A recursive method based on successive computations of perimeters of inscribed regular polygons for estimating π is formulated by employing the Pythagorean theorem alone without resorting to any trigonometric calcula...A recursive method based on successive computations of perimeters of inscribed regular polygons for estimating π is formulated by employing the Pythagorean theorem alone without resorting to any trigonometric calculations. The approach is classical but the formulation of coupled recursion relations is new. Further, use of infinite series for computing π is explored by an improved version of Leibniz’s series expansion. Finally, some remarks with reference to π are made on a relatively recently rediscovered Sumerian tablet depicting geometric figures.展开更多
This paper is concerned with the stochastically stability for the m-dimensional linear stochastic differential equations with respect to fractional Brownian motion （FBM） with Hurst parameter H ∈ （1/2, 1）. On the ...This paper is concerned with the stochastically stability for the m-dimensional linear stochastic differential equations with respect to fractional Brownian motion （FBM） with Hurst parameter H ∈ （1/2, 1）. On the basis of the pioneering work of Duncan and Hu, a Ito＇s formula is given. An improved derivative operator to Lyapunov functions is constructed, and the sufficient conditions for the stochastically stability of linear stochastic differential equations driven by FBM are established. These extend the stochastic Lyapunov stability theories.展开更多
文摘A recursive method based on successive computations of perimeters of inscribed regular polygons for estimating π is formulated by employing the Pythagorean theorem alone without resorting to any trigonometric calculations. The approach is classical but the formulation of coupled recursion relations is new. Further, use of infinite series for computing π is explored by an improved version of Leibniz’s series expansion. Finally, some remarks with reference to π are made on a relatively recently rediscovered Sumerian tablet depicting geometric figures.
基金Natural Science Foundation of Shanghai,China(No.07ZR14002)
文摘This paper is concerned with the stochastically stability for the m-dimensional linear stochastic differential equations with respect to fractional Brownian motion （FBM） with Hurst parameter H ∈ （1/2, 1）. On the basis of the pioneering work of Duncan and Hu, a Ito＇s formula is given. An improved derivative operator to Lyapunov functions is constructed, and the sufficient conditions for the stochastically stability of linear stochastic differential equations driven by FBM are established. These extend the stochastic Lyapunov stability theories.
