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2
votes
1answer
2k views

What do [] mean and what does it mean if it is used in an equation?

What do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x2+y(x)2]=50x+2y(dy/dx)=0$$
25
votes
7answers
2k views

Popular math books with depth

The most wonderful book I have ever read in my life was Fearless Symmetry by Avner Ash and Robert Gross, which is a good book that gives an intuition , and reasons behind the introducing fields, need ...
18
votes
6answers
953 views

Implication and Interpretation of Banach Tarski

As I understand, the Banach-Tarski paradox says a ball in 3-space may be decomposed into finitely many pieces and reassembled into two balls each of the same size as the original. Despite being called ...
2
votes
1answer
329 views

A very vague question about the cartesian product in mathematics.

Motivated by this question, I am wondering about Cartesian product analogs in various subfields of mathematics. The set theoretic Cartesian product creates an "output" set from a set of "input" sets, ...
12
votes
2answers
805 views

Is Mathematics graduation important for a Computer Scientist?

I know this might be a personal problem, but I often find some friends in the same problem as me so I think this might be helpful to them after all. I am going to graduate in Computer Science in ...
1
vote
1answer
73 views

Monty hosting a new show

I imagine the following setup. There is a contestant who has to pick one of three doors. How many prizes will be hidden is determined at random in the following way. Monty will toss a fair coin and ...
4
votes
6answers
588 views

Sleeping Mathematician (Sleeping Beauty)

I came across the following thought experiment, and I would like to understand whether the controversy around it is justified. Imagine an experiment in which a mathematician is put to sleep with some ...
57
votes
6answers
9k views

Chance of meeting in a bar

Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet? I am mainly ...
4
votes
1answer
198 views

KY Birds…which book is that from.

I am trying to find the name of this mathematical book, a smart student at my university gave a lecture about it a long time ago, and all I can remember of the talk was it was about mathematics it ...
4
votes
0answers
146 views

$\max_{y} \min_{x} f(x,y)$ as motif for exploring mathematics

It's been several years since my undergraduate math days, and I would like to spend a bit of time refreshing and then tackling a few things I never completely mastered. Rather than proceeding topic ...
2
votes
3answers
93 views

z-interval and sample size what is a normal sample size

Can a Z-interval be used when the sample size is between 15-30? does the variable play a role? I'm not too sure if it makes a difference. I know it can be used if the population is a normal or large ...
2
votes
2answers
164 views

A theorem about inductive inference

In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read: 'Inductive inference is well known to be a hazardous ...
6
votes
1answer
393 views

Ways to visualize the real numbers?

I was just wondering if there are any diagrams for visualizing subsets of the real numbers, or totally 'radically' different ways of looking at them as a real line? The model of the line relies on ...
2
votes
2answers
2k views

Book Reference for Calculus and Linear Algebra :: Engineer

I'm almost through with my (Mech.) engineering and am trying to touch some advanced concepts in Computational Science which requires me to study Calculus and Linear Algebra in a more theoretical ...
63
votes
6answers
2k views

Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
7
votes
0answers
821 views

How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?

I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
30
votes
3answers
1k views

What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
-1
votes
1answer
99 views

What would be the consequence of restricting multiplication by Zero to only Finite Cardinals? [closed]

What would be the consequence of restricting multiplication by Zero to only Finite Cardinals? Would this lead to contradictions? How could it be achieved?
8
votes
0answers
304 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
1
vote
1answer
192 views

What is an honest basis?

In a comment to this question, the commentator stated that "the monomials form an honest basis for your vector space". To be honest, I never heard of that. Is this something elementary?
-1
votes
1answer
2k views

What properties of the absolute value function should one remember? [closed]

When one begins to study real analysis, the absolute value function quickly enters and a large number of exercises involve manipulations with it. What are the basic properties of absolute value that ...
30
votes
19answers
3k views

Mathematics understood through poems? [closed]

Along with Diophantus mathematics has been represented in form of poems often times. Bhaskara II composes in Lilavati: Whilst making love a necklace broke. A row of pearls mislaid. One sixth fell to ...
37
votes
2answers
829 views

How do I find partners for study?

I want to find partners to study some post-graduate topics together online (because I'm pretty much out of steam as far as self-study goes, and I have problems finding a decent grad school). Are there ...
17
votes
4answers
858 views

Useful fibrations

What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to ...
33
votes
28answers
4k views

Classical texts that should not be missing from any shelf [closed]

It seems to me as if many modern texts are rather streamlined. They are designed not to expect too much from the reader but they often miss the depth of respective classical literature. The purpose ...
22
votes
1answer
966 views

Expository articles on Analysis and Probability theory

When I come across a notion from algebra or number theory which I don't know I usually check Keith Conrad's page to see if he has written something about it. Key features of his articles are a very ...
2
votes
1answer
98 views

Is there a use for this technique?

I remember reading once about the following algorithm: Consider a lattice grid and $N$ houses situated at grid points, in which live the town elders. They want to choose a lattice point location ...
10
votes
4answers
1k views

Why are nets not used more in the teaching of point-set topology?

I just finished working through a proof of Tychonoff's Theorem that uses nets (specifically, as a corollary of the fact that a net in a product space converges iff the projected nets in the components ...
8
votes
0answers
351 views

The difference between 10 and 9.99999 … (recurring) [duplicate]

Possible Duplicate: Does .99999… = 1? At supper today my daughter was discussing her maths (she's 13) - she had been studying putting decimal numbers into what she called standard ...
13
votes
2answers
966 views

What is the smallest constant that has explicitly appeared in a published paper? [closed]

You can read a lot about what large number in mathematics are, like Skewes' number, Moser's number or even Graham's number. So just for the sake of non-discrimination, I ask, what is the smallest ...
4
votes
1answer
2k views

Pigeonhole: Practical Applications in Computer Science

Most of the problems I've seen involving the pigeonhole principle have so far seemed fairly artificial. As I'm studying CompSci I'm interested what kind of practical, real world problems in CompSci ...
3
votes
1answer
291 views

Why is the Continuum Hypothesis not the Continuum Axiom?

If CH can be neither proved nor disproved, and to assume it true or false can yield different results, isn't it an axiom?
5
votes
4answers
513 views

From continuity to differentiability and analyticity- what's next?

Continuity is an intuitive concept. I will not dwell on the precise definitions of continuity and the rest here. Note that differentiability is a more restrictive condition than continuity, while ...
2
votes
2answers
11k views

Identifying Independent And Dependent Variables In Differential Equations

I've been given a second order differential equation $$x^2y'' + xy' + (x^2 - v^2)v = 0$$(where $v$ is a parameter)* and asked to identify the dependent and independent variables. Question is How do i ...
8
votes
2answers
1k views

Mathematical places to visit [closed]

There are certain buildings and places on this planet where mathematicians can find delight because of the history, the art, the architecture, and for other reasons. For example, the Alhambra with ...
14
votes
6answers
2k views

What is a field?

I've always wondered about what a field is meant to represent. For example, group automorphisns naturally represent symmetry in many areas. I'm not looking for a solid answer, just an idea.
3
votes
3answers
1k views

Ring theory exercises at the graduate level

Do you know any book or an online source that contains exercises on ring theory? I've solved some exercises of Lang's Algebra and Dummit & Foote's Abstract Algebra but there is a huge gap between ...
26
votes
1answer
891 views

Useful mathematical fora

I want to ask about various mathematical fora and discussion boards available online. I think it might be useful to have such a list here at Math.SE. If I may suggest, it could be useful to keep one ...
28
votes
7answers
17k views

Is there any difference between mapping and function?

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to ...
10
votes
3answers
832 views

The pronuncation of “Tychonoff” and “Alaoglu”

I am not quite sure this is the place to ask this sort of question, but I am gonna give a talk on Banach algebra in which I will use theorems named after these two mathematicians whose names I can not ...
6
votes
5answers
2k views

Side-stepping contradiction in the proof of ; ab = 0 then a or b is 0.

Suppose we need to show a field has no zero divisors - that is prove the title - then we head off exactly like the one common argument in the reals (unsurprisingly as they themselves are a field). ...
1
vote
2answers
240 views

If we don't know the actual value of PI then how can we use it in formulations or rely on it for real world calculations? [duplicate]

Possible Duplicate: Do We Need the Digits of $\pi$? Are we at best, estimating things when we use formulas involving pi to describe the area, etc. of things?
2
votes
0answers
114 views

Lectures of many valued logic

I am looking for a good introduction to this topic... something with lots of examples and models would be nice. I am specially interested by the case where the truth values are open sets in a ...
11
votes
3answers
604 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
41
votes
7answers
8k views

What does it take to get a job at a top 50 math program in the U.S.? [closed]

I'm a senior undergrad right at a small liberal arts college right now who is applying to math PhD programs in the U.S. I would like to eventually become a professor at a relatively good university ...
101
votes
4answers
16k views

Books that every student “needs” to go through

I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on ...
11
votes
1answer
2k views

Early specialization and career development [closed]

I'm in the last year of my undergraduate studies. I recently decided to become a mathematician and I will apply for a two year master degree as a preparation for a PhD. Looking to various master ...
121
votes
5answers
9k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
1
vote
1answer
392 views

Infinite induction “valid”

As you may know, induction works only when we have a statement involving natural numbers. For instance, For every $n$, the intersection of $n$ open sets is open. Now, the corresponding statement for ...
13
votes
6answers
652 views

Read old articles instead books.

I'd like to know if there is a site, or maybe a collection of books, where I can read old articles in mathematics in order to study topics directly from the source, instead reading books in the field. ...