All Questions
542 questions
1
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1
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Maximum/Minimum operator precedence
Is there any standard preceding order for the operators $a \wedge b = \min{(a,b)}$ and $a \vee b = \max{(a,b)}$ with respect to the arithmetic operators.
For example
$$ a \wedge b + c = (a \wedge b)...
- 279
20
votes
2
answers
4k
views
Teaching stochastic calculus to students who know no measure theory (or PDE, or...)
I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...
- 61.9k
7
votes
2
answers
1k
views
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 ...
- 2,437
22
votes
1
answer
33k
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vector to diagonal matrix [closed]
For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector.
Is there a simple way to write this transformation ...
- 247
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 ...
- 14.3k
1
vote
0
answers
103
views
Confusion optimal control abuse notation
I'm currently reading this paper describing a numerical scheme for the approximating optimal policy of a stochastic control problem. However, I run into a confusion directly on the first page where ...
- 5,405
5
votes
1
answer
548
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Question about denoting/designating of algebraic structures
I saw this image on Wikipedia (Template:Group-like structures, current revision):
Since there are five "properties" that we can have (in this context), namely: totality, associativity, identity, ...
user114642
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 ...
-1
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1
answer
118
views
Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) [closed]
I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.
In chapter IV proposition 4.5 he states if K is an algebraic number ...
- 13
9
votes
3
answers
3k
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Math History Question about the exponential function
While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...
- 297
13
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2
answers
2k
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teaching higher algebra
Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The kind of ...
- 534
10
votes
4
answers
2k
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Reference for working with the implicit function theorem
I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal ...
- 156k
5
votes
1
answer
1k
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Generalizing Big O notation to arbitrary vector spaces
I'm constructing a Coq library for Big-O notation. Naturally, I'd like it to be as general as possible. The Wikipedia page on Big-O notation says
The generalization to functions taking values in ...
- 153
4
votes
4
answers
3k
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Is there a standard notation for binary relations in category theory?
In set theory, I learned that a binary relation is simply a subset of a Cartesian product. Moving on to category theory, it seems that this definition is not enough. Just as a function is no longer ...
11
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4
answers
3k
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Topological examples of profinite groups
I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first ...
- 113
28
votes
3
answers
3k
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Why is "h" the notation for class numbers?
A student asked me why $\mathcal{O}_K$ is the notation used for the ring of integers in a number field $K$ and why $h$ is the notation for class numbers. I was able to tell him the origin of $\...
- 50.6k
7
votes
3
answers
877
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Origin of the notation s=\sigma+it in analytic number theory
I was wondering if the standard notation of denoting a complex variable by "$s$" had an interesting origin, or if it dates back to Riemann or Weierstrass. Almost every book in analytic number theory ...
- 899
22
votes
4
answers
5k
views
What is the best way explain to undergraduates that all 1-dimensional manifolds are orientable?
Let's suppose that $M$ is a connected $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely $\...
- 3,540
0
votes
0
answers
39
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Terminology: Almost stable states
I have a question about fixed points which are almost stable.
I have an increasing transition function $f:[0,1]\rightarrow[0,1]$ where $f(0)>0$ and $f(1)<1$ but I don't necessarily have ...
- 188
10
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7
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2k
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Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
9
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7
answers
1k
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Mathematics seminar for "non-mathematicians"
Next term I am leading a seminar for students, who will become teachers for elementary school i.e. for kids of age 6-10. The students in the seminar will have no mathematical background beyond the "...
1
vote
1
answer
308
views
Basic question regarding notation of summation over primitive characters
This seems like a very standard notation in analytic number theory, and I see it a lot. But I was confused with it and I would greatly appreciate any clarification.
When one writes sum of the shape
$$...
- 3,625
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
0
votes
0
answers
45
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Notation of $P^+$-families - bibliography searching
have you ever met with notation of $P^+$-families in other papers than Iian B. Smythe "A local Ramsey theory for block sequences" and his phd?
Thank you in advance
16
votes
2
answers
3k
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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 ...
- 45k
8
votes
1
answer
1k
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Notation for a representable functor
For an object $X$ of a category, $h_X$ is the contravariant functor represented by $X$, i.e. $h_X = Hom(-,X)$.
Question a) Who invented this notation? (My guess: Grothendieck)
b) Is there a special ...
- 63.1k
0
votes
1
answer
82
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Computability Theory Notation For Entering A Set At A Stage
Is there a standard (or at least common) symbol in computability theory used to indicate that $x$ enters the c.e. set $W_e$ at stage $s$, i.e., $x \in W_{e,s} - W_{e,s-1}$ (at least for $s \neq 0$)?
...
- 3,029
1
vote
0
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93
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Notation/definition for the state of a FIFO queue [closed]
A first-in first-out queue is filled up by tokens $t \in T$. The state of the queue $q \in Q$ is being changed by two operations,
\begin{equation}
\mathrm{push} : Q \times T \rightarrow Q
\end{...
- 135
6
votes
2
answers
588
views
Applications of isotropic quadratic forms
I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...
19
votes
3
answers
2k
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Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
7
votes
2
answers
767
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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 ...
- 5,544
11
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2
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536
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Historical quotation search: Equations/formulae in (Latin?) prose, before modern symbolic notation
I have been trying, without success, to find a vaguely-remembered quotation: the quadratic equation (or perhaps the quadratic formula), given in (Latin?) prose, along lines like “Consider that ...
- 21.3k
7
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3
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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
0
votes
1
answer
54
views
Writing a set of all possible (symmetric) products condensely? [closed]
I have a set of elements $\{a_1, a_2, a_3...\}$ and $\{b_1, b_2, b_3...\}$ and I want to condensely formally write the set of all possible products of these elements, where the ordering does not ...
- 1,465
14
votes
0
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919
views
Grothendieck construction and coends
In category theory, both the Grothendieck construction and coends are represented by a sort of "integral sign", respectively:
$$
\int F
$$
for a functor $F:C\to\mathbf{Cat}$,
and:
$$
\int^x G(x,x)
$$
...
- 2,129
13
votes
3
answers
2k
views
History surrounding Gauss Theorema Egregium and differential geometry
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
- 193
2
votes
2
answers
247
views
Technical term for representing object of a presheaf determined by a left-adjoint?
If $\mathcal{D}$ is a locally-small category, then a functor $F\colon\mathcal{C}\rightarrow\mathcal{D}$ has a right-adjoint if and only if for each object $d$ of $D$, the presheaf $$\mathcal{C}^{\...
- 6,051
6
votes
3
answers
1k
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An application of Maschke's theorem
I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...
- 1,472
7
votes
3
answers
3k
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The etale fundamental group of a field
Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...
- 7,338
1
vote
2
answers
865
views
Cayley-Dickson form of a quaternion
It is known that using the Cayley-Dickson construction a quaternion $q$ can be written in a symplectic form as $q=x+\mathbf{i}y$ with $x,y \in \mathbb{C}$.
I read in a couple of references that $x$ is ...
- 177
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4
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2k
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Help me find good math questions for my students [closed]
I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...
11
votes
2
answers
3k
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Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
- 2,201
4
votes
4
answers
4k
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Variation on the Sobolev space $H^1_0$
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
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
7
votes
5
answers
2k
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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 ...
10
votes
8
answers
2k
views
Undergraduate Probability Topics
I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or ...
5
votes
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
9
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3
answers
3k
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notation for formal Laurent series
I've found a few articles that write the ring of formal Laurent series in $t$ as $R((1/t))$, but what's the underlying meaning of $\cdot ((\cdot))$?
A mathematician of my acquaintance swears that $R((...
- 19.7k
1
vote
0
answers
84
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Basic notation question involving Lie Groups and Lie algebras
I just started reading "On the functional equations satisfied by Eisentstein series" by Langlands http://publications.ias.edu/sites/default/files/Eisenstein-ps.pdf . I wasn't sure of some notation/...
- 3,625
9
votes
4
answers
1k
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Notation for eventually less than
Is there some existing notation for
\[f(n)\leq g(n)\] for sufficiently large n
Apart from just writing that itself?
I'm thinking of something compact like the ...
- 7,013
-2
votes
1
answer
5k
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Looking for the name of a mathematical symbol that looks remotely like 1 (answer: indicator function) [closed]
Original question:
The symbol looks like a numeral 1 written like an R in $\mathbb{R}$. It has a double vertical line and a serif at the bottom. It represents a function of a parameter: $1_{\{0,1\}}(x)...
- 121