All Questions
542 questions
7
votes
0
answers
1k
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Conventions for Riemann curvature tensor
I am aware of two conventions for the Riemann curvature tensor, namely the expression
$$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$
is either declared to be $R(X,Y,Z,W)$ or $...
- 18.7k
5
votes
4
answers
957
views
Notation for $\log \log \cdots \log n$? [closed]
Is there some accepted, more concise notation for expressions like $\log \log \log n$?
I just noticed an arXiv posting that quotes the bound
$$
\frac{\log X \log \log X \log \log \log \log X}
{ \log \...
- 151k
27
votes
5
answers
5k
views
Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties
This really is two questions, but they are kind of related so I would like to ask them at the same time.
Question 1:
In a question asked by Amitesh Datta, BCnrd commented that it is important to ...
6
votes
0
answers
283
views
Interesting things you learned while grading/marking? [closed]
What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...
9
votes
0
answers
887
views
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 ...
- 16.9k
0
votes
1
answer
158
views
Unknown notation in "Boolean function complexity" by Stasys Jukna [closed]
I am currently reading Boolean Function Complexity - Advances and Frontiers by Stasys Jukna and on page 7 of the latest edition there is a paragraph titled Boolean functions as set systems with the ...
- 11
27
votes
2
answers
3k
views
Teaching the fundamental group via everyday examples
This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...
4
votes
2
answers
3k
views
Is the notation $f(x)$ "deprecated by professional mathematicians" (as claimed by Wolfram)? [closed]
Wolfram's MathWorld website, at the page on functions, makes the following claim about the notation $f(x)$ for a function:
While this notation is deprecated by professional mathematicians, it is ...
- 167
2
votes
0
answers
221
views
What is the p-adic Plancherel measure?
What I know as the Plancherel measure for a group is a measure on the spectrum of $G$, aka the set of irreducible representations - at least for finite groups, this makes perfect sense.
Now, this ...
- 7,746
0
votes
0
answers
290
views
A question about chaining Vinogradov notation
This is not a research question, but I hope it is still legitimate to ask for this platform. Suppose $A(x)$, $B(x)$, $C(x)$, $D(x)$ are positive-valued functions of $x$, and
$A(x) \ll B(x)$ and $ C(x) ...
- 6,224
2
votes
0
answers
323
views
Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem
I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \...
- 161
1
vote
0
answers
211
views
Are measures better thought of as densities than differentials?
The standard notation for integrating with respect to a measure $\mu$ is:
$$\int f(x)\,d\mu(x).$$
But I've wondered if it could be better written as:
$$\int f(x)\mu(x)\,dx$$
where $\mu(x)$ is now ...
- 4,943
8
votes
2
answers
3k
views
What is the standard notation for reversing the order of vector's components? [closed]
If we have a vector $x=(x_1,x_2,\ldots,x_n)$, is there any standard way to denote the vector $(x_n,x_{n-1},\ldots,x_1)$?.
I think that $x^{-1}$ could be a good option.
- 89
0
votes
1
answer
125
views
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, ...
- 479
30
votes
3
answers
4k
views
Nearly all math classes are lecture+problem set based; this seems particularly true at the graduate level. What are some concrete examples of techniques other than the "standard math class" used at the *Graduate* level?
In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier ...
7
votes
8
answers
4k
views
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?...
0
votes
1
answer
235
views
Is there any math notation for `be denoted by`? [closed]
The sentence s "In many supervised learning problems one has an output variable $y$ and a vector of input variables $x$ described via a joint probability distribution $P(x,y)$" from wiki
Here ...
- 111
9
votes
4
answers
1k
views
Characterization of the Poisson law
This semester, I teach an introduction to probability course tailored for students with no science background and so with very very little prerequisites. We started with the basics of analytic ...
- 10.9k
7
votes
1
answer
372
views
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 ...
- 7,197
2
votes
0
answers
314
views
Notation for induced subgraphs
For a graph $G=(V,E)$, is there a standard notation for the induced subgraph on $V \setminus \{v,w\}$ where $v,w$ are the endpoints of some edge $e$? I know $G[V \setminus \{v,w\}]$ is an option, but ...
- 19.7k
1
vote
0
answers
134
views
What benefits of math can be conveyed to mid/high schoolers? [closed]
I'm teaching mathematical proof writing to a few of math teachers (in the US) this summer. In the beginning of class, I send a survey asking them why they are here. Most of them are here for getting ...
- 5,230
16
votes
5
answers
1k
views
Permission to use Online Notes
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
20
votes
5
answers
2k
views
How and how much do the notations and diagrams influence our understanding of mathematical concepts?
How and how much do the notations and diagrams influence
our understanding of mathematical
concepts?
This question was stimulated by the MathOverflow questions Thinking and Explaining and ...
8
votes
2
answers
6k
views
Equal signs with fancy marks
Some people use $\stackrel{\mathrm{def}}{=}$, $:=$ or $\stackrel{\Delta}{=}$ for definitions.
In more informal contexts, I have also seen $\stackrel{?}{=}$, for "I wish to prove this equality, which ...
- 20.2k
8
votes
2
answers
447
views
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-...
1
vote
0
answers
102
views
Notation question: bigraded direct sum of graded objects
In some work I'm doing I have two graded modules $M$ and $N$ (graded on $\mathbb Z$, say) and need to take, not the usual direct sum, but the bigraded sum consisting of all $M_p \oplus N_q$ (so graded ...
- 2,062
3
votes
0
answers
649
views
Does the Polish character Ł have an established mathematical meaning
I was suggested to use the slashed letter $\L$ (the European character Ł, which looks like the English letter L with a small bar crossing its vertical part) to denote the left half-plane. To avoid ...
- 81
11
votes
3
answers
729
views
Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]
In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\...
- 802
17
votes
2
answers
3k
views
How useful/pervasive are differential forms in surface theory?
Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...
- 3,059
24
votes
1
answer
4k
views
What does the σ in σ-algebra stand for?
I was tutoring someone in analysis and realized I have no idea where this notation comes from (or analogous terms: σ-additive, σ-ring, etc). I would like to know why the letter σ was chosen. I can't ...
- 1,793
10
votes
2
answers
23k
views
What is the definition of the $\uplus$ symbol?
Hi,
I have what I hope is a very simple question related to unfamiliar notation.
I am looking through a maths paper on a topic related to set theory which contains a symbol,
$\uplus$,
and I ...
- 141
8
votes
2
answers
2k
views
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
1
vote
1
answer
276
views
Symbol for monotone relationship between two probability distributions
Motivation:
At the present time it really isn't clear to me why this question might be inappropriate for the MathOverflow. However, it appears that some people are down-voting this question even if ...
- 3,871
13
votes
5
answers
2k
views
How to make a lecture series useful
I have been to a number of advanced lecture courses (of between 3 and 10 lectures) over the years, given (in principle) by experts to graduate students and experts in neighbouring fields. Examples of ...
5
votes
4
answers
1k
views
Lecture on Fractals for Middle School Students
I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject.
I'll show some fractal images and a few ...
- 87
0
votes
1
answer
60
views
Name for matrix associated to smooth continuation
Is there an established name for the matrices that establish the conditions for a linear combination of $n$ functions $\lbrace f_1(x),\dots,f_n(x)\rbrace$ being the $n$-times smoothly differentiable ...
- 13.2k
17
votes
6
answers
7k
views
Explaining the concept of projective space: notes for students
This is a question on teaching.
I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
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
28
votes
2
answers
5k
views
The Origin of the Musical Isomorphisms
In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are ...
- 1,202
11
votes
3
answers
448
views
Easy proof that reflections generate $N(T)/T$ for connected compact group?
I'm teaching a course on Coxeter groups and I'd like to provide an overview of the connection to compact Lie groups. Let $G$ be a compact connected Lie group, $T$ a maximal torus and $N(T)$ the ...
- 156k
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 ...
- 1,147
24
votes
2
answers
9k
views
Explanation why $x,y,z$ are always variables
I heard or have read the following nice explanation for the origin of the convention that one uses (almost) always $x,y,z$ for variables. (This question was motivated by question
Origin of symbol *l* ...
- 17.9k
7
votes
2
answers
1k
views
Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction
This is cross-posted in MSE.
I have seen two different kinds of definitions of the notation $C^k(\overline{\Omega})$ — by "extension" of functions on $\Omega$ or by "restriction" of functions on $\...
user14319
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
0
votes
1
answer
62
views
Is there a common notation to indicate the final form of a simplified definition? [closed]
I'm trying to become better with using proper terminologies and standard notation when taking notes, which lead me to think:
Similar to the indication of a completed proof by use of the Q.E.D. mark, ...
- 23
1
vote
0
answers
176
views
Arithmetization of Syntax: Can any semantic be encoded as syntax?
It is my understanding that Gödel Encoding and "Arithmetization of Syntax" can be used to represent any logical system. This is exemplified by the encoding of a Universal Turing Machine.
"According ...
- 111
5
votes
9
answers
2k
views
Suggestions for teaching advanced high school students
Hi all,
I'm a grad student and just joined a mentoring program in which I will visit a group of advanced year ten high school students (around 16 years old) from a group of schools in the area. I don'...
1
vote
0
answers
114
views
Notation for the geometric quotient of a separated Deligne-Mumford stack?
Suppose that $X$ is a separated Deligne-Mumford stack, say over a base scheme.
Is there some standard notation for the geometric quotient of $X$? I've tried using $[X]$ but have had complaints.
- 3,147
27
votes
3
answers
3k
views
Is “problem solving” a subject to be taught?
I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...
1
vote
0
answers
227
views
What does it mean for two natural numbers to be *approximately equal*?
This is related to this other question of mine about a paper of Colin and Honda.
I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the ...
- 1,409