480 research outputs found
Tiling a Rectangle with the Fewest Squares
AbstractWe show that a square-tiling of ap×qrectangle, wherepandqare relatively prime integers, has at least log2psquares. Ifq>pwe construct a square-tiling with less thanq/p+Clogpsquares of integer size, for some universal constantC
Flip invariance for domino tilings of three-dimensional regions with two floors
We investigate tilings of cubiculated regions with two simply connected
floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected
component for such tilings, and provide an algebraic invariant that "almost"
characterizes the flip connected components of such regions, in a sense that we
discuss in the paper. We also introduce a new local move, the trit, which,
together with the flip, connects the space of domino tilings when the two
floors are identical.Comment: 33 pages, 34 figures, 2 tables. We updated the reference lis
Counting Berg partitions
We call a Markov partition of a two dimensional hyperbolic toral automorphism
a Berg partition if it contains just two rectangles. We describe all Berg
partitions for a given hyperbolic toral automorphism. In particular there are
exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same
connectivity matrix (k, l, m, n)
An Improved Search Method for Gravitational Ringing of Black Holes
A black hole has characteristic quasi-normal modes that will be excited when
it is formed or when the geometry is perturbed. The state of a black hole when
the quasi-normal modes are excited is called the gravitational ringing, and
detections of it will be a direct confirmation of the existence of black holes.
To detect it, a method based on matched filtering needs to be developed.
Generically, matched filtering requires a large number of templates, because
one has to ensure a proper match of a real gravitational wave with one of
template waveforms to keep the detection efficiency as high as possible. On the
other hand, the number of templates must be kept as small as possible under
limited computational costs. In our previous paper, assuming that the
gravitational ringing is dominated by the least-damped (fundamental) mode with
the least imaginary part of frequency, we constructed an efficient method for
tiling the template space. However, the dependence of the template space metric
on the initial phase of a wave was not taken into account. This dependence
arises because of an unavoidable mismatch between the parameters of a signal
waveform and those given discretely in the template space. In this paper, we
properly take this dependence into account and present an improved, efficient
search method for gravitational ringing of black holes.Comment: 19 pages, 9 figure
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