7,610 research outputs found
On the Asymptotic Existence of Hadamard Matrices
It is conjectured that Hadamard matrices exist for all orders ().
However, despite a sustained effort over more than five decades, the strongest
overall existence results are asymptotic results of the form: for all odd
natural numbers , there is a Hadamard matrix of order ,
where and are fixed non-negative constants. To prove the Hadamard
Conjecture, it is sufficient to show that we may take and . Since
Seberry's ground-breaking result, which showed that we may take and
, there have been several improvements where has been by stages
reduced to 3/8. In this paper, we show that for all , the set of
odd numbers for which there is a Hadamard matrix of order
has positive density in the set of natural numbers.
The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that
there are enough Paley Hadamard matrices to give the result.Comment: Keywords: Hadamard matrices, Asymptotic existence, Cocyclic Hadamard
matrices, Relative difference sets, Riesel numbers, Extended Riemann
hypothesis. (Received 2 August 2008, Available online 18 March 2009
An explicit construction for neighborly centrally symmetric polytopes
We give an explicit construction, based on Hadamard matrices, for an infinite
series of floor{sqrt{d}/2}-neighborly centrally symmetric d-dimensional
polytopes with 4d vertices. This appears to be the best explicit version yet of
a recent probabilistic result due to Linial and Novik, who proved the existence
of such polytopes with a neighborliness of d/400.Comment: 9 pages, no figure
Uniform Mixing and Association Schemes
We consider continuous-time quantum walks on distance-regular graphs of small
diameter. Using results about the existence of complex Hadamard matrices in
association schemes, we determine which of these graphs have quantum walks that
admit uniform mixing.
First we apply a result due to Chan to show that the only strongly regular
graphs that admit instantaneous uniform mixing are the Paley graph of order
nine and certain graphs corresponding to regular symmetric Hadamard matrices
with constant diagonal. Next we prove that if uniform mixing occurs on a
bipartite graph X with n vertices, then n is divisible by four. We also prove
that if X is bipartite and regular, then n is the sum of two integer squares.
Our work on bipartite graphs implies that uniform mixing does not occur on
C_{2m} for m >= 3. Using a result of Haagerup, we show that uniform mixing does
not occur on C_p for any prime p such that p >= 5. In contrast to this result,
we see that epsilon-uniform mixing occurs on C_p for all primes p.Comment: 23 page
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