10,171 research outputs found
On the X-rays of permutations
The X-ray of a permutation is defined as the sequence of antidiagonal sums in
the associated permutation matrix. X-rays of permutation are interesting in the
context of Discrete Tomography since many types of integral matrices can be
written as linear combinations of permutation matrices. This paper is an
invitation to the study of X-rays of permutations from a combinatorial point of
view. We present connections between these objects and nondecreasing
differences of permutations, zero-sum arrays, decomposable permutations, score
sequences of tournaments, queens' problems and rooks' problems.Comment: 7 page
Exact sampling and counting for fixed-margin matrices
The uniform distribution on matrices with specified row and column sums is
often a natural choice of null model when testing for structure in two-way
tables (binary or nonnegative integer). Due to the difficulty of sampling from
this distribution, many approximate methods have been developed. We will show
that by exploiting certain symmetries, exact sampling and counting is in fact
possible in many nontrivial real-world cases. We illustrate with real datasets
including ecological co-occurrence matrices and contingency tables.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1131 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org). arXiv admin note: text overlap with
arXiv:1104.032
A direct link between the quantum-mechanical and semiclassical determination of scattering resonances
We investigate the scattering of a point particle from n non-overlapping,
disconnected hard disks which are fixed in the two-dimensional plane and study
the connection between the spectral properties of the quantum-mechanical
scattering matrix and its semiclassical equivalent based on the semiclassical
zeta function of Gutzwiller and Voros. We rewrite the determinant of the
scattering matrix in such a way that it separates into the product of n
determinants of 1-disk scattering matrices - representing the incoherent part
of the scattering from the n disk system - and the ratio of two mutually
complex conjugate determinants of the genuine multi-scattering kernel, M, which
is of Korringa-Kohn-Rostoker-type and represents the coherent multi-disk aspect
of the n-disk scattering. Our result is well-defined at every step of the
calculation, as the on-shell T-matrix and the kernel M-1 are shown to be
trace-class. We stress that the cumulant expansion (which defines the
determinant over an infinite, but trace class matrix) imposes the curvature
regularization scheme to the Gutzwiller-Voros zeta function and thus leads to a
new, well-defined and direct derivation of the semiclassical spectral function.
We show that unitarity is preserved even at the semiclassical level.Comment: 23 pages, latex with IOP journal preprint style, no figures; final
version - considerably shortene
Correlated fractal percolation and the Palis conjecture
Let F1 and F2 be independent copies of correlated fractal percolation, with
Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question:
does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 will
contain an interval? The well known Palis conjecture states that `generically'
this should be true. Recent work by Kuijvenhoven and the first author
(arXiv:0811.0525) on random Cantor sets can not answer this question as their
condition on the joint survival distributions of the generating process is not
satisfied by correlated fractal percolation. We develop a new condition which
permits us to solve the problem, and we prove that the condition of
(arXiv:0811.0525) implies our condition. Independently of this we give a
solution to the critical case, yielding that a strong version of the Palis
conjecture holds for fractal percolation and correlated fractal percolation:
the algebraic difference contains an interval almost surely if and only if the
sum of the Hausdorff dimensions of the random Cantor sets exceeds one.Comment: 22 page
- …