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

Xenon/CT Blood Flow Mapping of the Liver using Multidetector-Row Computed Tomography: Compensation of Respiratory Misregistration by Volume Data

コンピュータ断層撮影(CT)検査における非放射性キセノン血流動態検査(キセノンCT検査)は脳血流測定方法として古くから用いられている手法であり,躯幹部臓器への応用に関してもいくつかの報告が見られる。しかしキセノンCT検査では経時的に撮像を繰り返すため,肺や上腹部領域において,呼吸深度の差異に起因する臓器の位置の不一致が血液マップ作成上の問題点として指摘されている。本研究では多列検出器型CT装置(MDCT)のらせん走査を用いた呼吸性移動の補正を目的とし,ファントム実験による撮像条件の検討に引き続き,臨床例への応用と妥当性の検証を行った。ファントム実験では,MDCTにおける軸位撮像のCT値の変動は単検出器型CTによる軸位撮像と比較して大きかった。一方,らせん走査ではCTの変動は小さかったが,大きな寝台移動速度ではアーチファクトが著明であった。管電圧に関しては80kV と120kVでCT値の変動に有意な差は認められなかった。従って臨床応用では管電圧80kV,小さな寝台移動速度によるらせん走査を採用することとした。臨床応用では,7別に対してらせん走査による呼仮性移動の補正の有用性を検討した。全例において呼仮性移動が認められたが,いずれの症例についても,らせん走査により得られた再構成画像を使用することにより呼仮性移動の補正が可能であった。算出された肝血流マップ上の有効なピクセル数は,呼仮性移動の補正を用いることで補正のない場合と比較して1.1-46.0%増加し,特に肝辺縁部でのピクセル数の増加が著明であった。MDCTは高いCT値の再現性を有し,連続したデータ収集により呼吸性移動の補正が可能であることから,キセノンCT検査による肝局所血流測定法の有用性を向上する可能性が示唆された。For improvement in the accuracy of xenon/CT blood flow mapping of the liver, this study was conducted to assess whether volume-data obtained by multidetector-row helical CT could compensate for the slice misregistration caused by inconstant breath depth. Using imaging phantoms scanned on both multidetector-row and single-detector CT, suitable parameters were decided. In the phantom studies, axial scan with multidetector-row CT showed significantly greater variance of CT value than that of SDCT. On multidetector-row CT, variances of the CT values maintained in low (80kV) tube voltage, which is advantageous for detecting subtle enhancement of the liver. Images reconstructed with 10-mm thickness showed smaller variance than those of 5-mm thickness on low-pitch helical scan. Remarkable helical artifacts were seen on the high-pitch helical scan. Following the phantom studies, 7 examinees were scanned using xenon/CT with a predetermined scan protocol (80 kV, collimation 5 mm, thickness 10 mm, low-pitch helical scan). In all cases, slice compensation was necessary and was successfully performed. The number of pixels which constituted blood flow map increased after the compensation. No patients showed any significant adverse effects. In conclusion, multidetector-row helical CT has the potential for providing accurate quantification of xenon/CT blood flow mapping of liver by compensating for respiratory misregistration

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

Bifurcation analysis of a simple 3D oscillator and chaos synchronization of its coupled systems

Tamaševičius et al. proposed a simple 3d chaotic oscillator for educational purpose. In fact the oscillator can be implemented very easily and it shows typical bifurcation scenario so that it is a suitable training object for introductory education for students. However, as far as we know, no concrete studies on bifurcations or applications on this oscillator have been investigated. In this paper, we make a thorough investigation on local bifurcations of periodic solutions in this oscillator by using a shooting method. Based on results of the analysis, we study chaos synchronization phenomena in diffusively coupled oscillators. Both bifurcation sets of periodic solutions and parameter regions of in-phase synchronized solutions are revealed. An experimental laboratory of chaos synchronization is also demonstrated