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13
votes
2answers
971 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 ...
13
votes
9answers
879 views

Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
12
votes
3answers
735 views

How to google search mathematical notions and expressions?

It is usually not difficult to google search mathematical notions; for example, one can search (with quotation marks) the term "brunnian braid" and find the definition and other related materials. ...
12
votes
0answers
218 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
11
votes
2answers
740 views

What's the difference between “duality” and “symmetry” in mathematics?

Motivated by the answer to this question--"What kind of “symmetry” is the symmetric group about?", I read the article about dual graph. It is said in this article that "the term 'dual' is used because ...
11
votes
2answers
393 views

Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$

I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work ...
11
votes
1answer
40k views

Why is a full circle 360° degrees? [duplicate]

What's the reason we agreed to setting the number of degrees of a full circle to 360? Does that make any more sense than 100, 1000 or any other number? Is there any logic involved in that particular ...
11
votes
6answers
823 views

Suggestions for topics in a public talk about art and mathematics [closed]

I've been giving a public talk about Art and Mathematics for a few years now as part of my University's outreach program. Audience members are usually well-educated but may not have much knowledge of ...
10
votes
4answers
477 views

Application of computers in higher mathematics

Currently the main application of computers in mathematics seems to be to compute things, i.e. to solve equations, evaluate integrals, etc. It is at all possible to delegate the thinking of a ...
10
votes
2answers
906 views

Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) being used in computer science? Any research I have done on this topic leads me to some sort of applied math concept. I know that there ...
9
votes
4answers
504 views

What is the difference between analytic combinatorics and the theory of combinatorial species?

Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't ...
9
votes
3answers
1k views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
9
votes
3answers
2k views

The most common theorems taught in Abstract Algebra

I am self learning abstract algebra. I want to know which theorems are a must to understand. Now these are limits I have to deal with (please consider when answering): I have limited ...
8
votes
1answer
739 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
7
votes
1answer
2k views

Journals that publish papers quickly [closed]

I have written two papers in Mathematics and want to get them published. Can you suggest some journals that publish quickly? Besides, how can I know if a journal is well-regarded or not? I know there'...
7
votes
2answers
781 views

What are Free Objects?

I've read the wikipedia article, but couldn't grasp the concept. Is there an informal definition? Are there examples of uses of free objects in calculus? Are free objects somehow connected to ...
7
votes
1answer
345 views

Recommendations for an “illuminating” (explained in the post) group theory/abstract algebra resource?

I recently asked a question regarding why homomorphisms and isomorphisms are important. The best answer to that question was actually a comment, which referred me to Brian M. Scott's answer here: http:...
7
votes
4answers
464 views

What sequence should I study these topics in? [closed]

I will shortly list a series of topics that amount to what is essentially the first two years of an undergraduate degree. I'd like to know what is considered best order in which to study these ...
6
votes
2answers
228 views

Are there any theories being developed which study structures with many operations and many distributive laws?

"Algebraic structure" will mean a set with some n-ary operations defined on it. This does not include vector spaces for example. During my study of algebra I have encountered mostly algebraic ...
6
votes
1answer
470 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
5
votes
2answers
183 views

The two distinct cultures in mathematics

I once read an article about two distinct culture within mathematics, "analysts" and "algebraists" when I was in high school. I am not still a graduate student, but I really want to hear what it feels ...
-1
votes
2answers
654 views

Is it to the students' advantage to learn the language of infinitesimals? [closed]

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...
22
votes
5answers
1k views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?
20
votes
3answers
1k views

What are the applications of finite calculus

I'm reading through Concrete Mathematics [Graham, Knuth, Patashnik; 2nd edition], and in the section regarding Summation, they have a sub-section entitled "Finite and Infinite Calculus". In this ...
16
votes
3answers
7k views

Importance of Linear Algebra

In one of his online lectures Benedict Gross comments that one can never have too much Linear Algebra. Also, looking around it seems like I can find comments to the effect that Linear Algebra has ...
16
votes
4answers
604 views

Banach spaces over fields other than $\mathbb{C}$?

Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ...
14
votes
3answers
1k views

Transcendental number

While reading on Wikipedia about transcendental numbers, i asked myself: Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e, \frac{\pi}{e}$ etc. are transcendental numbers? ...
12
votes
9answers
1k views

Strangest Notation? [closed]

While this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?
12
votes
7answers
7k views

Why don't we use base 6 or 11?

Another question on this site asks why we have chosen our number system to be decimal base 10. There are others asking basically the same thing as well. I'm not really satisfied with any of the ...
12
votes
4answers
5k views

Importance of eigenvalues

I know how to find eigenvalues and eigenvector .But I dont know what to do with that. What is there use? Can anyone explain me that?
11
votes
2answers
978 views

Abstract algebra book with real life applications

Is there an abstract algebra book that emphasizes the applications to "real world" problems? Update: By real world, I mean mostly related to physics or other sciences. But references to coding theory ...
11
votes
4answers
3k views

Angle brackets for tuples

I've recently noticed that use of angle brackets for writing tuples, e.g. $\langle x, y \rangle$ instead of the usual round brackets in a few books I've been reading — Lawvere's Sets for Mathematics, ...
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 ...
10
votes
7answers
775 views

Advice on self study of category theory

I'm very intrested in start to study category theory with aim to use this in algebraic geometry. I already took courses (besides the basics) on commutative algebra and general topology with a soft ...
10
votes
4answers
253 views

Is there a “natural” / “categorical” definition of the “parity” of a permutation?

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...
9
votes
3answers
438 views

Need Suggestions for beginner who is in transition period from computational calculus to rigorous proofy Analysis

I have completed basic calculus 1,2,3 courses, Linear Algebra, etc. I have not, however, got into rigorous Analysis yet, which I am planning to do now. I have three books in mind. They are : Terence ...
9
votes
4answers
1k views

What is linearity? [duplicate]

Once someone asked me the question "What is linearity?" in a proficiency exam. I went hot and cold all over. Although, I heard and even used the term linearity many many times, I had not really ...
9
votes
4answers
1k views

Publishing elementary proofs of theorems

I'm an undergraduate student and I believe I found another proof of Heron's formula. I have a bunch of questions: I would like to publish this "proof" (I haven't found mistakes yet) in some magazine. ...
9
votes
2answers
2k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
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?
8
votes
4answers
704 views

What are some elementary results (number theory) using theorems that went long-unproven?

Firstly, I do not mind if there are examples from fields other than number theory! This was just the first field, and where I think the richest examples, may come from. Now to elaborate on the title, ...
8
votes
3answers
9k views

What on earth is the difference between Calculus and Analysis [duplicate]

I noticed that there isn't a word for Calculus in my native language, Dutch. So I just went to the English wikipedia entry on Calculus, and tried searching for the Dutch article, and as I suspected, ...
8
votes
2answers
251 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
7
votes
1answer
2k views

Symmetric vs. Positive semidefinite

This may be a trivial or irrelevant question. I'm very sorry if so. Why do mathematicians use only symmetric matrices when they want positive semi/definite matrices? I mean haven't seen using non-...
7
votes
1answer
145 views

Could there be an “$n$-th root” of the category $\mathsf{Set}$?

Here is a thought experiment: Suppose we did not know what sets and functions are. The general idea of a topos is, that it somehow serves as a foundation for mathematics. So let there be an alternate ...
6
votes
1answer
1k views

Explain Zermelo–Fraenkel set theory in layman terms

What does Zermelo–Fraenkel set theory mean? According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if ...
6
votes
6answers
3k views

Best Intermediate/Advanced Computer Science book

I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's "Introduction to the Theory of Computation"). I know ...
6
votes
2answers
638 views

Organization of the Learning Process

Sorry for off topic. I'll delete this topic immediately when community decides it's useless, however if anyone finds it's interesting, share your opinion with us. I just want to know your opinion ...
6
votes
4answers
2k views

What is a great book to read about sequences, sums and products?

I want to learn everything about sequences, sums and products from A - Z. Is there one book that stands out from the rest quality wise? I want to start my research off right!
6
votes
2answers
712 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

I'm searching for some material (books or lecture notes) that extensively uses a geometric approach to explain the meaning of the concepts realted regarding to vector spaces, matrices, and linear ...