All Questions
542 questions
1
vote
1
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732
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Notations for open and closed sets
I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
- 4,786
63
votes
22
answers
44k
views
What are the worst notations, in your opinion? [closed]
With which notation do you feel uncomfortable?
- 12.9k
2
votes
0
answers
284
views
Notation for spectral sequences [closed]
Every single spectral sequence I have seen in my life was denoted by $E$. Even when there is more than one spectral sequence, people tend to use the same letter with some workaround (e.g. a fourth ...
- 2,152
21
votes
8
answers
5k
views
Examples of bad notation and its consequences [closed]
An example of bad mathematical notation that comes in my mind and has caused complications throughout history is the notation for imaginary numbers. The original notation used to represent imaginary ...
- 2,858
0
votes
0
answers
102
views
Merging two composable walks in a graph
Let $G$ be a graph (i.e., an undirected graph in which we allow for loops and parallel edges). Denote by $V$ the vertex set, by $E$ the edge set, and by $\psi$ the incidence function of $G$, and let $\...
- 10.5k
192
votes
81
answers
33k
views
Suggestions for good notation
I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:
Iverson introduced the notation [X] to mean 1 if X is ...
- 156k
1
vote
0
answers
78
views
Minus sign inside derivative operator, notation problem
Hello fellow mathematicians. Can anybody help me understand what the minus (-) sign in this derivative means? Its the usual d/dy but with a minus added d-/dy. I can't find references, the book cited ...
- 11
2
votes
0
answers
182
views
What do you call $x$ such that $\textrm{dim} f^{-1}(f(x))>0$?
Let $f:V\to W$ be a morphism between varieties, with $\dim \overline{f(V)} = \dim V$. What do you call the closed proper subvariety $S$ of $V$ consisting of points $x$ such that $\textrm{dim} f^{-1}(f(...
- 20.2k
86
votes
16
answers
9k
views
Teaching homology via everyday examples
What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory?
To be more precise, I am teaching a short course on homology, from ...
- 82.7k
2
votes
1
answer
472
views
First use of corner quotes for Gödel numbers
Who first used the corner quotes, $\ulcorner$ and $\urcorner$, for the notion of Gödel number? They can also be written as\Godelnum with Sam Buss's macro.
They were ...
- 3,574
53
votes
7
answers
8k
views
Zorn's lemma: old friend or historical relic?
It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
- 12.9k
84
votes
12
answers
21k
views
Is Euclid dead?
Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (...
- 12.9k
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?
- 1,451
16
votes
3
answers
4k
views
How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting?
Most mathematical notation is designed with handwriting in mind in the first place, and typography must then try to follow, not always very successfully. However there is a particular type of notation ...
- 11.8k
5
votes
1
answer
589
views
what do empty parens symbol mean?
Quick easy question: what is the meaning of the symbol $(\space\space )$. I've seen it now in two papers, one of which is Milgram's Group Representations and the Adams Spectral Sequence, available at ...
- 3,065
3
votes
0
answers
176
views
What is the meaning of big-O of a random variable?
I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below:
screenshot of the book
In the excerpt, the big-O notation $O(\xi^...
- 6,367
7
votes
1
answer
723
views
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{ ...
- 82.7k
1
vote
1
answer
117
views
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 ...
- 30.3k
25
votes
2
answers
3k
views
What is the origin/history of the following very short definition of the Lebesgue integral?
Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
- 32.5k
1
vote
0
answers
190
views
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
92
votes
8
answers
16k
views
Has incorrect notation ever led to a mistaken proof?
In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, ...
- 18.7k
1
vote
2
answers
865
views
Cayley-Dickson form of a quaternion
It is known that using the Cayley-Dickson construction a quaternion $q$ can be written in a symplectic form as $q=x+\mathbf{i}y$ with $x,y \in \mathbb{C}$.
I read in a couple of references that $x$ is ...
- 63.9k
19
votes
2
answers
11k
views
Meaning of $\Subset$ notation
The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
- 259
42
votes
16
answers
5k
views
Justifying/Explaining math research in a public address
I have been chosen by my university to give a 1 hour public research lecture. Every year a researcher is chosen for this honour. Traditionally people explain their own research about designing ...
- 4,713
1
vote
1
answer
249
views
Name for extension of the symplectic group
Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\...
- 12.9k
2
votes
1
answer
248
views
What is the meaning of $\alpha^{+L}$ for $\alpha$ an infinite countable ordinal?
Condition (a) of lemma 3.4 in the paper “Countable ranks at the first and second projective levels” [M. Carl, P. Schlicht, P. Welch] is
$\alpha^{+L} = \omega_1,$
where $\alpha$ denotes any infinite ...
- 930
8
votes
2
answers
1k
views
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 ...
- 2,494
1
vote
0
answers
85
views
Notation for function that is constant with respect to a parameter
I am wondering if there is a common notation for a function that does not depend on a particular parameter. I am wondering about notation both for applying the function ($f(x, y)$) as well as defining ...
- 111
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 ...
- 4,786
1
vote
1
answer
147
views
"Variable and fixed" in categories
We often find in Grothendieck terminology the words variable and fixed (or absolute).
For example in SGA 4 studies variable topological spaces, groups, and categories as examples of morphisms of topos....
- 2,369
2
votes
0
answers
311
views
Degree of a morphism between affine varieties
(Context: rewriting a joint paper with a coauthor.)
We are defining the degree of a morphism $f:A^m\to A^{n}$ to be $\max_{1\leq i\leq n} \deg(f_i)$, for $f_1,f_2,\dotsc,f_{n}$ the polynomials ...
- 20.2k
1
vote
1
answer
171
views
Notation for infinite cartesian products
This is a soft question, feel free to delete it if deemed inappropriate for the site. What is the best notation for the cartesian product of an infinite number of copies of the same set $E$? Maybe one ...
- 63.9k
6
votes
0
answers
466
views
What is the "permanence relation" really?
I have come across the words "permanence relation" in a 1969 paper by Keith Hannabuss The Dirac equation in de Sitter space. The only other similar google hit for this phrase appears in ...
- 1,446
14
votes
9
answers
2k
views
math circles video lectures for school children?
Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...
- 4,713
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
32
votes
20
answers
6k
views
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
1
vote
0
answers
48
views
Notation for dominating (or uniformly bounded) function
While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function.
A situation like this. For some true function $f:\mathbb{R} \to \...
- 284
4
votes
1
answer
784
views
Notation diversity
This morning I had a brief discussion about different notations of trigonometric functions in Europe, so I looked for an online resource dealing with these diversities in mathematical notation. I ...
- 188k
3
votes
1
answer
202
views
Reference request: Dictionary of the Leibniz notation
Is there any published, somewhat comprehensive, list of (almost?) all the many ways in which the Leibniz notation ($dx,$ $P(dx),$ $d\mu(x),$ $du\wedge dv,$ etc., etc.) gets used in the various areas ...
- 188k
1
vote
0
answers
103
views
Are there standard short notations for ascending and descending cyclic permutations?
In a paper I am currently writing I use cyclic permutations of the form
$$
(k,k+1,\dots,\ell)
$$
and
$$
(\ell,\ell-1,\dots,k)
$$
of consecutive elements quite a lot (I added the commas to avoid ...
- 7,127
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 ...
- 12.9k
2
votes
0
answers
234
views
Why is $H$ the standard notation for mean curvature?
I am curious about the origin of the notation $H$ to denote the mean curvature of a surface in $\mathbb{R}^{3}$.
I suppose that the symbol $K$, which is commonly used to denote the Gaussian curvature, ...
- 985
22
votes
2
answers
2k
views
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
-3
votes
1
answer
222
views
What is the basis for the quantifier notation? [closed]
The symbols $\forall, \exists$ are the ones officially used to denote universal and existential quantifiers respectively. I understand that the choice of $\exists$ was made by Peano, while of $\forall$...
- 4,219
13
votes
1
answer
730
views
Who introduced the notation for $\beth$ numbers and when?
Georg Cantor, when developing the basics of set theory, noted that there are two ways to increase cardinality: power sets and successors (or, in modern terms, the Hartogs operation).1
Eventually the ...
- 188k
25
votes
19
answers
20k
views
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
2
votes
1
answer
740
views
What does the subscript 'x' of a matrix mean? [closed]
The 3x6 matrix G is as follows,
$\text{G} = [\text{V}_\times| I_{3\times3}]$
$\text{V}$ is a skew matrix of a vector with 3 elements about a 3D point. The dimension of $\text{V}$ is 3x3.
$I$ is the ...
- 5,429
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.
...
- 24.8k
17
votes
5
answers
3k
views
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
16
votes
2
answers
3k
views
Metric on one-point compactification
Is there a standard construction of a metric on one-point compactification of a proper metric space?
Comments:
A metric space is proper if all bounded closed sets are compact.
Standard means found in ...
- 4,713