Tadashi Shima

On a spectral analysis for the Sierpinski gasket

A complete description of the eigenvalues of the Laplacian on the finite Sierpinski gasket is presented. We then demonstrate highly oscillatory behaviours of the distribution function of the eigenvalues, the integrated density of states (for the infinite gasket) and the spectrum of the Laplacian on the infinite gasket. The method has two ingredients: the decimation method in calculating eigenvalues due to Rammal and Toulouse and a simple description of the Dirichlet form associated with the Laplacian.

On eigenvalue problems for Laplacians on P.C.F. self-similar sets

AbstractWe formulate and study a strong harmonic structure under which eigenvalues of the Laplacian on a p.c.f. self-similar set are completely determined according to the dynamical system generated by a rational function. We then show that, with some additional assumptions, the eigenvalue counting function ρ(λ) behaves so wildly that ρ(λ) does not vary regularly, and the ratio

On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals

\rho (\lambda )/\lambda ^{d_s /2}

Infinite-horizon multi-leader-follower incentive stackelberg games for linear stochastic systems with H ∞ constraint

is bounded but non-convergent as λϖ∞, whereds is the spectral dimension of the p.c.f. self-similar set.

H ∞ constraint incentive Stackelberg game for discrete-time stochastic systems

We work with increasing finite setsVm called pre-gaskets approximating the finite Sierpinski gasket located inRN−1 (N ≥ 3). The eigenvalues of the discrete Laplacian onVm under the Dirichlet and Neumann boundary conditions are completely determined using the decimation method due to Rammal.

H∞ Constraint Pareto Optimal Strategy for Stochastic LPV Systems

We consider a general finitely ramified fractal set called a nested fractal which is determined byN number of similitudes. Basic properties of the integrated density of statesN(x) for the discrete Laplacian on the associated nested prefractal are investigated. In particulardN is shown to be purely discontinuous ifM<N, whereM is the number of branches of the inverse of the rational function involved in the spectral decimation method due to Rammal-Toulouse. Sierpinski gaskets and the modified Koch curve are special examples.

Photoneutron cross sections for neodymium isotopes: Toward a unified understanding of (γ,n) and (n,γ) reactions in the rare earth region

An incentive Stackelberg game for a class of Markov jump linear stochastic systems with multiple leaders and followers is investigated in this letter. An incentive structure is developed that allows the leader’s Nash equilibrium to be achieved. In the game, the followers are assumed to behave in two ways under the leader’s incentive strategy set. One involves achieving a Pareto-optimal solution, and the other involves achieving Nash equilibrium. Consequently, it can be verified that irrespective of how the followers behave, they can be induced to achieve the leader’s Nash equilibrium by using a corresponding incentive strategy set. It is shown that the incentive strategy set can be obtained by solving the cross-coupled stochastic algebraic Riccati-type equations. As another important contribution, a novel concept of incentive possibility is proposed for a special case. In order to demonstrate the effectiveness of the proposed scheme, a numerical example is solved.

Large deviations and related LIL's for Brownian motions on nested fractals

In this paper, an infinite-horizon incentive Stackelberg game with multiple leaders and multiple followers is investigated for a class of linear stochastic systems with H∞ constraint. In this game, an incentive structure is developed in such a way that leaders achieve Nash equilibrium attenuating the disturbance under H∞ constraint. Simultaneously, followers achieve their Nash equilibrium ensuring the incentive Stackelberg strategies of the leaders while the worst-case disturbance is considered. In our research, it is shown that by solving some cross-coupled stochastic algebraic Riccati equations (CCSAREs) and matrix algebraic equations (MAEs) the incentive Stackelberg strategy set can be obtained. Finally, to demonstrate the effectiveness of our proposed scheme, a numerical example is solved.