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7
votes
3answers
90 views

What does one study to increase understanding of the $P \stackrel{?}{=} NP$ problem?

If one were to learn more about the $P \stackrel{?}{=} NP$ problem, where would one start? I understand what the problem is—but not enough to be able to read anything technical about it. ...
23
votes
4answers
1k views

Why is “mathematical induction” called “mathematical”?

One of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from the ...
0
votes
0answers
39 views

Books on contemporary set theory [duplicate]

I have gone through Halmos' Naive Set Theory. Now, could you recommend me a good follow-up book for a rigorous treatment of contemporary set theory? (For example, I've been suggested to look at ...
2
votes
1answer
65 views

Popularization of mathematics: inspiring quotes (or interviews) on the fascination of mathematics

I would like to compile a big list of references to particularly inspiring interviews with important mathematicians or famous quotes on the fascination that mathematics holds for ...
2
votes
3answers
178 views

“Methods of Theoretical Physics for Mathematicians”

I read in the Princeton Companion to Mathematics that even pure mathematicians should know some theoretical physics. However, I see that there are many reference books of mathematical methods for ...
2
votes
0answers
69 views

A Book recommendation for double Integrals?

I have a really hard time learning Double Integrals, which I attempted to understand when I first saw the use of polar co-ordinates for Integrals. So my goal is to learn double Integrals and also ...
8
votes
1answer
164 views

Websites that promote co-operation and social networking among mathematicians

Are there some websites that could be defined as social networks for mathematicians and scientists? What I have in mind is something similar to Academia.edu or ResearchGate, but more specific towards ...
0
votes
2answers
49 views

What is <any number>^i?

I think I understand what imaginary numbers are, that $i$ is basically the name we give to $\sqrt{-1}$. Does $n^i$ have any sort of meaning? Is it used for anything? You can't really multiply $n$ by ...
1
vote
1answer
36 views

Neural networks and mathematical modelling

Could you point out some reference books [accessible to an undergrad math student] that deal with the mathematical modelling aspect of neural networks?
4
votes
1answer
125 views

Is Hoffman-Kunze a good book to read next?

I'm planning on self-studying linear algebra, and trying to decide on a book. I'm thinking of using Hoffman and Kunze. What sort of experience is required to handle Hoffman and Kunze? So far, I've ...
13
votes
1answer
215 views

Collections of undergraduate research projects

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics: calculus; analysis; abstract algebra; linear algebra; number theory; geometry; ...
0
votes
2answers
58 views

Connections of theory of computability and Turing machines to other areas of mathematics

The question is quite straightforward: Could you point out some reference papers that highlight (in a way that is fairly accessible) the connections between (1) theory of computability, algorithms, ...
2
votes
2answers
175 views

Little, unknown, English or French research journals with good mathematics

In this article by Gian-Carlo Rota, you can read: "I bought a copy of Frederick Riesz' Collected Papers as soon as the big thick heavy oversize volume was published. [...] It was clear that ...
51
votes
25answers
6k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
38
votes
15answers
5k views

Do we really need reals?

It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions. ...
5
votes
2answers
131 views

What are some mathematical problems which have been forgotten?

As mathematicians continue to study mathematics, often times they run into a problem which takes a considerable amount of effort to solve. For instance, trying to factor polynomials has lead to a ...
1
vote
0answers
27 views

Importance of metrization theorem?

I wonder if there is a case metrization theorems(such as Nagata-Smirnov, Bing, Urysohn) pave a way to do a theory. What would be a nice application of metrization theorems?
2
votes
1answer
62 views

Can an electronic computer (stochastic) simulation be used as a “formal” proof of a tedious mathematical problem?

It seems like math purely "on paper" vs. purely on a computer both have their advantages and disadvantages. However, when combined together, they seem to greatly increase the possibilities and help ...
7
votes
3answers
343 views

On progress in mathematics: some long-open problems and long-standing conjectures

I would like to ask a question here on Math Stack Exchange taking inspiration (and therefore combining) from two well-known threads on MathOverflow: (1) Not especially famous, long-open problems which ...
2
votes
1answer
108 views

Natural progression in a curriculum for self-study of analysis

Would you list what is a natural and effective progression to self-study topics in analysis in order to gain a broad knowledge of the enormous corpus of knowledge that modern analysis involves. As a ...
1
vote
1answer
41 views

Probability and Measure Theory by Ash

Has anyone used the textbook above? If so how does it compare with billingsley, Chung and similar such books in terms of rigor, coverage, and ease if use for self study?
2
votes
1answer
74 views

Learning concepts in mathematics

Apologies for the soft question, but I was wondering whether it is a good idea, in mathematics, to learn/study things simply for the sake of studying it. A very good example comes from category ...
1
vote
1answer
99 views

Philosophers who became mathematicians — how did they do it? And who were they?

(I hope this is not too personal. If you want to get to the point scroll down to the end, where my questions are.) I'm a philosopher who's been -- gradually -- coming around to mathematics. I have ...
4
votes
2answers
115 views

Solving min-max optimization problems in original ways (that is, avoiding the frenzy of differentiation)

As I see from the students I'm tutoring, once faced with a min-max problem, the average student is taken by the frenzy of differentiation. I would like to show that sometimes it is better to use ...
0
votes
1answer
153 views

I need help organising these books by topic

Okay, this should be a quick and easy question for those of you who've studied calculus. I have a list of books that I want to order by topic, the books are as follows: Michael Spivak - ...
6
votes
2answers
42 views

Question about loss of generality in proofs

My concern is with choosing specific conditions within a proof to arrive at a general result. As an example, I'll use the proof that $\mathbb{Q}$ is dense in $\mathbb{R}$. The proof I know goes as ...
0
votes
1answer
44 views

Combinatorial techniques, methods, and ideas in (“undergraduate”) real analysis

This question is dual to Probabilistic techniques, methods, and ideas in ("undergraduate") real analysis: I would like to collect some examples of combinatorial arguments to undergraduate ...
0
votes
0answers
60 views

Reference request: calculus of variations

I am searching for a good book to self-study calculus of variations. It should be fairly complete; build up gradually from the very basics; offer detailed explanations; have some emphasis on ...
3
votes
3answers
143 views

Examples of combinatorial/probabilistic proofs of theorems in linear algebra

Are there any examples of combinatorial/probabilistic proofs of theorems in linear algebra? Motivation: I see here, the inverse is true.
4
votes
1answer
87 views

Analytic solutions to a simple math trick

As proven here $3816547290$ is the only positive integer in which every digit is used; each digit is used only once; the first $n$ digits are divisible by $n$, for $n=1,...,10$. ...
15
votes
7answers
481 views

What are some mathematically productive ways to waste time? [closed]

What are some productive things that can be done (other than directly studying Mathematics) during leisure time that has a side effect to improve oneself at Mathematics? For example, reading ...
6
votes
4answers
156 views

Math newbie: what to read? [closed]

Quick question for you all: what should a high school senior who intends to major in CS and math read to become familiar with proofs, calc, algorithms, etc? I know that's incredibly broad, basically ...
2
votes
3answers
116 views

A mathematically mature introduction to Turing Machines and Computability [reference-request]

In the computer science course for mathematicians held at my university Turing Machines have been presented very briefly. So much so that I didn't quite get why they are relevant to mathematics. I did ...
11
votes
4answers
253 views
+100

Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I have enjoyed the subject and I would like to know "what's next". In other words, I would like to know ...
6
votes
4answers
179 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard F. Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
1
vote
0answers
36 views

Looking for a partial converse of Rolle's theorem

Let $f: [a,b] \to \mathbb R $ be a continuous function differentiable in $(a,b)$ such that $f(b)=0$ and for some $c \in (a,b) , f'(c)=0$ ; then under what additional conditions can we conclude that ...
2
votes
1answer
66 views

Some questions about prime divisors and number of primes

For an integer $n \ge 2$, let $\omega (n)$ denote the number of distinct prime divisors of $n$ and $\pi (n) $ be number of primes not exceeding $n$. Let $a_1, \ldots, a_k$ be integers greater than ...
1
vote
1answer
78 views

Book comparison, Linear Algebra

so next semester (Spring 2015) I'm taking a Linear Algebra class. I was wondering if anyone who's had this book "Linear Algebra and Its Applications, 4th Edition - by David C. Lay" can give me an ...
15
votes
4answers
1k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
0
votes
1answer
27 views

Restriction estimates

What is the defining property of what someone in the harmonic analysis community would call a "restriction estimate?" I see sobolev norms, Fourier transforms, and inequalities relating these. The ...
0
votes
0answers
32 views

On subgroups of the form $HZ(G)$ where $H$ is abelian subgroup of non-abelian group $G$ such that $H \rlap{\;\,/}\subseteq Z(G)$

Let $H$ be a an abelian subgroup of a non-abelian group $G$ such that $H \rlap{\;\,/}\subseteq Z(G) Z(G)$ ; then I can prove that $HZ(G)$ is an abelian subgroup such that $Z(G) \subset HZ(G) \subset ...
1
vote
2answers
50 views

Is there a good short phrase for a point where a function is continuous but not smooth?

Given a point $x_0$ where a function $f$ is $C^0$ but not $C^1$, how could one call this point intuitively? I am not looking for a technically precise term (like a point where $f'$ is ...
0
votes
1answer
68 views

math books of undergraduate/graduate level without formula? [closed]

Just wondering, is there a math book talks deep into the math ideas (maybe undergraduate or graduate level, so not the pre-algebra content), but comes with no or very few formula?
21
votes
20answers
2k views

What are some interesting sole exceptions or counterexamples? [duplicate]

Many theorems assert that a particular property holds for all objects in a class except those in a given list of exceptions. Examples of rules that admit precisely one exception include: All primes ...
2
votes
0answers
117 views

Can anyone explain this quote about how mathematicians think?

I found this quote by Stephen Wolfram on page 1177 of his book A New Kind of Science. Yet of the limited set of people exposed to higher mathematics, different ones often seem to think in ...
0
votes
0answers
58 views

What is some reason that there are no book bridge the gap of these three books

I am referring to the (beginner's text- Stochastic Calculus by Mircea Grigoriu and Introduction to Stochastic Calculus by klebaner.) and the advanced texts - stochastic differential equation by ...
2
votes
0answers
70 views

Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
3
votes
1answer
43 views

What is the discrete log used for?

Perusing Wikipedia, I stumbled on the discrete logarithm. I looks interesting that we'd be able have a function that could solve $b^k=g$ for integers $b,k,$ and $g$. However, Wikipedia says "No ...
1
vote
1answer
75 views

subtle/annoying fallacious proofs [duplicate]

I've been invited to a maths themed Xmas after party. I need to prepare a selection of interesting, and relatively simple fallacious proofs which other guests will try and find the flaw in. I'm trying ...
8
votes
2answers
149 views

How to select the right books?

As the saying goes, "Give a man a fish, feed him for a day. Teach a man how to fish, feed him for life." I've always had a problem with selecting appropriate books. It could be a problem that I'm a ...