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1500 questions
78
votes
5 answers
How are mathematicians taught to write with such an expository style?
I wasn't sure if this question was appropriate for MSE.
One of the major complaints we see in industry is a person's ability to communicate which includes writing.
We see the same thing on questions that are posted on MSE by younger students.…
Amzoti
- 56,629
78
votes
1 answer
Gerrymandering on a high-genus surface/can I use my powers for evil?
Somewhat in contrast to this question.
Let's say the Supreme Court has just issued a ruling that the upper and lower roads of an overpass need not be in the same congressional district. This makes states with lots of overpasses into high-genus…
Alex Becker
- 61,883
78
votes
22 answers
An example of a problem which is difficult but is made easier when a diagram is drawn
I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, word problems) can be done fairly easily when…
Jeel Shah
- 9,560
77
votes
3 answers
An explanation of the Kalman filter
In the past 3 months, I've been trying to understand the Kalman filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, predicate, etc.). However, I still am unable to…
xsari3x
- 207
77
votes
1 answer
Difference between Analytic and Holomorphic function
A function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be holomorphic in an open set $A \subset \mathbb{C}$ if it is differentiable at each point of the set $A$.
The function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be analytic if…
Supriyo
- 6,329
- 7
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77
votes
3 answers
What do prime ideals in $k[x,y]$ look like?
Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like?
As far as I know, the maximal ideals of $k[x,y]$ are of the form $(x-a,y-b)$ where $a,b\in k$. What can we say about the…
user14242
- 3,030
77
votes
4 answers
Why does the google calculator give $\tan 90^{\circ} = 1.6331779e^{+16}$?
I typed in $\tan 90^{\circ}$ in Google and it gave $1.6331779\mathrm{E}16$. How did it come to this answer? Limits? Some magic?
Gizmo
- 939
77
votes
2 answers
Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus):
the indefinite integral $\int f$ (also known as the…
user9464
77
votes
4 answers
Difference between basis and subbasis in a topology?
I was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?
Grobber
- 3,336
77
votes
4 answers
Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z...: unified treatment of transforms?
I understand "transform methods" as recipes, but beyond this they are a big mystery to me.
There are two aspects of them I find bewildering.
One is the sheer number of them. Is there a unified framework that includes all these transforms as…
kjo
- 14,904
77
votes
10 answers
Is learning (theoretical) physics useful/important for a mathematician?
I'm starting to read The Princeton Companion to Mathematics, at the beginning it says:
A proper appreciation of
pure mathematics requires some knowledge of applied
mathematics and theoretical physics.
Some of my professors have told me that…
Vicfred
- 2,957
77
votes
15 answers
Dot Product Intuition
I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ).
For example, suppose I have vector $\mathbf{a} =…
nerdy
- 3,456
77
votes
6 answers
Proof a graph is bipartite if and only if it contains no odd cycles
How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
Asinomás
- 107,565
77
votes
14 answers
What are some mathematical topics that involve adding and multiplying pictures?
Let me give you an example of what I mean. Flag algebras are a tool used in extremal graph theory which involve writing inequalities that look like:
(It's not too important to my question what this inequality means, but let me give you some…
Misha Lavrov
- 159,700
77
votes
5 answers
Why should I care about adjoint functors
I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global…
DBr
- 5,010