In this paper, we investigate some relations between Bernoulli numbers and FrobeniusEuler numbers, and we study the values for p-adic l-function

We obtain the general solutions of the cubic functional equation 3[g(x+y)+g(x−y)+6g(x)]=2g(2x+y)+2g(2x−y)+g(−x−y)+g(−x+y)+6g(−x) and the Jensen-quadratic functional equation f((x+y)/2,z+w)+f((x+y)/2,z−w)=f(x,z)+f(x,w)+f(y,z)+f(y,w).

We obtain the general solution and the stability of the-variable quadratic functional equation The quadratic form is a solution of the given functional equation.

We propose a simple and intuitive method to derive the exact convergence rate of global $L_2-norm$ error for strong numerical approximation of stochastic differential equations the result of which has been reported by Hofmann and $M\text{"}uller-Gronbach\left(2004\right)$. We conclude that any strong numerical scheme of order $\gamma >1/2$ has the same optimal convergence rate for this error. The method clearly reveals the structure of global $L_2-norm$ error and is similarly applicable for evaluating the convergence rate of global uniform approximations

In this paper, we obtain the general solution and the stability of the 2-variable quadratic functional equationf(x+y,z+w)+f(x−y,z−w)=2f(x,z)+2f(y,w). The quadratic form f(x,y)=ax2+bxy+cy2 is a solution of the above functional equation.？

We introduce the inverse shadowing property of geometric Lorenz flows and prove that the geometric Lorenz flows do not have the inverse shadowing property.

We report that the prefactor for the migration of neutral indium has been successfully determined
via an ab-initio calculation in combination with classical rate theory for the first time.
Our theoretical approach starts with a search for the saddle points of indium in the transition
state from the initial state to final state by using a climbing image nudged elastic band (CINEB)
method, which enables us to extract the migration energy, 0.79 eV, for indium. Thereafter, we
obtain the prefactor of neutral indium by calculating the effective frequencies in the initial and the
final states by using a dynamical matrix method.

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.