In this paper,a new basic method of constructing smooth production functions isgiven,and many ordinary production functions are drawn. It is obvious thatsome more production functions can be obtained from the method.
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
In this paper, we investigate the continuous dependence of solutions of the functional dif-ferential equation with infinite delay x'(t) = f(t, x_t) on initial functions. Endowing the phasespare a g-norm as well as...In this paper, we investigate the continuous dependence of solutions of the functional dif-ferential equation with infinite delay x'(t) = f(t, x_t) on initial functions. Endowing the phasespare a g-norm as well as a supremum norm. we show that if the equation satifies a mild fadingmemory dondition, then the continuity of f in respect to the topology induced by the supremumnorm can yield the continuity of solutions of the equation in respect to the topology induced bythe g-norm which is stronger than the ahead one.展开更多
文摘In this paper,a new basic method of constructing smooth production functions isgiven,and many ordinary production functions are drawn. It is obvious thatsome more production functions can be obtained from the method.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
基金This research was supported in part by an NSF grant with number NSY-DMS-8521408. On leave from South China Normal University, Guangshou, PRC. This research was supported in part by the National Science Foundation of PRC
文摘In this paper, we investigate the continuous dependence of solutions of the functional dif-ferential equation with infinite delay x'(t) = f(t, x_t) on initial functions. Endowing the phasespare a g-norm as well as a supremum norm. we show that if the equation satifies a mild fadingmemory dondition, then the continuity of f in respect to the topology induced by the supremumnorm can yield the continuity of solutions of the equation in respect to the topology induced bythe g-norm which is stronger than the ahead one.
