All Questions
495 questions
1
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1
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Resources on blended teaching and flipped classroom in undergraduate mathematics education [closed]
I'd like to learn about the implementation of "blended teaching" in general and "flipped classroom" in particular for the teaching of undergraduate mathematics. Can anyone ...
- 141
26
votes
3
answers
3k
views
Why is the standard definition of a $(p, q)$-tensor so bizarre?
At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows.
Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{...
- 1,389
17
votes
4
answers
3k
views
Languages beyond enumerable
A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...
- 151k
36
votes
1
answer
3k
views
Hilbert's Hotel
Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943).
Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?
- 411
23
votes
12
answers
15k
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Textbook for undergraduate course in geometry
I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
71
votes
11
answers
9k
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How to introduce notions of flat, projective and free modules?
In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...
- 65.4k
35
votes
14
answers
4k
views
Where have you used computer programming in your career as an (applied/pure) mathematician?
For background: I'm working on a book to help mathematicians learn how to program. However, I need to see some examples from people in the field that have done different kinds of things than I have.
...
34
votes
6
answers
3k
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Does seeing beyond the course you teach matter? The case of linear algebra and matrices
This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...
27
votes
17
answers
9k
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Using slides in math classroom
I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the ...
27
votes
10
answers
4k
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What (fun) results in graph theory should undergraduates learn?
I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...
25
votes
11
answers
5k
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Learning through guided discovery
I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page: http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/
I've ...
23
votes
4
answers
5k
views
Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?
I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
- 1,437
33
votes
11
answers
13k
views
Lecture notes on representations of finite groups
Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck "...
45
votes
12
answers
20k
views
Teaching undergraduate students to write proofs
In my experience, there are roughly two approaches to teaching (North American) undergraduates to write proofs:
Students see proofs in lecture and in the textbooks, and proofs are explained when ...
18
votes
17
answers
6k
views
What is your favorite isomorphism? [closed]
The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...
52
votes
9
answers
26k
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Is Galois theory necessary (in a basic graduate algebra course)?
By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) ...
16
votes
5
answers
3k
views
Integrating powers without much calculus
I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) to ...
- 30.1k
32
votes
5
answers
7k
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The interrelationship problem of modern mathematics – How to deal with it in first year graduate courses?
I was reading recently online Peter May's complaints (I'm a fan, you can tell, I'm sure) about teaching the third quarter of the graduate algebra sequence at the University of Chicago. This course ...
1
vote
0
answers
190
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what belongs in a first university-level geometry course? [closed]
I know this is not really a research question, but I would like to ask it of research mathematicians, to see if there is a consensus. In a recent discussion on this topic, someone suggested that if ...
- 99
0
votes
0
answers
303
views
Is Baire's theorem stronger than needed for functional analysis?
Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...
- 109
15
votes
2
answers
5k
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What areas of algebra could be interesting to probability theorists?
I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me ...
- 16.9k
30
votes
6
answers
11k
views
Mathematics for machine learning
I would like to know what mathematics topics are the most important to learn before actually studying the theory on neural networks.
I ask that because I will start to learn about neural networks and ...
- 419
22
votes
16
answers
6k
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What are your experiences of handouts in mathematics lectures?
There are many different styles of lecturing, and many different aspects that are blended together to give a whole "lecturing style". That said, I'm particularly interested in hearing people's ...
35
votes
2
answers
2k
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Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 156k
51
votes
6
answers
5k
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What does it take to run a good learning seminar?
I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might ...
24
votes
11
answers
8k
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The role of the mean value theorem (MVT) in first-year calculus
Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
14
votes
4
answers
5k
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Which edition of Philosophiae Naturalis Principia Mathematica of Isaac Newton would you recommend to me?
I'm searching for a good edition of Philosophiae Naturalis Principia Mathematica of Isaac Newton in English. Which edition of the Principia can you suggest me? If it's possible, cheap and similar to ...
- 141
12
votes
12
answers
2k
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What are fun elementary subjects in probability?
I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....
17
votes
17
answers
3k
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Readings for an honors liberal art math course
Our university has an Honors section of our "liberal arts mathematics" course. Typically 10-20 students enroll each Fall, with most of them extremely bright, but lacking the interest and/or ...
16
votes
5
answers
2k
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"Classical" consequences of Bezout's theorem in dimensions $>2$
By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
- 14.3k
13
votes
11
answers
5k
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Math History books
I'm teaching a course over the summer (it's a sort of make-your-own course for non-majors) and I'm planning on organizing it as a math history course, hitting on major threads through about 1900, and ...
- 16k
67
votes
6
answers
4k
views
Good ways to engage in mathematics outreach?
Greetings all, I have often heard that it would be good if we as a community did more in the way of mathematics outreach: more to explain what it is we do to the community at large, more to expose ...
-4
votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 43.2k
47
votes
10
answers
10k
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Possibility of an Elementary Differential Geometry Course
I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.
I've found that in talking to professional physicists and ...
30
votes
6
answers
5k
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Euclid with Birkhoff
I'm looking for a short and elementary book which does Euclidean geometry with Birkhoff's axioms.
It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.
...
- 45k
17
votes
3
answers
2k
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Axioms for constructive Euclidean geometry
In the summer I will be teaching a course in (plane) Euclidean geometry to future high school teachers and I am looking for a suitable axiom system (unlike College (Euclidean) geometry textbook ...
- 2,050
12
votes
1
answer
521
views
Source of a quote by Ferdinand Rudio
I am looking for the source and context of this quote, found e.g. at St Andrews:
Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was ...
- 31.5k
28
votes
4
answers
3k
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The function $\sum_{0}^{\infty} x^n/n^n$
The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...
- 679
24
votes
7
answers
8k
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How do professional mathematicians learn new things? [closed]
How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues?
39
votes
4
answers
2k
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Important open exposition problems?
Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it ...
18
votes
12
answers
10k
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Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course
I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject.
I wonder whether there are some basic algebraic geometry ...
14
votes
2
answers
5k
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A certain mathematical competition in the UK
There is a foreword, written by professor Snow, to the book A mathematician's apology.
In the foreword, it is written some thing like the following:
"Hardy was opposed to a certain mathematical ...
- 366
24
votes
7
answers
4k
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Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof?
This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, by bounty rules, otherwise one would be accepted automatically. I can't ...
- 3,203
22
votes
13
answers
8k
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Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
8
votes
2
answers
693
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Seeking a combinatorial proof for a binomial identity
Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means:
$$\sum_{j=0}^m(-1)^j\binom{n+1}j2^{m-j}
=\sum_{j=0}^m(-1)^j\binom{n-m+j}j.$$
QUESTION. Can you provide a ...
- 43.2k
21
votes
10
answers
6k
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Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
9
votes
7
answers
7k
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Review papers in mathematics
Are there review papers, literature reviews in mathematics that describe the recent developments in various fields for a newcomer? Or is the prerequisite knowledge always provided in research ...
3
votes
2
answers
222
views
Which W W Sawyer titles exist in non-English language editions?
In this community question asking about books that teach the practice of mathematics, the author mentions the works of W W Sawyer.
Which of Sawyer's books were translated into languages other than ...
user159323
50
votes
4
answers
7k
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Motivation for concepts in Algebraic Geometry
I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.
**
Question
**
Are there any well-motivated introductions to ...
- 12.1k
25
votes
3
answers
7k
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Analysis from a categorical perspective
I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
- 5,759