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$^{\natural}$-convexity, and that it characterizes submodularity if we restrict the endpoints of the line segments to be at $\ell_\infty$-distance one. By considering discrete midpoint convexity for all pairs at $\ell_\infty$-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$^{\natural}$-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$^{\natural}$-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 $\sim$ 12' (or $160h_{70}^{-1}$ kpc; $H_0 = 70 h_{70}$ km s$^{-1}$Mpc$^{-1}$) in radius of the cluster center. The azimuthally averaged surface brightness profile from the ROSAT PSPC exhibits a central excess above an isothermal $\beta$ 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 $\sim 1.5$ 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