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location Oxford, United Kingdom
age 50
visits member for 3 years, 9 months
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Sep
3
comment Is this problem on weighted bipartite graph solvable in polynomial time or it is NP-Complete
Why do you talk about finding a sequence? The order you pick up your $v_i$'s does not matter. Better talk about a subset of A. Do you allow weights to be negative? If not, then you trivially will have to take the whole A.
Sep
3
comment On the positivity of matrices
The vector space of matrices commuting with G has a basis {$I$,$\pi+\pi^4$,$\pi^2+\pi^3$}, where $\pi$ is the permutation matrix corresponding to the cyclic permutation (1,2,3,4,5). Thus $P$ will be a linear combination of these 3 matrices.
Sep
3
comment On the positivity of matrices
$G$ is able to guarantee that $P_{ij}=p$, as the dimension of the centraliser algebra (i.e. the vector space of matrices commuting with every element in $G$) is 3, which can be e.g. checked directly, or by computing the permutation character of $G$. All these things I don't define may be found in books on permutation groups, e.g. P.Cameron's books.google.co.uk/books/about/…
Sep
2
comment On the positivity of matrices
G depends on M only, and can be checked to be of size 10.
Sep
2
comment On the positivity of matrices
One way to conclude that for $P$ being the average over an orbit all these $P_{ij}=p$, would be to note that $P$ commutes with all $\Pi\in G$. One can check that the algebra of all matrices commuting with $G$ coincides with the algebra of symmetric circulant matrices. Thus $P$ has the claimed form.
Sep
2
comment On the positivity of matrices
@Hansen: I don't see how you would complete a proof "by hand" without being able to reduce the number of parameters significantly.
Sep
2
comment On the positivity of matrices
here it is: journals.cambridge.org/article_S0305004100036185
Sep
1
comment On the positivity of matrices
What I am saying is: take an offending P, and average it over G. This gives a symmetric circulant matrix which is still offending, and which only depends on one parameter, $p$.
Aug
31
comment On the positivity of matrices
@gondolf - you're welcome. Please click on "accept this answer" if you're happy with it.
Aug
31
comment On the positivity of matrices
@Hansen - see my latest edit.
Aug
31
comment On the positivity of matrices
@Hansen - see my latest edit for a proof and a reference.
Aug
30
comment On the positivity of matrices
yes, it is true for $n\leq 4$. This was shown by P.Diananda in 1962 (Proc. Cambridge Phil.Soc., vol 58). The counterexample for n=5 (see my answer) is also from this text.
Aug
30
comment On the positivity of matrices
IIRC, the minimal size for counterexample is $n=5$.
Aug
30
comment On the positivity of matrices
This computational technique is called semi-definite programming.
Aug
25
comment Invariant planes of a nilpotent matrix with two Jordan blocks of size two
The 2-dimensional subspaces are parametrised by the points on a quadric $Q$ in $P^5$, via Plucker coordinates. You can write down how $N$ acts there, and take the fixed vectors of linear map which satisfy the quadratic relation defining $Q$. Thus you will have them parametrised by the intersection of $Q$ and the linear subspace of fixed points.
Aug
24
comment Real solutions for systems of monomial equations
this is a particular case of a problem involving binomial ideals; the latter are quite well-studied, in the context of toric varieties.
Aug
18
comment How to prove a Proposition of Rouquier?
how can it be so that some statements in the text indicate that it's not clear how to prove some other statement in the same text?!
Aug
18
comment What is the minimal girth of a cayley graph for Alt(n) in which the girth relator is not a proper power?
6? Why? Why not 42? :)
Aug
18
comment What is the minimal girth of a cayley graph for Alt(n) in which the girth relator is not a proper power?
from topology point of view, you might want to disregard length 2 loops.
Aug
13
comment Linkage between homotopy equivalence and identification of algorithms
this is called "univalent foundations of mathematics", see e.g. homotopytypetheory.org