In this paper,a new proof of the stolz theorem is given by the infinite triangular matrix.

The main purpose of this paper is to study in full generality combinatorial inverse relations of arbitrary infinite lower-triangular matrices, namely, a pair of such matrices (F_(n,k)) and (G_(n,k)) thatGenerally speaking, our method relies on an expression in terms of determinants for each n-row and k-column entry G_(n,k) with the assumption that (F_(n,k)) is known.

By introducing assistant infinite triangle and Delaunay triangulation, this new algorithm assures that its running result is entirely Delaunay triangulation and overcomes Bowyer s algorithmic limitations.

The boundedness of the set of infinite matrix transformations from convergence-free space to sequence spaces is studied,and a general form of it is deducted.

Let λ and μ be sequence space and have both the signed-weak gliding hump property,(λ,μ) be the algebra of the infinite matrix operators which transform λ to μ.

This paper introduces the research development of the important effect algebra in quantum mechanics,and points out that it is of great significance to the establishment of mathematical foundation of quantum mechanics by making use of infinite matrix theory to study its convergent theory and invariants.

A necessary and sufficient condition is obtained that the continuous spectrum of upper-triangular-type infinite dimensional Hamiltonian operators is empty and its criterions are given.