# Volume 4 No. 2

July 2001## Some Reduction Formulas and the Characterization of Singular and Nonsingular Directed Fans

**pp 34-38 (Vol 4 No. 2 2001)**

**Severino V. Gervacio and lsagani B. Jos
**

*Department of Mathematics*

*De La Salle University*

*2401 Taft Avenue, 1004 Manila*

**ABSTRACT**

A digmph is called singular or nonsingular according as its adjacency matrix is singular or nonsingular. An expression of the determinant of the adjacency matrix of a digraph in terms of the determinant of smaller digraphs obtained from the given one is called a reduction formula. Reduction formulas are established in this paper. Furthermore, . using these reduction formulas,· we determine which of the directed fans are singular. Moreover, we show that if a directed fan Fn is nonsingular, then the determinant of its adjacency matrix is (-1r)n.

## A Closed Integral Form for the Background Gauge Connection

**pp 17-21 (Vol 4 No. 2 2001)**

**Emmanuel T. Rodulfo and Joseph Ambrose G. Pagaran
**

*Physics Department, De La Salle University,*

*2401 Taft Avenue, Manila*

*November 2000*

**ABSTRACT**

By the appropriate use of the Pock-Schwinger gauge properties, we derive the closed integral form of the ‘point-split’ non-local background gauge connection originally expressed as a finite sum. This is achieved in the limit when the finite sum becomes infinite. With this closed integral form of the connection, we obtain the same exact results in the calculation of one-loop effective Lagrangian accommodating arbitrary orders of covariant field derivatives in quantum field theory of arbitrary spacetime dimensions and of arbitrary gauge group. Particularly, we display the one-loop effective Lagrangian for real boson fields up to 8 mass dimensions-the same result obtained when the connection was yet in the finite sum form.

## On Semi-Continuous Functions

**pp 22-25 (Vol 4 No. 2 2001)**

**Sergio R. Canoy, Jr. **

** Julius V. Benitez
**

*Department of Mathematics*

*College of Science and Mathematics*

*MSU-Iligan Institute of Technology*

*Iligan City 9200*

**ABSTRACT**

This paper gives equivalent statements of semi-continuity; a concept introduced by N. Levine [4] in 1 ?63. In particular, we give a characterization of semi-continuity which utilizes the concept of semi-closure of a set defined by one of the authors in [1]. Also, we characterize semi-continuity of maps _into the space of real numbers with the standard topology.

## McShane Integral of Functions With Values in a Ranked Countably Normed Space

**pp 26-33 (Vol 4 No. 2 2001)**

**Sergio R. Canoy, Jr.**

*Department of Mathematics *

*College of Science and Mathematics *

*MSU-Iligan Institute of Technology *

*Iligan City 9200*

**ABSTRACT**

We shall define McShane integral of functions with values in a complete mnked countably normed space. We shall relate this definition to the definition given by Gordon for Banach-valued functions /2}. Further, we give some simple properties of the integral and state its Cauchy criterion. As particular examples, we shall show that r-continuous functions and simple functions are McShane integrable.

## Inference for Long-memory Processes Using Local Lyapunov Exponents

**pp 5-16 (Vol 4 No. 2 2001)**

**Alex Gonzaga
**

*Department of Physical Sciences and Mathematics*

*University of the Philippines Manila*

*Padre Faura Street, Manila*

**ABSTRACT**

Local Lyapunov exponent (LLE) is a finite-time version of Lyapunov exponent, a tool for analyzing chaos. In this paper, we propose a new approach in analyzing long-memory time series. We apply LLE in the context of long-memory processes. The distribution function of the LLE for ARFIMA(p,d,q) process is derived, and an unbiased estimator and some uniformly most powerful tests for long-memory are proposed.

## A Closer Look on the Components Disconnected (n,k)-Cubes

**pp 1-4 (Vol 4 No. 2 2001)**

**Severi no V. Gervacio ^{1}**

*Department of Mathematics*

*De La Salle University*

*2401 Taft Avenue, 1004 Manila*

**Gloria A. Rosalejos ^{2}**

*Department of Mathematics*

*Mindanao State University*

*Marawi City*

**ABSTRACT**

This paper presents some properties of the components of the graph called the (n, k)-cube, written Q(n, k), whenever the said graph is disconnected.