It is well known that for almost all real number x, the geometric mean of the first n digits di（x） in the Lüroth expansion of x converges to a number K0 as n→∞. On the other hand, for almost all x, the arithm...It is well known that for almost all real number x, the geometric mean of the first n digits di（x） in the Lüroth expansion of x converges to a number K0 as n→∞. On the other hand, for almost all x, the arithmetric mean of the first n Lüroth expansion digits di（x） approaches infinity as n→∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin＇s inequalities, relating the 1/k-th powers of the k-th elementary symmetric means of n numbers for 1≤k≤n. In this paper, we investigate what happens to the means of Lüroth expansion digits in the limit as one moves f（n） steps away from either extreme. We prove sufficient conditions on f（n） to ensure divergence when one moves away from the arithmetic mean and convergence when one moves f（n） steps away from geometric mean.展开更多
In this paper I propose a method for founding solutions of Navier-Stokes equations. Purpose of the research is to solve equations giving form to relations between pressure, velocity and stream. Starting from the fact ...In this paper I propose a method for founding solutions of Navier-Stokes equations. Purpose of the research is to solve equations giving form to relations between pressure, velocity and stream. Starting from the fact we do not know the form of functions we give a general representation in Maclaurin Series and prove that with reasonable values of parameters, representation holds and therefore has meaning in continuum. Then we solve the system of equations with respect to the pressure and match equations relation between parameters: matches of equations are possible because of the physical dimensions of equations. Then values of Continuity Equation are verified. The result is a polynomial finite and that coincides with the function in continuum, or is anyway one of its representation. The result under hydrostatic condition returns Stevino formula.展开更多
In this paper, multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with ...In this paper, multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with their Euler-Maclaurin asymptotic expansions of the errors. The presented method is suit for solving multidimensional and singular integrals by comparing with Gauss quadrature rule. The error asymptotic expansions show that the convergence order of the initial quadrature rules is , where . The order of accuracy can reach to by using extrapolation and splitting extrapolation, where h0 is the maximum mesh width. Some numerical examples are constructed to show the efficiency of the method.展开更多
基金Supported by the National Natural Science Foundation of China(11271148)
文摘It is well known that for almost all real number x, the geometric mean of the first n digits di（x） in the Lüroth expansion of x converges to a number K0 as n→∞. On the other hand, for almost all x, the arithmetric mean of the first n Lüroth expansion digits di（x） approaches infinity as n→∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin＇s inequalities, relating the 1/k-th powers of the k-th elementary symmetric means of n numbers for 1≤k≤n. In this paper, we investigate what happens to the means of Lüroth expansion digits in the limit as one moves f（n） steps away from either extreme. We prove sufficient conditions on f（n） to ensure divergence when one moves away from the arithmetic mean and convergence when one moves f（n） steps away from geometric mean.
文摘In this paper I propose a method for founding solutions of Navier-Stokes equations. Purpose of the research is to solve equations giving form to relations between pressure, velocity and stream. Starting from the fact we do not know the form of functions we give a general representation in Maclaurin Series and prove that with reasonable values of parameters, representation holds and therefore has meaning in continuum. Then we solve the system of equations with respect to the pressure and match equations relation between parameters: matches of equations are possible because of the physical dimensions of equations. Then values of Continuity Equation are verified. The result is a polynomial finite and that coincides with the function in continuum, or is anyway one of its representation. The result under hydrostatic condition returns Stevino formula.
文摘In this paper, multidimensional weakly singular integrals are solved by using rectangular quadrature rules which base on quadrature rules of one dimensional weakly singular and multidimensional regular integrals with their Euler-Maclaurin asymptotic expansions of the errors. The presented method is suit for solving multidimensional and singular integrals by comparing with Gauss quadrature rule. The error asymptotic expansions show that the convergence order of the initial quadrature rules is , where . The order of accuracy can reach to by using extrapolation and splitting extrapolation, where h0 is the maximum mesh width. Some numerical examples are constructed to show the efficiency of the method.
