This tag is for questions where the poster seeks references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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9
votes
6answers
2k views

Best books on Representation theory

What are some of the best books on Representation theory for a beginner? I would prefer a book which gives motivation behind definitions and theory.
1
vote
0answers
27 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
17
votes
3answers
725 views

Exceptional books on real world applications of graph theory.

What are some exceptional graph theory books geared explicitly towards real-world applications? I would be interested in both general books on the subject (essentially surveys of applied graph ...
3
votes
1answer
69 views

Textbooks on graph theory

I've read the textbook Groups and Their Graphs by Grossman, and I'm interested in learning more about graphs. I know about O. Ore's book in the same series (Graphs and Their Uses), but I'm interested ...
3
votes
1answer
40 views

Book Recommendations for Picard Big and Little Theorems

Does anybody have book recommendations for reading about Picard's Little and Big Theorems? Preferably, I am looking for a book that is intended for an undergraduate/first year graduate student who ...
2
votes
1answer
2k views

Probability of finding at least k consecutive heads in N coin tosses?

There are quite a few topics on this question already but I couldn't find a well-explained solution. Please point me towards some relevant literature or theory to analyze this problem. $K$ ...
2
votes
1answer
35 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
64
votes
22answers
5k views

The Best of Dover Books (a.k.a the best cheap mathematical texts)

Perhaps this is a repeat question -- let me know if it is -- but I am interested in knowing the best of Dover mathematics books. The reason is because Dover books are very cheap and most other books ...
0
votes
0answers
33 views

Koblitz - Are chapters III & IV independent of I & II

I am interested in learning about Modular forms and have heard many great things about Neal Koblitz's Introduction to Elliptic Curves and Modular Forms. However, Koblitz doesn't discuss modular forms ...
3
votes
1answer
42 views

Undistinguishable elements in posets

Given a finite partially ordered set $P = (V, <)$, I say that $x$ and $y$ in $V$ are indistinguishable in $P$ if for all $z \in V \backslash \{x, y\}$, I have $z < x$ iff $z < y$, and $x < ...
1
vote
1answer
38 views

References for mathematics enthusiasts?

I'm willing to give a mathematics Olympiad [syllabus] and I can't buy all the book(or even some of them) and am looking for some online resources like questions, articles, or more prefably some single ...
1
vote
1answer
43 views

What is the difference between reinforcement learning, trial and error, and fictitious play?

I have three question about three algorithms. I have a game with $n$ players. The action space of player $i$ is given by $\mathcal{A}_i=\{a_1, a_2, \cdots, a_m\}=\mathcal{A}$. The joint action space ...
1
vote
2answers
72 views

Typed version of Newton's Principia Mathematica

I need a typed pdf version of Newton's Principia. Is it available for free online? And I also need the proof of universal law of gravity and the elliptical orbits of planets(If there's no typed ...
1
vote
0answers
35 views

For starting the studies in Mathematical-Biology

I will take an undergraduate course in Mathematical Biologylearn themes like: population dynamics; the emergence of patterns of Philotaxia; Turing´s bifurcation; Genetics; Chaos; Neural networks; ...
5
votes
1answer
65 views

Limit everywhere, limit function is continuous, specific proof.

Suppose $f:[a,b] \to R$ is a function such that $\lim_{t\to x} f(t) = g(x)$ exists $\forall x \in [a,b]$. It can be shown that $g(x)$ is a continuous function. I seem to remember that there was a ...
2
votes
0answers
43 views

Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following. "Convert a partial differential equation of higher order into a canonical system of the first order" What does the above statement mean/imply? ...
1
vote
1answer
40 views

Where can I find introductory video lectures about calculus and analysis?

I am having calculus classes that are titled as Calculus for Mathematicians, for the rest of the students who are studying calculus, they use Stewart's book. In our classes, we're having something ...
4
votes
1answer
55 views

References for mathematical theory of summability of divergent series

Once in a while, I can't help it to ask very broad questions. I have read (a portion of) Hardy's Divergent Series. Back then, I think besides in mathematics, divergent series and the need to assign ...
0
votes
1answer
27 views

Wasserstein metric: conditions for the existence of minimizer and duality

Let $(X,d)$ be a metric space and let $\mathcal P(X)$ be the set of all Borel probability measures on $(X,d)$. The Wasserstein distance on $\mathcal P(X)$ is given by $$ W_d(\mu,\bar\mu):=\inf_{M\in ...
1
vote
1answer
22 views

References dealing with function spaces in pde like $C^k(\mathbb R^n)$, $C^\infty_c(\mathbb R^n), \ldots$?

What would be nice references for function spaces like $C^k(\mathbb R^n$), $C_0(\mathbb R^n)$, $C^\infty_c(\mathbb R^n), \ldots$ and most common function spaces which are offen employed in partial ...
0
votes
2answers
23 views

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients?

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients? Where can I read a proof? p.s. I don't even see the answer to the first question with a google ...
3
votes
2answers
85 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
1
vote
2answers
42 views

Reference about the Conley index thoery

I'm reading "Isolated invariant sets and the Morse index" by Charles Conley.But I'm lost in some of the concise description or definition.Could you recommend me some references or textbooks for the ...
0
votes
0answers
35 views

Which Trigonometry Book is Recommended? [duplicate]

I'm taking trigonometry for this upcoming fall, and I want to get a good head start like I did with statistics a while back. I was recommended Cynthia Young' s Trigonometry book and Loney's book. ...
4
votes
0answers
45 views

Adjunction between cocomplete categories

Let $C$ be a small category. Let $D,E$ be cocomplete categories. Let us denote by $\hom$ (resp. $\hom_c$) the category of (cocontinuous) functors. Then there is an equivalence of categories ...
5
votes
2answers
261 views

Curious gamma identity

I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc). The identity is as follows: $$\Gamma(t) = x^t ...
3
votes
2answers
302 views

List of functions not integrable in elementary terms

When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...
0
votes
0answers
31 views

Reference Request: Fubini's theorem for non-negative functions

I have never seen this (1st page) formulation of Fubini's theorem in the literature. Does anyone know where I can find it? In every calculus book (e.g. Apostol, Courant, etc.) I looked, the authors ...
1
vote
1answer
62 views

References for a notion of “restricted adjoint”

A construction that I've been finding all over the place in studying the category of NF (Quine's New Foundations) sets and functions is a situation like the following: there's a functor ...
12
votes
2answers
447 views

Pure mathematics in our society

Is there some book or essay which deals with the sociological and economical justification of doing and funding pure mathematics? I'm looking for a modern version of Hardy's A Mathematician's Apology, ...
7
votes
2answers
1k views

Prerequisites for Atiyah Macdonald

I am currently doing a one semester course on groups and rings where we have learned about (so far): Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphisms, ...
9
votes
1answer
128 views

Is there any curriculum based on recreational mathematics?

I'm a high school physics teacher. Next year, I'll be teaching mathematics for middle school students so I was wondering if there's a curriculum based on recreational mathematics which not only ...
9
votes
3answers
206 views

Areas of contemporary Mathematical Physics

I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc have had a significant impact on pure Mathematics especially geometry ...
4
votes
0answers
50 views

Does anyone have a good reference on calculating contour integrals around the unit circle (numerically or otherwise)?

I am looking for a reference that will help me calculate contour integrals around the unit circle or other curve. I have a particularly ugly function which isn't likely to have a nice closed form so I ...
1
vote
1answer
63 views

The relationship between each harmonic numbers

In Knuth's "Concrete Mathematics" in chapter about numbers below equality is given $$H_n = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + \frac{\epsilon_n}{120n^4} $$ where $0 < \epsilon_n < ...
12
votes
8answers
411 views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
-1
votes
0answers
36 views

Behavior of groups under extension

We know that an extension of a solvable group by a solvable group is solvable. Similarly we can find other properties of group extensions here Can someone provide a reference to these statements where ...
0
votes
0answers
102 views

Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
0
votes
0answers
29 views

Reference for understanding Frechet and Gateaux derivatives

In multivariable calculus, when we were discussing directional derivatives, we were told that the fact that the directional derivative equals the gradient times the direction vector $( \partial^{\vec ...
6
votes
1answer
96 views

best intuitive books/video lectures to read topology and functional analysis

What are the best intuitive books/video lectures to read topology and functional analysis ? I am aware of basic linear algebra, analysis and measure theory.
4
votes
6answers
229 views

Books for studying Mathematical Physics?

Currently I'm doing Advanced Classicial Mechanics courses.I'm finding it hard to understand due to the lack of knowledge in linear algebra, multi variable calculus and other chapters. Can anyone ...
3
votes
2answers
79 views

Best textbook for Geometric Measure Theory

I was wondering what is the best textbook for Geometric Measure Theory for self study. I am looking for one that isnt excessively detailed or long either as I found Rana's Introduction to measure ...
5
votes
2answers
117 views

How was the $3x+1$ problem checked up to $5 \times 2^{60}$?

The Wikipedia article for the Collatz conjecture states that: The conjecture has been checked by computer for all starting values up to $5 \times 2^{60} \approx 5.764 \times 10^{18}$. It gives ...
0
votes
1answer
36 views

Reference - formal characterization and analysis of Koch curve

I am studying the Koch curve but most resources I have seen do not describe the Koch curve formally and are similar to the Wikipedia page on the subject. For example, I have looked at books like ...
0
votes
1answer
19 views

Book on Lipschitz pointwise constant

Does anyone know of a book (or possibly an accessible paper) discussing Lipschitz pointwise constants and perhaps including some examples? Thank you
0
votes
3answers
156 views

Book on modular arithmetic

I am searching for some good book which section is devoted to modular arithmetic. I am self learner so I strongly prefer that book has exercises best with answers or solutions. I have CS background ...
4
votes
4answers
2k 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? ...
0
votes
1answer
64 views

What is the metric spaces needed to motivate concepts of general topology?

I intend to start learning some topology on my own. I wonder How much metric spaces I should know in order to motivate the concepts of topology? I know it's possible to learn topology without any ...
10
votes
1answer
340 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
5
votes
4answers
767 views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...