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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11
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2answers
1k views

References on Linear Algebraic Groups/Lie Theory

I am currently doing a course on Lie groups, Lie Algebras and Representation theory based on Brian Hall's book of the same name. We should cover upto chapter 4/5 in this book by the end of the ...
2
votes
1answer
1k views

Application of various fields of mathematics to others

I have, only in recent times, taken seriously to mathematics.While reading a bit of real analysis(to be precise, flipping through the pages of Professor Abbott's Understanding Analysis) I asked myself ...
4
votes
1answer
866 views

Where can I find a comprehensive guides to linear algebra and calculus?

I'm a software engineer with a keen interest in all sorts of artificial intelligence and machine learning applications, and also quantum computing. Both areas require quite a bit of linear algebra ...
0
votes
0answers
189 views

Fixed finite index subgroups

Let $G$ be a group and $H$ a finite index subgroup, one could study the cardinality of the set of subgroups $K$ of $H$ of fixed finite index $[H:K]=n$. At first I thought this was always a finite set,...
3
votes
3answers
688 views

Need a book/series/online resource to learn college algebra to calculus

I keep running into real world problems where I need to use math that I don't understand, but I know I need a good base to start with. I would really like a good series that goes from college algebra ...
5
votes
1answer
214 views

Is there any good reads for Goodwillie calculus out there?

I have a basic understanding of combinatorial species and category theory and now I am curious about this functor calculus or Goodwillie calculus. Can anyone kindly recommend me a nice place to start. ...
9
votes
1answer
732 views

Sex, Crime and Functional Analysis?

Ever since a friend told me about this book titled Sex, Crime and Functional Analysis: Part I. Functional Analysis, I have been looking for a copy of this book, but in vain. SO does this book ...
4
votes
1answer
667 views

Open problems in Algebraic Topology, Geometric Topology and related fields

I've been reading about the Arf invariant and came across the following conjecture (see here): Each framed bordism class contains a manifold which admits a (possibly different) framing with zero Arf ...
3
votes
5answers
696 views

Does $i^i$ and $i^{1\over e}$ have more than one root in $[0, 2 \pi]$

How to find all roots if power contains imaginary or irrational power of complex number? How do I find all roots of the following complex numbers? $$(1 + i)^i, (1 + i)^e, (1 + i)^{ i\over e}$$ EDIT:: ...
4
votes
0answers
1k views

Proving that $\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong \operatorname{Pic}^0(X \times Y)$

Let $k$ be a field of arbitrary characteristic and let $X$ and $Y$ be projective varieties over $k$. I have come across the formula $$\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong \...
8
votes
2answers
957 views

Prerequisite reading for Weil's Basic Number Theory

I'm interested as to what would constitute prerequisite reading for André Weil's book Basic Number Theory. For simplicity (and generality), you can assume that the reader can read anything that ...
2
votes
1answer
170 views

Is the integral closure of $k[[t]]$ in a finite extension of $k((x))$ necessarily a free module?

In Milne Prop 2.29, it is said that the integral closure $B$ of a PID $A$ in a separable finite extension of its fraction field is a free $A$-module. On the other hand, I have read here that if the ...
3
votes
0answers
350 views

Standard parabolic Lie subalgebras and conjugacy

Let $\mathfrak g$ be a given semisimple Lie algebra with corresponding adjoint Lie group $G$. A parabolic subalgebra is any subalgebra containing a Borel subalgebra. We can pick a Borel $\mathfrak{b}...
6
votes
2answers
344 views

Reference for a derivative formula for matrices

I found the following identity: $$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$ on the matrix cookbook. It is equation 47 on page 8. Note that $X$ is an $n \times ...
10
votes
1answer
368 views

van der Waerden's original proof

I am looking for a book/site which has the English translation of the proof of van der Waerden's theorem as presented by van der Waerden himself. In other words is the translation of the paper: ...
1
vote
0answers
43 views

CAS for counting points of varieties over finite fields

I am looking for a computer algebra system, which is able to some of the following (in theory equivalent) things for a smooth projective variety defined over a finite field: -Count the number of ...
10
votes
7answers
12k views

Good First Course in real analysis book for self study

Does anybody know of a good book in real analysis for self study for a beginner? What about Analysis 1 by Terence Tao?
0
votes
1answer
912 views

Autocovariance of white noise convolved with function on L2

I want to compute the autocovariance of a Gausian zero-mean white noise process convolved with a function $f(t) \in L^2$. Could anyone show me how I'd do this or provide a reference? $L^2$ is the ...
4
votes
1answer
1k views

Where can I find lots of worked examples of proof by induction?

This may not be an appropriate question. If so I apologise. I struggle largely with proofs by induction. Can someone please point me toward some learning materials (text-books would be ideal) where ...
3
votes
3answers
92 views

How to expand a representation

If there is a finite group with a normal subgroup and a representation of this subgroup over a finite field. How can one expand this representation to a representation of the whole group? Are there ...
3
votes
0answers
80 views

Siegel's theorem

I want to learn the proof of the following theorem by Siegel. The statement of the theorem is taken from "Symmetric bilinear forms" by Milnor and Husemoller (pp. 44). They say that the proof is due to ...
1
vote
2answers
2k views

Optimality conditions and Directions in Simplex method

I am trying to understand the optimality conditions in Simplex -method, more in the chat here -- more precisely the terms such as "reduced cost" i.e. $\bar{c}_j=c_j-\bf{c}'_B \bf{B}^{-1} \bf{A}_j$ and ...
1
vote
1answer
87 views

Interactive Vizualizer of different Simplex -methods?

My book [1] around the pages 80-100 outlines the theories behind different simplex methods such as Naive-Simplex, Revised-SImplex, Full-tableau-Simplex, Dual Simplex, etc-simplex --. It is very dry ...
10
votes
7answers
2k views

Lecture notes for measure theoretic probability theory

I'm looking for good lecture notes (or concise books) that develop probability theory from a measure theoretic point of view. In particular, I'm looking for a text where the measure theoretic part is ...
3
votes
0answers
78 views

Question, need help finding the source.

I was hoping someone could tell me if they knew the source to this problem: Let S be a subset of {1, 2, 3, 4,..., 10, 11}. We say that S is LUCKY if no two elements of S differ by 4 or 7. The ...
11
votes
0answers
323 views

Measure-driven differential equations

Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
0
votes
1answer
353 views

Proof of Archimedes Lemma about the Center of Mass

I am looking for a proof of the following statement, known as Archimedes' Lemma: If an object is divided into two smaller objects, the center of mass of the compound object lies on the line segment ...
2
votes
3answers
2k views

how to show that power series is analytic inside the radius of convergence?

Let $f(z) = \sum a_n z^n$ be a power series with radius of convergence $R$. How do we show that $f$ is analytic in the circular region of radius $R$?
3
votes
0answers
338 views

Chebyshev Equioscillation Theorem in $L_{\infty}[a,b]$?

Let $a,b\in\mathbb{R}$, $a<b$. Consider \begin{align} C[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is continuous}\}\text{,} \\ L_{\infty}[a,b] & :=\{f\in\mathbb{R}^{[a,b]}:f\text{ is ...
1
vote
1answer
146 views

how do we know that integral is non-elementary? [duplicate]

Possible Duplicate: How can you prove that a function has no closed form integral? Is there a condition that states that the indefinite integration is non-elementary?
8
votes
2answers
2k views

Finding a paper by John von Neumann written in 1951

There's a 1951 article by John von Neumann, Various techniques used in connection with random digits, which I would really like to read. It is widely cited, but I can't seem to find an actual copy of ...
9
votes
6answers
11k views

Math Textbooks for High School

I'm a high school student who is trying to figure out a complete course of self study for each year of high school. Is there a way to self learn grades of math without devoting too much time? For ...
3
votes
5answers
1k views

Mathematical notation for computer science

Can anyone point me in the direction of good introductory material on the use of mathematical notation in the field of computer science? I often come across notation in research papers that I don't ...
4
votes
5answers
317 views

topology - analysis Book

I need some notion about topology(I'm very interested in boundary points, open sets) and few examples of solved exercises about limits of functions($f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$) using $\...
2
votes
2answers
291 views

Resources for IMO

I am seeking an online resource or any book where I can find the questions of International Mathematical Olympiad questions chapter wise eg Number Theory problems grouped together. Like this site ...
10
votes
2answers
279 views

Articles on ideas in the history of mathematics notation?

I'm teaching a course this term on the history of scripts (writing systems) and rather than talking interminably about Semitic and Chinese and their spawn, I'd like to give students a more varied diet,...
9
votes
4answers
5k views

Advanced Linear Algebra courses in graduate schools

After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course? And would an advanced linear algebra course be taught in graduate schools?
16
votes
10answers
2k views

Problem books in higher mathematics

Are there any problem book series on different topics with proper solutions? I have already found some in analysis (Problems in Mathematical Analysis 1-3) but not any in less popular topics.
9
votes
1answer
6k views

Pollard-Strassen Algorithm

I'm aware that the Pollard-Strassen algorithm can be used to find all prime factors of $n$ not exceeding $B$ in $O\big(n^{\epsilon} B^{1/2}\big)$ time. This is really useful because I need to find all ...
2
votes
1answer
2k views

about a good book - Vector Calculus[by Jerold E. Marsden, Anthony J. Tromba ]

I start reading Vector Calculus by Jerold E. Marsden, Anthony J. Tromba and I want to know if there is a book with the answers of the exercises. I like a lot this book, it seems to be made for a ...
2
votes
2answers
246 views

Reference: Finite Groups and Geometry

The examples $C_n, D_n, A_4, S_4, A_5$ are the first nice examples of groups to relate with 2-D and 3-D Euclidean geometry. These groups can be investigated by studying action on sphere as described ...
1
vote
1answer
131 views

Localizations of the Boolean Ring P(X)

Given a set $X$, we can construct the Boolean ring whose elements are the power set of $X$. The multiplication therein is intersection, and the addition is symmetric difference. I am interested in ...
15
votes
1answer
638 views

Hilbert's Original Proof of the Nullstellensatz

Does anyone have a link to Hilbert's Original Proof of the Nullstellensatz, or know a book where it's printed? I'd be interested to see what it was like. I only really know the Noether normalisation ...
4
votes
1answer
402 views

Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag

I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can't seem ...
6
votes
0answers
311 views

Resource: Group Theory

There is a website providing recent thesis in finite geometry. Is there any website with collection of recent thesis on (finite) group theory. I want to see the Ph.D. thesis of Raymond T. Shepherd, "...
4
votes
4answers
181 views

Why Does Finitely Generated Mean A Different Thing For Algebras?

I've always wondered why finitely generated modules are of form $$M=Ra_1+\dots+Ra_n$$ while finitely generated algebras have form $$R=k[a_1,\dots, a_n]$$ and finite algebras have form $$R=ka_1+...
4
votes
2answers
105 views

What do I need to know to simulate many particles, waves, or fluids?

I've never had a numerical analysis course so I don't know what I need to know. I'm just wondering what kind of books I should get to make me able to simulate these things. I'm wanting to simulate ...
1
vote
0answers
90 views

Where can I find a database with premade generalizations of mathematical structures?

Last question, I needed to use the general definition of a polynomial which is: $$a_nt^n+a_{n-1}t^{n-1}+\cdots+a_1t+a_0$$ But I needed to type it all, if I could find templates it would be easier/...
2
votes
0answers
44 views

Fibered Product of Subcategories

Is there a general construction or existence theorem for the fibered product of two subcategories of some ambient category? What sort of problems might one run into? Does this require a 2-categorical ...
10
votes
0answers
200 views

Reference suggestion: eigenvalues of tridiagonal matrices

I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals). I have seen authors use continued ...