interested. I've always been more interested in the pure math side of my courses, and love ...
7
votes
2answers
2k views

Complex analysis book with a view toward Riemann surfaces?

I am considering complex analysis as my next area of study. There are already a few threads asking about complex analysis texts (see Complex Analysis Book and What is a good complex analysis textbook?...
9
votes
3answers
180 views

unramified extensions

Let $K$ be a number field with ring of integers $O_K$. It is well known that for almost all prime $p\in\mathbb{Z}$, the prime $p$ is unramified in $K$, that is, if $pO_K=\mathfrak{p}_1^{e_1}\ldots \...
3
votes
1answer
191 views

Complex tori as elliptic curves

I have a question about the proof of the following theorem: A complex torus is conformally equivalent (so isomorphic as Riemann surface) to a complex elliptic curve I used the book "N.Koblitz, ...
3
votes
2answers
124 views

What letter should I use to denote an ideal?

In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$. By itself, it is hardly surprising - after ...
2
votes
1answer
112 views

Reference for similarity tests for triangles

For triangles there are the standard similarity tests: AA, SAA, SAS, SSS and SSA (with angle opposite to the longer site). I am looking for a good reference of those elementary theorems with ...
1
vote
2answers
112 views

Generalization of monotonicity and condition

Consider a function $f: \mathbb{R} \to \mathbb{R}$. As usual, $f$ is non-increasing if $f(x) \geq f(y)$ for all $x < y$. We also have the condition $f'(x) \leq 0$ $\forall x$, provided that $f$ is ...
6
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5answers
370 views

Books with fun exercises

As university is a bit slow, we organize a group activity of doing math exercises (after learning the basics of the subject separately). Thus I am looking for a book with many exercises that are ...
5
votes
1answer
243 views

Methods of Multilinear Algebra in Representation Theory

I have been interested in representation theory lately in particular on that of Lie algebras. Now I have noticed that one way of building representations is to take tensor/exterior/symmetric powers. I ...