Linked Questions

48 votes
11 answers
7k views

In "splendid isolation"

While browsing the Net for some articles related to the history of the Whittaker-Shannon sampling theorem, so important to our digital world today, I came across this passage by H. D. Luke in The ...
48 votes
4 answers
5k views

How to constructively/combinatorially prove Schur-Weyl duality?

How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring $\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\...
darij grinberg's user avatar
21 votes
3 answers
6k views

What are the current breakthroughs of Geometric Complexity Theory?

I've read from Wikipedia about Geometric Complexity Theory (GCT) which (if I understood correctly) is a program for coping with the $ P=NP $ problem using algebraic methods. That program seems ...
47 votes
2 answers
4k views

The two ways Feynman diagrams appear in mathematics

I've heard about two ways mathematicians describe Feynman diagrams: They can be seen as "string diagrams" describing various type of arrows (and/or compositions operations on them) in a monoidal ...
Simon Henry's user avatar
  • 39.9k
15 votes
3 answers
2k views

Integration of a function over 7-sphere

Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$. The problem is finding or approximating the ...
Hrushikesh Pawar's user avatar
8 votes
3 answers
3k views

What is higher dimensional algebra?

Could anyone explain what higher dimensional algebra is? I tried to look on the web but I couldn't find a satisfactory definition, the ones that I found are too vague. What I'm looking for is a good ...
Giorgio Mossa's user avatar
5 votes
2 answers
1k views

Formula for n-th iteration of dx/dt=B(x)

Let $B(x)$ be infinitely differentiable with respect to $x$. Drop the use of parentheses on $B$ to delimit the argument $x$ and use them instead to hold the order of the derivative with respect to $x$....
resolvent's user avatar
  • 133
9 votes
2 answers
662 views

Сlosed formula for $(g\partial)^n$

The objective is to obtain a closed formula for: $$ \boxed{A(n)=\big(g(z)\,\partial_z\big)^n,\qquad n=1,2,\dots} $$ where $g(z)$ is smooth in $z$ and $\partial_z$ is a derivative with respect to $z$. ...
Wakabaloola's user avatar
7 votes
2 answers
916 views

Rigorous definition of the commutator $[a(k_1), a^\ast(k_2)]$ of creation and annihilation operators in boson quantum field models

In their lecture notes "Boson Quantum Field Models" (in "Mathematics of Contemporary Physics", R.Streater (ed.)), Glimm and Jaffe define an annihilation operator $a(k), k \in \mathbb{R}$ on a certain ...
Kevin McLeod's user avatar
7 votes
3 answers
554 views

GIT quotients for linear representations of $SL(2,\mathbb C)$

Let $V$ be the standard two-dimensional representation of $SL(2,\mathbb C)$ and let ${\rm Sym}^2V$ be its symmetric square. Let $n$ be a positive integer and consider the following two representations ...
aglearner's user avatar
  • 14k
3 votes
1 answer
188 views

Generalized tensor-train decomposition

If $U\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ and $U^{(k)}\in\mathbb R^{r_{k-1}}\otimes\mathbb R^{n_k}\otimes\mathbb R^{r_k}$ with$^1$ $$U_{i_1,\:\ldots\:,i_p}=\sum_{j_0=1}^{r_0}\cdots\sum_{j_p=1}^{r_p}U^...
0xbadf00d's user avatar
  • 161
5 votes
0 answers
225 views

Schröder and graphical logic?

I was actually surprised by a comment by John Baez over at the n-Category Cafe about his surprise that Ernst Schröder, a mathematician of whom he had known through Schröder's work on mathematical ...
Tom Copeland's user avatar
  • 9,937