151 research outputs found

### Local Warming

Using 55 years of daily average temperatures from a local weather station, I
made a least-absolute-deviations (LAD) regression model that accounts for three
effects: seasonal variations, the 11-year solar cycle, and a linear trend. The
model was formulated as a linear programming problem and solved using widely
available optimization software. The solution indicates that temperatures have
gone up by about 2 degrees Fahrenheit over the 55 years covered by the data. It
also correctly identifies the known phase of the solar cycle; i.e., the date of
the last solar minimum. It turns out that the maximum slope of the solar cycle
sinusoid in the regression model is about the same size as the slope produced
by the linear trend. The fact that the solar cycle was correctly extracted by
the model is a strong indicator that effects of this size, in particular the
slope of the linear trend, can be accurately determined from the 55 years of
data analyzed.
The main purpose for doing this analysis is to demonstrate that it is easy to
find and analyze archived temperature data for oneself. In particular, this
problem makes a good class project for upper-level undergraduate courses in
optimization or in statistics.
It is worth noting that a similar least-squares model failed to characterize
the solar cycle correctly and hence even though it too indicates that
temperatures have been rising locally, one can be less confident in this
result.
The paper ends with a section presenting similar results from a few thousand
sites distributed world-wide, some results from a modification of the model
that includes both temperature and humidity, as well as a number of suggestions
for future work and/or ideas for enhancements that could be used as classroom
projects.Comment: 12 pages, 5 figures, to appear in SIAM Revie

### Splitting dense columns in sparse linear systems

AbstractWe consider systems of equations of the form AATx = b, where A is a sparse matrix having a small number of columns which are much denser than the other columns. These dense columns in A cause AAT to be very (or even completely) dense, which greatly limits the effectiveness of sparse-matrix techniques for directly solving the above system of equations. In the literature on interior-point methods for linear programming, the usual technique for dealing with this problem is to split A into a sparse part S and a dense part D, A = [S D], and to solve systems involving AAT in terms of the solution of systems involving SST using either conjugate-gradient techniques or the Sherman-Morrison-Woodbury formula. This approach has the difficulty that SST is often rank-deficient even when AAT has full rank. In this paper we propose an alternative method which avoids the rank-deficiency problem and at the same time allows for the effective use of sparse-matrix techniques. The resulting algorithm is both efficient and robust

### New Orbits for the n-Body Problem

In this paper, we consider minimizing the action functional as a method for
numerically discovering periodic solutions to the $n$-body problem. With this
method, we can find a large number of choreographies and other more general
solutions. We show that most of the solutions found, including all but one of
the choreographies, are unstable. It appears to be much easier to find unstable
solutions to the $n$-body problem than stable ones. Simpler solutions are more
likely to be stable than exotic ones.Comment: 16 pages, 2 tables, 6 figure

### Two-Mirror Apodization for High-Contrast Imaging

Direct detection of extrasolar planets will require imaging systems capable
of unprecedented contrast. Apodized pupils provide an attractive way to achieve
such contrast but they are difficult, perhaps impossible, to manufacture to the
required tolerance and they absorb about 90% of the light in order to create
the apodization, which of course lengthens the exposure times needed for planet
detection. A recently proposed alternative is to use two mirrors to accomplish
the apodization. With such a system, no light is lost. In this paper, we
provide a careful mathematical analysis, using one dimensional mirrors, of the
on-axis and off-axis performance of such a two-mirror apodization system. There
appear to be advantages and disadvantages to this approach. In addition to not
losing any light, we show that the nonuniformity of the apodization implies an
extra magnification of off-axis sources and thereby makes it possible to build
a real system with about half the aperture that one would otherwise require or,
equivalently, resolve planets at about half the angular separation as one can
achieve with standard apodization. More specifically, ignoring pointing error
and stellar disk size, a planet at $1.7 \lambda/D$ ought to be at the edge of
detectability. However, we show that the non-zero size of a stellar disk pushes
the threshold for high-contrast so that a planet must be at least $2.5
\lambda/D$ from its star to be detectable. The off-axis analysis of
two-dimensional mirrors is left for future study.Comment: 21 pages, 7 figures. For author's webpage version see
http://www.orfe.princeton.edu/~rvdb/tex/piaa/ms.pdf This version has improved
figures and addresses comments of a refere

### Rectangular-Mask Coronagraphs for High-Contrast Imaging

We present yet another new family of masks for high-contrast imaging as
required for the to-be-built terrestrial planet finder space telescope. The
``best'' design involves a square entrance pupil having a 4-vane spider, a
square image-plane mask containing a plus-sign shaped occulter to block the
starlight inside 0.6 lambda/D, and a Lyot-plane mask consisting of a
rectangular array of rectangular opennings. Using Fraunhofer analysis, we show
that the optical system can image a planet 10^{-10} times as bright as an
on-axis star in four rectangular regions given by {(xi,zeta): 1.4 < | xi | <
20, 1.4 < | zeta | < 20}.
Since the design involves an image plane mask, pointing error is an issue. We
show that the design can tolerate pointing errors of about 0.05 lambda/D.
The inclusion of a 4-vane spider in the entrance pupil provides the
possibility to build a mirror-only on-axis system thereby greatly reducing the
negative effects of polarization.
Each of the masks can be realized as two masks consisting of stripes of
opaque material with the stripes oriented at right angles to each other. We
call these striped masks barcode masks. We show that it is sufficient for the
barcode masks by themselves to provide 10^{-5} contrast. This then guarantees
that the full system will provide the required 10^{-10} contrast.Comment: 12 pages, 5 figure

### Diffraction Analysis of 2-D Pupil Mapping for High-Contrast Imaging

Pupil-mapping is a technique whereby a uniformly-illuminated input pupil,
such as from starlight, can be mapped into a non-uniformly illuminated exit
pupil, such that the image formed from this pupil will have suppressed
sidelobes, many orders of magnitude weaker than classical Airy ring
intensities. Pupil mapping is therefore a candidate technique for coronagraphic
imaging of extrasolar planets around nearby stars. Unlike most other
high-contrast imaging techniques, pupil mapping is lossless and preserves the
full angular resolution of the collecting telescope. So, it could possibly give
the highest signal-to-noise ratio of any proposed single-telescope system for
detecting extrasolar planets. Prior analyses based on pupil-to-pupil
ray-tracing indicate that a planet fainter than 10^{-10} times its parent star,
and as close as about 2 lambda/D, should be detectable. In this paper, we
describe the results of careful diffraction analysis of pupil mapping systems.
These results reveal a serious unresolved issue. Namely, high-contrast pupil
mappings distribute light from very near the edge of the first pupil to a broad
area of the second pupil and this dramatically amplifies diffraction-based edge
effects resulting in a limiting attainable contrast of about 10^{-5}. We hope
that by identifying this problem others will provide a solution.Comment: 23 pages, 13 figures, also posted to
http://www.orfe.princeton.edu/~rvdb/tex/piaaFresnel/ms.pd

- â€¦