A practical preparation of highly versatile N-acylpyrroles from 2,4,4-trimethoxybutan-1-amine

A novel method for the preparation of N-acylpyrrole is described. The method involves condensation of carboxylic acids with 2,4,4-trimethoxybutan-1-amine, followed by acid-mediated cyclization to form the pyrrole ring. The preparative procedure is highly tolerant of a variety of functional groups.     

Embedding of trees in euclidean spaces

It is proved that for any treeT the vertices ofT can be placed on the surface of a sphere inR ³ in such a way that adjacent vertices have distance 1 and nonadjacent vertices have distances less than 1.     

Every Graph Is an Integral Distance Graph in the Plane

We prove that every finite simple graph can be drawn in the plane so that any two vertices have an integral distance if and only if they are adjacent. The proof is constructive.     

On Graver's conjecture concerning the rigidity problem of graphs

We show that the four-dimensional case of Graver's conjecture is not true.     

A pinhole-plate ultrasonic atomizer

A new pinhole-plate ultrasonic atomizer has been proposed and developed. A pinhole-plate having one or more pinholes is joined to a cavity filled with liquid and is vibrated by a circular ring piezoelectric element bonded onto it. The vibrating plate generates ultrasonic waves only in the vicinity of the pinholes to jet and atomize the liquid from them. The atomizer is very simple and compact, and extremely low in power consumption.     

Continuous relaxation for discrete DC programming

Discrete DC programming with convex extensible functions is studied. A natural approach for this problem is a continuous relaxation that extends the problem to a continuous domain and applies the algorithm in continuous DC programming. By employing a special form of continuous relaxation, which is named “lin-vex extension,” the produced optimal solution of the extended continuous relaxation coincides with the solution of the original discrete problem. The proposed method is demonstrated for the degree-concentrated spanning tree problem, the unfair flow problem, and the correlated knapsack problem.     

A single-shot game of multi-period inspection

This paper deals with an inspection game of Customs and a smuggler during some days. Customs has two options of patrolling or not. The smuggler can take two strategies of shipping its cargo of contraband or not. Two players have several opportunities to take an action during a limited number of days but they may discard some of the opportunities. When the smuggling coincides with the patrol, there occurs one of three events: the capture of the smuggler by Customs, a success of the smuggling and nothing new. If the smuggler is captured or no time remains to complete the game, the game ends. There have been many studies on the inspection game so far by the multi-stage game model, where both players at a stage know players’ strategies taken at the previous stage. In this paper, we consider a two-person zero-sum single-shot game, where the game proceeds through multiple periods but both players do not know any strategies taken by their opponents on the process of the game. We apply dynamic programming to the game to exhaust all equilibrium points on a strategy space of player. We also clarify the characteristics of optimal strategies of players by some numerical examples.