All Questions
495 questions
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De-Nesting Absolute Value Function into Linear Combination of Absolute Value Functions
Context: In formulating problems for secondary school mathematics teachers (and students) about absolute value functions, which we define as functions $\mathbb{R} \rightarrow \mathbb{R}$ that send $x \...
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9
votes
1
answer
725
views
Popular mistakes in probability
$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
9
votes
3
answers
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Where can I find questions motivating important ideas in math?
I would like questions that demonstrate why a mathematical tool or technique is useful, and which can be used to introduce that idea. Ideally, this would be a compilation of problems organized by the ...
9
votes
1
answer
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Topology, the board game
Edit: I am reposting this question fom math.stackexchange.com; there may be some professors here who have more experience teaching topology.
This is a math education question that I've been thinking ...
9
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4
answers
2k
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Problem suggestions for polymath for undergraduates research
I'm inspired by the polymath project. It might be great for few undergraduates to work together on a research topic.
What are some research problems with the following properties(Experimental ...
9
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0
answers
887
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How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
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8
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12
answers
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How do I explain the number e to a ten year old? [closed]
Hardly a research level question, but interesting nonetheless, I hope. Pi is easy, but not e. Where could I start?
8
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2
answers
2k
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What is the best *general triangle*?
During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...
user112109
8
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4
answers
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Choice of adviser
Not sure how to tag this one so feel free to edit and add tags.
When I initially started graduate school my choice for an area of study was quite nebulous. I had only figured out enough to know that ...
8
votes
4
answers
788
views
Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$
Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...
- 43.2k
8
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4
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How to teach introductory statistic course to students with little math background?
Next semester I will teach an elementary statistic course for the first time (which I am actually quite excited about). A brief description can be found here. I am told to expect very little math ...
8
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2
answers
1k
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Mathematics of sustainable development and energy sobriety in the classroom
Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am ...
8
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6
answers
1k
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Seemingly emergent structures in mathematics
I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so...
I'm curious about examples of mathematical structure that seems to arise "...
8
votes
3
answers
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Applications of Group Theory Which Motivate Theoretical Questions?
I'm going to be a teaching assistant for an undergraduate class in abstract algebra next semester, for students who have not taken abstract algebra before. It will deal with group theory and linear ...
8
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4
answers
2k
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Differential Equation Examples for Calculus Students
I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for ...
8
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1
answer
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Who is this guy : Z.A. Melzak (wrote Companion to Concrete Mathematics) ? [closed]
Author : Z.A. Melzak
Book Title : Companion to Concrete Mathematics.
Publication : Dover renewed 2004 2 volumes in one. Copyright 1972/1976.
I found this book extremely nice.
To whet your appetite ...
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8
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4
answers
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Name for a basic principle of calculus?
$$
[\text{size of boundary}] \times [\text{rate of motion of boundary}] = [\text{rate of change of size of bounded region}]
$$
This differs from the fundamental theorem of calculus in that it does not ...
8
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3
answers
2k
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The harmonic (series) beetle: live illustrations of mathematical theorems
In my analysis class I use the following problem to illustrate the divergence
of the harmonic series (consider this as a hint for solving it).
Exercise.
A beetle creeps along a 1-meter infinitely ...
8
votes
2
answers
679
views
To what extent can algorithms in undergraduate linear algebra be made continuous/polynomial/etc.?
I feel like many of the algorithms that I learned — indeed, that I have taught — in undergraduate linear algebra classes depend sensitively on whether certain numbers are $0$. For example,...
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2
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2k
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Examples of analytic functions to motivate a first course in complex variables
[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...
8
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4
answers
1k
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Multivariable Calculus Lecture Ideas
I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...
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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 ...
8
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2
answers
693
views
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 ...
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8
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2
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447
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Big ideas and big ways of thinking in statistics?
I'm moving to a new university for the fall semester, and I'll be teaching a statistics class for the first time. I'm familiar enough with doing statistics (my dissertation in math ed was a mixed-...
8
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2
answers
2k
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Which universities teach true infinitesimal calculus? [closed]
My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
8
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0
answers
416
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Pedagogical question on Lie groups vs. matrix Lie groups
There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
- 28.1k
8
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0
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554
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Lower semicontinuity of naive fiber size
I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...
- 156k
7
votes
5
answers
3k
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What does a mathematician expect from mathematics education? [closed]
Consider that my question is not a personal and/or subjective question. Perhaps, you have hired a mathematics educator in your department and you are interested in finding a way to communicate with ...
7
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8
answers
4k
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Graduate School
How does one apply to graduate school when he has been working for sometime? I am interested in pursuing a PhD in math and making a career switch. Would my work experience benefit my application (I am ...
7
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8
answers
4k
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Mathematical Advice for Interested Highschool Students
This may not be a research level math question, but I believe it is still relevant to Math Overflow.
What general resources exist for students in highschool who are very interested in Mathematics?...
7
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3
answers
3k
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Specializing early
Topic: this is a mathematics education question (but applies to other sciences too).
Assumptions: my first assumption is that most mathematical concepts used in research are not intrinsically more ...
7
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5
answers
6k
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Advantages of the sequence definition of limits
I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...
7
votes
4
answers
841
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Easy to state applications of dimension theory in algebraic geometry
Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages.
Shafarevich gives two nice applications of the theory. 1) A proof ...
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7
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5
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Commutative algebra final project
I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following ...
7
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3
answers
3k
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The etale fundamental group of a field
Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...
- 7,338
7
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6
answers
1k
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Another chicken or egg: sequence or series
This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...
7
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3
answers
1k
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Higher dimensional Bezout via Hilbert polynomials: a reference
For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...
- 14.3k
7
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2
answers
1k
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Maximal Ellipsoid
John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
- 1,588
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4
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Help me find good math questions for my students [closed]
I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...
7
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4
answers
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What would be good to know before starting my undergraduate studies to become a good mathematician?
First of all, I'm sorry if this isn't the kind of question that should be made in MathOverflow. I read the FAQ and I didn't consider this (that) inappropriate. I couldn't resist! People here are ...
7
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3
answers
3k
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Problems reducing to a graph-theory algorithm
This is essentially a question in pedagogy -- the answers could be useful to teach (or rather, motivate) graph theory, and especially the algorithmic side of it.
I have been very impressed with this ...
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2
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How should you respond to a student who asks whether a very nice physical example constitutes a proof? [closed]
"Is this really a proof?" is the exact question e-mailed to me today from an undergraduate mathematics student whom I know as a highly competent student. The one sentence question was accompanied with ...
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7
votes
2
answers
740
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How quickly will billiard trajectories cluster?
Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic billiard trajectories
inside a convex polygon. Assume they all have the same speed.
Define an $\epsilon$-cluster as a ...
- 151k
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2
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830
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Virtual algebraic calculation within proofs
It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically. Since I have in mind doing ...
7
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2
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878
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What is the best way to read advanced textbooks in Pure Mathematics (PhD Level)? [closed]
This question was asked earlier on Mathstackexchange but was closed very very soon without any answer and then deleted by the system!
I am a PhD student (1st year) in a poor country with a corrupt ...
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7
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1
answer
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Self-taught undergrad math: ordering of topics?
After some initial research on math topics, it seems there are about 4 main streams as follows:
1) calculus -> analysis -> complex variables
2) linear algebra -> abstract algebra -> topology
3) ...
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7
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1
answer
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Theory of surfaces in $\mathbb{R}^3$ as level sets
Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
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1
answer
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Alternate algorithms for Chinese remainder theorem
I was teaching Discrete this semester and set the students loose on a system of linear congruences. One of them came up with this solution. Say $$ x \equiv 1 \textrm{ mod } 3 $$ $$ x \equiv 3 \textrm{ ...
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1
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Five cubes, Hadamard and Shklyarskiy
Here is my(=bad) translation of from the paper about Shklyarskiy by Golovina:
... in 1937/38 Dodik presented to school students a complete proof of Abel's theorem about equations of degree 5. He ...
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2
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Where can I find resources for creating a mathematics "bridge course"?
My department is in the very early stages of developing a "bridge course" or "introduction to proofs" course, motivated by our lower-level courses not currently doing a good job of preparing our ...
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