[4] **viXra:1202.0071 [pdf]**
*submitted on 2012-02-21 22:27:58*

**Authors:** Choe Ryong Gil

**Comments:** 23 pages

In this paper we have introduced a new topology and a convergence in Banach space, which would be called a L-topology and a L-convergence. It is similar to the weak topology and weak convergence, but there are some essential differences. For example, the L-topology is stronger than weak topology, but weaker than the strong one. On the basis of the notion, we have considered the problem on the separability and reflexibility of Lipschitz (Lip-) dual space. Furthermore, we have introduced a new topology of Lip-dual space, which is similar to the weak* (W*-) topology of linear dual of Banach space and would be called an L*-topology, and we have considered the problems on the metrizability of L*-topology and on the L*-separability of Lip-dual space, too.

**Category:** Functions and Analysis

[3] **viXra:1202.0069 [pdf]**
*replaced on 2012-02-23 03:28:33*

**Authors:** Choe Ryong Gil, Kim Myong Il

**Comments:** 17 pages

In this paper we have introduced a new concept on the convergence of a sequence of the nonlinear Lipschitz (Lip-) functionals, which would be called an L*-convergence, and we have considered its applications in Banach spaces. This convergence is very similar to the weak* (W*-) convergence of the sequence of the bounded linear functionals, but there are some differences. By the L*-convergence, we have considered the problem on the relations of the compactness between the Lip-operator and its Lip-dual operator, and we have obtained the mean ergodic theorems for the Lip-operator.

**Category:** Functions and Analysis

[2] **viXra:1202.0060 [pdf]**
*submitted on 2012-02-19 02:03:52*

**Authors:** Choe Ryong Gil

**Comments:** 17 pages

In this paper we have obtained a new theorem that a nonlinear Lipschitz (Lip-) functional defined on the closed subset of Banach spaces can be extended to the whole space with Lip-continuity and maintenance of Lip-constant, which would be called an extension theorem (ET). This theorem is a generalization to the Lip-functional of the famous Hahn-Banach theorem on the bounded linear functional. By the ET, we have completely solved the open problem on the relation of the invertibility between the Lip-operator and its Lip-dual operator.

**Category:** Functions and Analysis

[1] **viXra:1202.0015 [pdf]**
*submitted on 2012-02-06 15:20:56*

**Authors:** Richard J. Mathar

**Comments:** 12 Pages. Includes complete C++ source listing.

The volume inside intersecting spheres may be computed by a standard method which computes
a surface integral over all visible sections of the spheres. If the visible sections are divided in simple
zonal sections, the individual contribution by each zone follows from basic analysis. We implement
this within a semi-numerical program which marks the zones individually as visible or invisible.

**Category:** Functions and Analysis