All Questions
542 questions
1
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1
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163
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The meaning of $L_p^l(\Omega)$ in a paper of Bogovskii on Sobolev spaces
On the first page of the old paper Solution of the first boundary value problem for an equation of continuity of an incompressible medium of Bogovskii, the notations $W_p^l(\Omega)$ and $L_p^l(\Omega)$...
- 6,367
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
8
votes
1
answer
1k
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Why aren‘t op and co switched?
When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^...
- 355
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
votes
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
26
votes
12
answers
2k
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Examples of improved notation that impacted research?
The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.
I am aware that there is a related post ...
- 13.2k
15
votes
3
answers
3k
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History of the pullback corner notation
Where/when did the convention originate of marking pullback (and/or pushout) squares by that little right-angle symbol in the corner?
The earliest instance I’ve been able to find is in Paul Taylor’s ...
- 8,481
1
vote
0
answers
113
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Common notation for function over infinitely many variables? [closed]
For a document about reinforcement learning, I want to write the joint probability density over the entire trajectory of states and actions like $p(s_0, a_0, s_1, a_1, s_2, \dotsc)$. However, this ...
- 12.9k
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
votes
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 ...
- 188k
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
9
votes
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 ...
- 7,024
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
0
votes
2
answers
250
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Is it improper to define matrices as being $n \times m$ rather than $m \times n$? [closed]
For whatever reason, I have always defined matrices as being $n \times m$, and that is how I have been defining matrices throughout my dissertation. Recently however, I have noticed that nearly every ...
- 263
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
5
votes
2
answers
445
views
About the maximum number of leaves adjacent to a vertex in a tree
Let $T$ be a finite tree graph with the set of vertices $V(T)$. For an arbitrary vertex $ v \in V(T)$, I define $l(v)$ to be the number of leaves connected to $v$.
In my study, I need to define the ...
- 7,226
5
votes
0
answers
640
views
What does $\omega^*$ mean? [closed]
I've recently found in some short article (source below) the symbol $\omega^*$ (generally, starred ordinal number), but without explanation what that symbol means. From the context I understood that ...
- 873
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
2
votes
0
answers
100
views
Name for the theory of words with equal length, prefix, successors
I've worked with this theory for a while, but I've never been quite sure what to call it:
$$(\Sigma^*, =_{el}, \preceq, (S_a)_{a \in \Sigma})$$
Where
$\Sigma^*$ is the set of finite words on finite ...
- 429
26
votes
18
answers
34k
views
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
1
vote
1
answer
182
views
Pronunciation: the Erdős–Rado partition notation
The Erdős–Rado notation $a \rightarrow (b)^c_d$ is common in partition calculus / combinatorial set theory, as well as its negation $a \not\rightarrow (b)^c_d$. In that field, is there a standard way ...
- 188k
6
votes
1
answer
222
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Reference request: Different definitions of Big O notation
This question might sound strange, but I would like to settle this problem once and for all.
For as long as I can remember, I was introduced to the Big O notation by this definition:
Def. 1: Let $f, g$...
- 477
2
votes
0
answers
74
views
Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.9k
2
votes
0
answers
316
views
Higher order Leibniz rule and ordered multiindex notation
Although I think this is probably known, I am making here a short exposition on the multiindex notations I am using to make this question self-contained. I note that there is at least two different ...
- 3,307
1
vote
0
answers
294
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What does square bracket superscript star mean in basic group theory typically?
I'm reading some paper where they haven't really defined their notation very well (or I've missed something). You can see the image below.
What does the square bracket and star mean precisely? The ...
- 322
2
votes
0
answers
177
views
Can NBG be interpreted in this system that use new notation for class-abstractions?
We introduce a new symbol $\lambda$ to denote class-abstractions, and we add the following rule:
if $\phi$ is a formula that use $``\mu"$, and in which the symbol $\sf y$ doesn't occur; then: $\lambda ...
- 11.3k
2
votes
1
answer
128
views
Notation for H is isomorphic to a subgraph of G
Is there a notation for the statement $H$ is isomorphic to a subgraph of $G$? I was thinking of using $H<G$, but I'd like to use standard notation if possible.
- 5,429
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
5
votes
2
answers
377
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What is meant by this notation of the real forms of $E_6$?
There are five real forms of the exceptional Lie group, $E_6$. Four of them are notated as in the following:
The split form as EI or $E_{6(6)}$
The quasi-split form as EII or $E_{6(2)}$
EIII or $E_{...
- 2,369
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
44
votes
10
answers
11k
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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
4
votes
0
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160
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Proof of Theorem 9.2 of the book Cubic Forms by Yu. I. Manin (end of page 37)
I warn that I first posted this question in Mathematics Stack Exchange but it got no attention at all. I think that it fits better there by its explanatory nature but maybe the book being reference is ...
- 573
3
votes
1
answer
244
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Finitely-generated conjugation action on a subgroup that is not normal... what is that?
If $H \lhd G$, then $G$ acts on $H$ by conjugation. I need to talk about this action but in a situation where $H$ is not (necessarily) normal. When $H \leq G$, there is a "partial action" of ...
- 38.6k
5
votes
3
answers
799
views
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
3
votes
0
answers
238
views
How to denote a partial derivative?
This question is related to Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? and Suggestions for good notation .
When there are two ...
- 41
11
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5
answers
2k
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Is there a reference containing standard mathematical notations?
Suppose you are writing a mathematical text (say an article) and you want to call an object (for example, a set) by a letter. It would be cool then to have some reference (optimally available on the ...
- 333
2
votes
0
answers
124
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Good notation for finite partial functions from $\omega$ to 2
I'm working in computability theory and need to use partial functions with finite domain from $\omega$ to 2 as approximations in my current paper. Normally this is simply done using $2^{< \omega}$ ...
- 63.9k
2
votes
1
answer
2k
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Chudnovsky algorithm and Pi precision
What are the precision/ number of correct Pi digits after N iterations of Chudnovsky algorithm. Looking for a formula (rather than a table) and reference.
- 39
3
votes
1
answer
243
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Temporal generalization of graphs: density vs $n$ and $m$?
In short: we generalize graphs to the temporal case, but fail to fully preserve the usual relation between density, number of vertices, and number of edges; how to make better?
Context.
We propose a ...
- 81
-1
votes
1
answer
187
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Typesetting of symbols and "operators" denoting sets [closed]
Question:
what are the conventions for typesetting sets of certain objects, especially the vertices and edges of a graph or those adjacent to an edge or vertex.
For vectors and matrices there is the ...
2
votes
0
answers
905
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Confusing notation for sets of unordered vs ordered pairs
Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$.
One may also consider ...
- 1,444
18
votes
3
answers
2k
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Where does the name "R-matrix" come from?
In quantum integrability and related topics a lot of not-so imaginative terminology is used. One may hear people talk about "Q-operators", "R-matrices", "S-matrices", "T-operators", as well as "L-...
- 1,996
4
votes
0
answers
180
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Ideals with certain properties
I recently isolated the following definition, which I believe it should have appeared somewhere.
Let $\kappa$ be a cardinal, and let $X$ be a set with $\kappa^+\leq |X|$.
Definition: An ideal
$\...
- 1,133
-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
150
votes
31
answers
70k
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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
0
votes
1
answer
259
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Explanation of a formula to calculate the zenith distance of sun and moon [closed]
I am studying tidal accelerations and referring to a well known paper by I M Longman :
Formulas for computing.." J Geophys Research 64 (12) Dec 1959.
At Eq 12 he writes a term "1336.rev"...
- 188k
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
9
votes
1
answer
2k
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Origin of the symbol for the tensor product
I have recently realised that the Paleo-Hebrew (and Phoenician) graph for the Hebrew letter ט (Teth) is $\otimes$. This made me wonder if there is any relation between the choice of the symbol and the ...
- 188k
17
votes
5
answers
5k
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Bourbaki's epsilon-calculus notation
Bourbaki used a very very strange notation for the epsilon-calculus consisting of $\tau$s and $\blacksquare$. In fact, that box should not be filled in, but for some reason, I can't produce a \Box.
...
- 687
23
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 ...
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