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Results tagged with gm.general-mathematics
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Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.
0
votes
The sum of triangular functions
Solving for definite $n\in\mathbb{Z}$ I find something a bit more complicated than what is written in the OP:
$$\frac{1}{2}+\sum _{q=1}^{N-1} \frac{\cos (\pi (n-1) q/N)+\cos (\pi n q/N)}{\cos (\pi q …
- 188k
2
votes
Accepted
Find the expansion of the exact solution (beyond Taylor)
So there are two parameters $\alpha$ and $\beta$ and a function $V(\alpha,\beta)$, obtained from your first equation by substituting $S=\alpha^2/\beta$ and $\mu=2\beta/\alpha$. With some effort we can …
- 188k
8
votes
Geometric meaning of unimodular matrix
If the matrix has integer elements, then the geometric meaning is that a unimodular transformation maps the integer lattice onto itself:
Consider a basis $B$ of an $m$-dimensional lattice
$L(B)= …
- 188k
1
vote
Accepted
Discrete summation of Gaussian functions. Decay time problem
the decay time in your "simple case" is well approximated by the large-$M$ limit [*]
$$\lim_{M\rightarrow\infty}M\sigma\tau=2.84$$
here is a plot of
$$f_M(s)=\left.\frac{F_M(t)}{1-F_M(\infty)}\righ …
- 188k
13
votes
Unexpected occurrences of the Sierpinski triangle
The moves leading to the solution of the Towers of Hanoi puzzle form a Sierpiński triangle, as nicely described in this blog:
It is worth pausing a moment to think about this. The Tower of …
40
votes
Fascinating moments: equivalent mathematical discoveries
Not particularly profound, but with far-reaching practical consequences, is this equivalence between results from 19th century astronomy and 20th century digital data processing: The efficient method …
-1
votes
Journey into a strange wilderness
The landscape of string theory is aptly characterized as a "wilderness" for mathematicians.
"String theory is a piece of 21st-century physics that has fallen by accident into the 20th century and the …
12
votes
Solution to $(A+x^2)e^x=B$ with Lambert W function
You seek a solution in $x$ of the transcendental equation
$$e^x(x-t_1)(x-t_2)=a.$$
(The coefficients $t_1,t_2$ are real for $A<0$, complex otherwise.) The solution $W(t_1,t_2;a)$ is referred to as the …
- 188k
25
votes
How to deal with an advisor that offers you nearly no advising at all?
In the German tradition a PhD advisor is called a "Doktorvater/Doktormutter" --- a thesis parent. It might help to think about your relationship with your advisor along those lines. Many parents like …
27
votes
History of right hand rule
John Ambrose Fleming is credited with the invention of the right-hand-rule in the context of electromagnetism.
The figure below, illustrating $X=Y\times Z$ in a right-handed coordinate system, is from …
- 188k
5
votes
Question on pure mathematics helping climate change research
Chris Budd is one mathematician who has made a career in numerical weather prediction. He describes some of the challenges in The Mathematics of Climate Change. He explains the need for ‘simple’ mathe …
- 188k
10
votes
Where can square roots come from when they are not distances?
The square root of SWAP (which is the only two-qubit operation needed to realize a universal quantum computation) has no "distance" interpretation I can think of.
6
votes
Permanent archival of errata/corrigenda for published papers
There are two issues here, one is ensuring that the erratum is still online somewhere many decades from now, the other is ensuring that it can still be found. The first issue is actually simpler to re …
- 188k
6
votes
Accepted
Why are discreteness and smoothness in physics inversed with respect to geometry?
The "manifold picture" can be applied to physics in the context of the Brillouin zone, see for example On Brillouin Zones. The reason that discreteness and smoothness appear inverted, is that the Bril …
- 188k
7
votes
Surprising applications of the theory of games?
For applications of game theory to elections, this is a classic reference:
A game-theoretic model of party affiliation of candidates and office holders
We develop a formal model of ambition theory, e …
- 188k