573 research outputs found
Discrete Midpoint Convexity
For a function defined on a convex set in a Euclidean space, midpoint
convexity is the property requiring that the value of the function at the
midpoint of any line segment is not greater than the average of its values at
the endpoints of the line segment. Midpoint convexity is a well-known
characterization of ordinary convexity under very mild assumptions. For a
function defined on the integer lattice, we consider the analogous notion of
discrete midpoint convexity, a discrete version of midpoint convexity where the
value of the function at the (possibly noninteger) midpoint is replaced by the
average of the function values at the integer round-up and round-down of the
midpoint. It is known that discrete midpoint convexity on all line segments
with integer endpoints characterizes L-convexity, and that it
characterizes submodularity if we restrict the endpoints of the line segments
to be at -distance one. By considering discrete midpoint convexity
for all pairs at -distance equal to two or not smaller than two,
we identify new classes of discrete convex functions, called local and global
discrete midpoint convex functions, which are strictly between the classes of
L-convex and integrally convex functions, and are shown to be
stable under scaling and addition. Furthermore, a proximity theorem, with the
same small proximity bound as that for L-convex functions, is
established for discrete midpoint convex functions. Relevant examples of
classes of local and global discrete midpoint convex functions are provided.Comment: 39 pages, 6 figures, to appear in Mathematics of Operations Researc
Recent Progress on Integrally Convex Functions
Integrally convex functions constitute a fundamental function class in
discrete convex analysis, including M-convex functions, L-convex functions, and
many others. This paper aims at a rather comprehensive survey of recent results
on integrally convex functions with some new technical results. Topics covered
in this paper include characterizations of integral convex sets and functions,
operations on integral convex sets and functions, optimality criteria for
minimization with a proximity-scaling algorithm, integral biconjugacy, and the
discrete Fenchel duality. While the theory of M-convex and L-convex functions
has been built upon fundamental results on matroids and submodular functions,
developing the theory of integrally convex functions requires more general and
basic tools such as the Fourier-Motzkin elimination.Comment: 50 page
Shapley-Folkman-type Theorem for Integrally Convex Sets
The Shapley-Folkman theorem is a statement about the Minkowski sum of
(non-convex) sets, expressing the closeness of the Minkowski sum to convexity
in a quantitative manner. This paper establishes similar theorems for
integrally convex sets and M-natural-convex sets, which are major classes of
discrete convex sets in discrete convex analysis.Comment: 13 page
Scaling and Proximity Properties of Integrally Convex Functions
In discrete convex analysis, the scaling and proximity properties for the class of L^natural-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n leq 2, while a proximity theorem can be established for any n, but only with an exponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discretely convex function in one dimension to the case of integrally convex functions in two dimensions. Furthermore, we identified a new class of discrete convex functions, called directed integrally convex functions, which is strictly between the classes of L^natural -convex and integrally convex functions but enjoys the same scaling and proximity properties that hold for L^natural -convex functions
Installation of the superconducting gravimeter CT(#043) at Syowa Station, Antarctica
During the wintering period of the 44th Japanese Antarctic Research Expedition (JARE-44: February 2003 to January 2004), a new superconducting gravimeter CT(#043) with a cryocooler was installed and tested to replace the former TT70(#016) at Syowa Station, Antarctica. The CT(#043) has design sensitivity of 1nGal (1×10^(-11)m/s^2) to study the Earth\u27s dynamics in tidal and longer-period bands. A new type of diaphragm was used to effectively isolate the vibration from the refrigerator cold-head and to prevent solid air contamination from entering the Dewar. A real-time remote monitoring system including a Web camera for diagnostics from Japan has also been installed
X-ray Measurements of the Gravitational Potential Profile in the Central Region of the Abell 1060 Cluster of Galaxies
X-ray spectral and imaging data from ASCA and ROSAT were used to measure the
total mass profile in the central region of Abell 1060, a nearby and relatively
poor cluster of galaxies. The ASCA X-ray spectra, after correcting for the
spatial response of the X-ray telescope, show an isothermal distribution of the
intra-cluster medium (ICM) within at least 12' (or kpc;
km sMpc) in radius of the cluster center. The
azimuthally averaged surface brightness profile from the ROSAT PSPC exhibits a
central excess above an isothermal model. The ring-sorted ASCA GIS
spectra and the radial surface brightness distribution from the ROSAT PSPC were
simultaneously utilized to constrain the gravitational potential profile. Some
analytic models of the total mass density profile were examined. The ICM
density profile was also specified by analytic forms. The ICM temperature
distribution was constrained to satisfy the hydrostatic equilibrium, and to be
consistent with the data. Then, the total mass distribution was found to be
described better by the universal dark halo profile proposed by Navarro, Frenk,
and White (1996;1997) than by a King-type model with a flat density core. A
profile with a central cusp together with a logarithmic radial slope of was also consistent with the data. Discussions are made concerning the
estimated dark matter distribution around the cluster center.Comment: 32 pages. Accepted: ApJ 2000, 35 pages, Title was correcte
E-Healthcare at an Experimental Welfare Techno House in Japan
An automated monitoring system for home health care has been designed for an experimental house in Japan called the Welfare Techno House (WTH). Automated electrocardiogram (ECG) measurements can be taken while in bed, in the bathtub, and on the toilet, without the subject’s awareness, and without using body surface electrodes. In order to evaluate this automated health monitoring system, overnight measurements were performed to monitor health status during the daily lives of both young and elderly subjects
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