We study the asymptotic behavior in time of solutions to the Cauchy problems for the nonlinear Schrödinger equation with a critical power nonlinearity and the Hartree equation. We prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the Cauchy problem with small initial data. This estimate is very important for the proof of the existence of modified scattering states to the nonlinear Schrödinger equations with a critical nonlinearity and the Hartree equation. In order to derive the desired estimates we introduce a certain phase function since the previous methods, based solely on a priori estimates of the operator x + it ∇ acting on the solution without specifying any phase function, do not work for the critical case under consideration. The well-known nonexistence of the usual L 2 scattering states shows that our result is sharp.
Most cognitive approaches for explaining cooperation in Prisoner's Dilemma games include the view that many people believe that mutual cooperation is generally a gainful strategy to all parties and will cooperate when they think their partner cooperates. Proceeding along these lines, we argue that many participants treat a Prisoner's Dilemma game as an assurance game, and respond in a reciprocal manner to the choice or expected choice of their partner. We examine two bases for the expectation of a partner's cooperation in one-shot games: `general trust' and a `sense of control'. Further, we discuss why we expect general trust and a `sense of control'. Further, we discuss why we expect general trust and a sense of control to play different roles in societies, particularly in Japanese society and American society. Specifically, we test a general hypothesis that a sense of control plays a relatively more important role as a foundation for expectations in Japanese society and general trust plays the more important role in American society.
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