All Questions
262 questions
5
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1
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393
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Not quite adjoint functors
What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...
- 47.3k
5
votes
3
answers
647
views
Looking for ideas concerning the teaching of lower-division differential equation courses...
I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking ...
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 1,174
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 ...
- 51
5
votes
2
answers
2k
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Any suggestions for a course in Mathematical Logic?
I am teaching a topics course for Mathematics majors (at Temple), and am considering Logic as the topic. I was wondering if people (a) have suggestions for an appropriate text and (b) how much might ...
5
votes
1
answer
521
views
How to find eigenvalues following Axler?
Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix ...
- 16.5k
5
votes
1
answer
1k
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Is Diagonalization worth to be taught? [closed]
When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
5
votes
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
5
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0
answers
2k
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A course on modern algebraic geometry from "The Stacks Project"
I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't.
For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...
- 59
4
votes
4
answers
971
views
Understanding reasons for best constants in inequalities
Why, in functional analysis, is so important to calculate best constant in an embedding inequality?
Cross-posted from "https://math.stackexchange.com/questions/727690/understanding-reasons-for-best-...
- 403
4
votes
4
answers
4k
views
Variation on the Sobolev space $H^1_0$
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\},
$$
and let $C^1_c(\Omega)$ be the space of ...
- 3,322
4
votes
1
answer
1k
views
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 ...
4
votes
3
answers
507
views
Defining negation
I'm currently coauthoring a book intended to teach first-year students basic proof techniques. One of the chapters, written by my coauthor, is about basic logic. In that chapter the negation of a ...
- 18.7k
4
votes
2
answers
869
views
Terminology question on covering spaces
I'm teaching an elementary class about fundamental groups and covering spaces. It was very useful to use "fool's covering spaces" of a space $X$, defined as
functors $\Pi_1(X)\to Sets$, where $\Pi_1(X)...
- 407
4
votes
1
answer
183
views
Notation for weak derivatives
I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and ...
- 4,459
4
votes
2
answers
287
views
Teaching suggestions for Kleene fixed point theorem
I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
- 41
4
votes
1
answer
441
views
How to teach generalizing the induction hypothesis? [closed]
I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...
- 9,201
4
votes
0
answers
176
views
Are injective modules flabby on basic open sets?
In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds:
Statement: If $A$ is a commutative ring and $...
- 1,064
4
votes
0
answers
795
views
Almost linear ODE: how node becomes a spiral
Most introductory ODE books contain a discussion of almost linear systems, and there are two cases when the behavior of an almost linear system near an equilbrium point can differ from the behaviour ...
- 29.1k
3
votes
6
answers
2k
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Teach a course in 1 month
I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?
3
votes
2
answers
651
views
Can this informal argument (for the fact that almost all reals in the unit interval are irrational) be saved?
In the textbook from which I am teaching a Discrete Math course, the authors propose randomly generating an infinite sequence of decimal digits $d_1, d_2, \dots$. We are to think of this as the ...
- 6,292
3
votes
3
answers
1k
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Pedagogical question concerning $\Gamma(z)$
Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
...
- 17.6k
3
votes
2
answers
957
views
Simple definition of the Hausdorff measure using squared paper
I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...
- 20.2k
3
votes
1
answer
271
views
Elementary classification of division rings
Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
- 26.5k
3
votes
3
answers
515
views
undergraduate handle decomposition. Reference
As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.
- 829
3
votes
3
answers
2k
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What to teach in a second graduate course in algebra? What textbook to use?
There is a standard syllabus for a first graduate course in algebra. One teaches groups,
rings, fields, perhaps a little bit of Galois theory, perhaps a little bit of
category theory, perhaps a ...
3
votes
2
answers
141
views
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
I am currently wondering whether it is realistically possible to choose the topic "Fractals and fractal dimensions" for a seminar aimed at undergraduate students in the 2nd semester, with ...
- 1,942
3
votes
2
answers
395
views
Integration in several variables and elementary applications
This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...
3
votes
1
answer
507
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What are some interesting grading/curving systems you have seen for a course? [closed]
It seems like every math course has something unique in how things are graded.
1) What are some interesting grading systems you have seen/used? (include curving types, etc.)
2) What are some pros ...
3
votes
0
answers
167
views
Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
3
votes
0
answers
873
views
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 ...
- 161
3
votes
0
answers
431
views
Concrete questions that turn into math problems [closed]
I'm writing an article about the way we teach math, trying to find out why so many people are discouraged from learning, and have no interest for math and logic.
At some point, I want to show that ...
- 131
2
votes
4
answers
4k
views
Best way to introduce the Chinese Remainder Theorem (to a high school student)
What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and ...
- 1,269
2
votes
4
answers
6k
views
Undergraduate Derivation of Fundamental Solution to Heat Equation
It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...
- 5,935
2
votes
4
answers
1k
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Eigenvalues of powers of linear mappings
Let $\tau$ be a linear map on a finite dimensional complex vector space. Clearly, if $\lambda$ is an eigenvalue of $\tau$ then $\lambda^n$ is an eigenvalue of $\tau^n$, for any natural (integer, on ...
- 4,096
2
votes
2
answers
6k
views
Examples of random variables
I'm looking for a list of examples of random variables to use in teaching a measure-theoretic probability course. For example, the Rademacher functions are an explicit construction of independent ...
- 5,227
2
votes
3
answers
410
views
Pedagogical notes on line bundles on complex projective manifolds
I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...
2
votes
1
answer
628
views
Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?
(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.)
Imagine an introductory probability course ...
2
votes
1
answer
897
views
Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis
Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
2
votes
1
answer
295
views
Examples of new results found via exams [closed]
I suspect that there have been many instances throughout history where a new proof of an existing result has been discovered by a student while taking an exam. Does anyone have an example of this?
2
votes
2
answers
1k
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Decomposition of $K_{10}$ in copies of the Petersen graph
It is a well-known and cute exercise in algebraic graph theory to show that $K_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose ...
- 10.9k
2
votes
1
answer
359
views
Defining integrals by residue theorem
I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....
- 341
2
votes
0
answers
1k
views
Linear Algebra Text Book [closed]
In our department we do not like our current linear algebra book and so we would want to find a better book. This is for the first course in linear algebra and the title of the course is
Elementary ...
2
votes
0
answers
3k
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What is the geometric meaning of the third derivative of a function at a point? [closed]
What is the geometric meaning of the third derivative of a function at a point?
This question is now asked on the sister site: https://math.stackexchange.com/questions/14841/what-is-the-meaning-of-...
- 61
2
votes
0
answers
526
views
How much of math could be taught without using mathematical notation? [closed]
Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
1
vote
2
answers
825
views
Simple yet interesting applications of Calculus or Linear Algebra to Economics [closed]
This is essentially a vast generalization of my previous question: Examples of separable ordinary differential equations in economics
I'm giving a talk to college-level math teachers on some ...
1
vote
1
answer
387
views
proof without words for logarithms [closed]
Does anyone know of any PROOF WITHOUT WORDS for logarithmic functions?
The only one I've seen in calculus based and I need one for high school math kids in MATH 1,2,3.
Any suggestions would be ...
1
vote
1
answer
117
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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
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
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