All Questions
495 questions
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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
32
votes
20
answers
6k
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What are your favorite puzzles/toys for introducing new mathematical concepts to students?
We all know that the Rubik's Cube provides a nice concrete introduction to group theory. I'm wondering what other similar gadgets are out there that you've found useful for introducing new math to ...
- 459
168
votes
37
answers
207k
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Too old for advanced mathematics? [closed]
Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, ...
- 12.4k
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 ...
CommunityBot
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40
votes
21
answers
16k
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Journals for undergraduates
Are there math journals that are aimed for undergraduates? I don't mean here journals where students can publish their papers, but journals that publish introductory articles that an undergraduate can ...
- 2,437
16
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5
answers
3k
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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 ...
- 12.9k
16
votes
3
answers
2k
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Solving a modified birthday problem at a glance
Modified Birthday Problem: a bunch of people line up, and the winner is the first person who shares their birthday with someone lined up ahead of them. What position in the line is optimal?
Three (...
- 4,713
195
votes
18
answers
17k
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Great graduate courses that went online recently
In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...
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2
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2k
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Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?
Motivation: let $A\in\mathbf{R}^{n\times n}$ be symmetric. Then by the method of Lagrange multipliers, a maximum of $x\mapsto x^tAx$ on the compact unit sphere $\mathbf{S}^{n-1}$ must be an ...
- 1,566
25
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19
answers
20k
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Math books for advanced high school students
I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...
- 1,181
87
votes
33
answers
24k
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Parodies of abstruse mathematical writing
Perhaps under the influence of a recent question
on perverse sheaves,
in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53),
I recalled a long-ago parody of abstruse mathematical language
...
- 2,454
63
votes
6
answers
12k
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Why isn't integral defined as the area under the graph of function?
In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
17
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4
answers
3k
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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.
...
- 24.8k
17
votes
5
answers
3k
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Teaching prime number theorem in a complex analysis class for physicists
This is a question about pedagogy.
I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis ...
- 3,699
0
votes
0
answers
148
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About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
48
votes
8
answers
5k
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Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (...
4
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1
answer
1k
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Chalkboard eraser [closed]
I just started my first year of university and because I'm visually impared I have trouble seeing what's written on the chalkboard.
I've partially solved this problem by purchasing chalk from hagoromo ...
- 188k
20
votes
2
answers
2k
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Bitcoin Research
I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...
22
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1
answer
3k
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What is so special about Chern's way of teaching?
First of all sorry for this non-research post.
I was watching Jeffrey Blitz Lucky documentary movie and it was interesting to me that a winner of Lottery was a math Ph.D. from Berkeley.
In the movie ...
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34
votes
5
answers
7k
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How should you explain parallel transport to undergraduates?
The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection.
This is in the vein of many other questions on mathoverflow:
What is ...
- 31
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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 ...
- 435
5
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0
answers
186
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Examples of partial adjoints
Recall that a functor $$R: D \to C$$ is said to have a partial left adjoint $L$ defined at an object $X \in C$ if the functor
$$D \to Sets, Y \mapsto Hom_C(X, R(Y))$$
is corepresentable by some object ...
- 2,040
30
votes
15
answers
6k
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Lunch seminars for PhD students
The problem that I would like to ask about is metamathematical, but I hope the question is appropriate.
I would like to know if there exist mathematical departments that run a regular seminar for all ...
- 2,284
9
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3
answers
1k
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Books on the relationship between the Socratic method and mathematics?
Apart from books on heuristics by George Polya.
When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real ...
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59
votes
5
answers
25k
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Are there any "related rates" calculus problems that don't feel contrived?
I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the ...
- 50.6k
11
votes
4
answers
2k
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Why do mathematicians prefer one definition over the other when they both define the same concept?
Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...
- 4,786
8
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1
answer
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What topics should be included in a calculus-for-the-liberal arts course?
I have in mind a course taken by liberal-arts students who will probably never take another math course. I would like such a course to convey some of the way mathematical thinking is done (i.e. not a ...
- 19.6k
6
votes
1
answer
4k
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Examples of separable ordinary differential equations in economics
I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...
- 91.8k
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
74
votes
21
answers
25k
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How should one present curl and divergence in an undergraduate multivariable calculus class?
I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should present curl and ...
CommunityBot
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333
votes
34
answers
96k
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Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
- 2,754
4
votes
4
answers
2k
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When did you "meet Polya"? [closed]
I guess most of us didn't meet Polya in person (this is the answer to the title)! Perhaps, it is much easier to guess that most of us have met one of his writings (or alike) on problem solving, and ...
- 91.8k
114
votes
34
answers
86k
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Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...
- 4,459
26
votes
18
answers
34k
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Undergraduate differential geometry texts
Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one?
(I know a ...
- 63.9k
5
votes
1
answer
208
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Seven Bridges of Königsberg for hypergraphs
I am teaching a course involving hypergraphs. I would like to have a physical analogy/motivating problem for hypergraphs similarly to how the Seven Bridges of Königsberg motivate graphs. Can you help ...
- 1,444
3
votes
0
answers
873
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Hard problems solving tricks
This question is motivated by this one that I posted on math.stackexchange.
When I fail to solve a hard math problem (like the ones I presented in the linked post), I read a solution and I noticed ...
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44
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10
answers
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What kid-friendly math riddles are too often spoiled for mathematicians?
Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples:
What is $1+2+\cdots+100$?
Is it possible to tile a mutilated chess board with dominoes?...
- 1,887
5
votes
3
answers
799
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Euclidean function of Euclidean domain defined at 0
In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as ...
- 63.9k
-8
votes
1
answer
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Why is it impossible to find work of John Tate online? [closed]
The work of John Tate belongs to mankind. Why is not online in pdf´s? Who is dirty enough to earn money on HIS work?
- 91.8k
-4
votes
2
answers
228
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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 $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 85.6k
55
votes
18
answers
9k
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How can an extremely mathematically talented young person be helped to fulfill his/her potential?
Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented $11$ years old student and I don't know how I can help him. For ...
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150
votes
31
answers
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What are the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
- 12.9k
45
votes
10
answers
4k
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effective teaching
Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, ...
- 4,713
26
votes
3
answers
3k
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Why is the standard definition of a $(p, q)$-tensor so bizarre?
At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows.
Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{...
- 2,369
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votes
14
answers
4k
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Math talk for all ages
I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
CommunityBot
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votes
1
answer
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Are there search algorithms that are competitive against (gradient based) optimization routines for continuous problems?
Suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function for which we want to minimize. We may arbitrarily impose good conditions for $f$, such as Lipschitzness, smoothness, convexity, ...
CommunityBot
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48
votes
12
answers
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How to explain to an engineer what algebraic geometry is?
This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most ...
1
vote
0
answers
167
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A taxonomy of proof methods [closed]
I am looking for a taxonomy of proof methods in mathematics.
For basic proof methods I would think of proof by contradiction, mathematical induction, structural induction (yes I am a computer ...
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0
answers
322
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Online courses for mathematics [closed]
I'm sorry if I'm posting this in the wrong forum. My background is in biology and medicine. I am looking to re-learn undergraduate-level mathematics, in particular discrete mathematics, calculus, and ...
- 3,065
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 ...
- 4,195