Volume 2,  Issue 2, 2001

Article 18

IMPROVED INCLUSION-EXCLUSION INEQUALITIES FOR SIMPLEX AND ORTHANT ARRANGEMENTS

DANIEL Q. NAIMAN AND HENRY P. WYNN

DEPARTMENT OF MATHEMATICAL SCIENCES
JOHNS HOPKINS UNIVERSITY
BALTIMORE, MD 21218
 E-Mail: daniel.naiman@jhu.edu

DEPARTMENT OF STATISTICS
WARWICK UNIVERSITY
COVENTRY CV4 7AL
UNITED KINGDOM
E-Mail: hpw@stats.warwick.ac.uk

 

Received 6 September, 2000; accepted 25 January, 2001.
Communicated by: S.S. Dragomir

ABSTRACT.  Improved inclusion-exclusion inequalities for unions of sets are available wherein terms usually included in the alternating sum formula can be  left out. This is the case when a key abstract tube condition, can be shown to hold.  Since the abstract tube concept was introduced and refined by the authors, several examples have been identified, and key properties of abstract  tubes have been described. In particular, associated with an abstract tube is an inclusion-exclusion identity which can be truncated to give an inequality that is guaranteed to be at least as sharp as the inequality obtained by truncating the classical inclusion-exclusion identity. 
We present an abstract tube corresponding to an orthant arrangement where the inclusion-exclusion formula terms are obtained from the incidence structure of the boundary of the union of orthants. Thus, the construction of the abstract tube is similar to a construction for Euclidean balls using a Voronoi diagram. However, the proof of the abstract tube  property is a bit more subtle and involves consideration of abstract tubes for arrangements of simplicies, and an intricate geometric arguments based on their Voronoi diagrams.
Key words:
Orthant Arrangments, Inclusion-Exclusion

2000 Mathematics Subject Classification:
52C99, 52B99, 60D05.

Download this article (PDF):

Suitable for a printer:    

Suitable for a monitor:        

 To view these files we recommend you save them to your file system and then view by using the Adobe Acrobat Reader. 

That is, click on the icon using the 2nd mouse button and select "Save Target As..." (Microsoft Internet Explorer) or "Save Link As..." (Netscape Navigator).

See our PDF pages for more information.

Other papers in this issue

Power-Monotone Sequences and Fourier Series with Positive Coefficients
J. Nemeth

On Some Fundamental Integral Inequalities and their Discrete Analogues
B.G.  Pachpatte

Subharmonic Functions and their Riesz Measure
Raphaele Supper

A Priori Estimate for a System of Differential Operators
Chikh Bouzar

Improved Inclusion-Exclusion Inequalities for Simplex and Orthant Arrangements
Daniel Q. Naiman and Henry P. Wynn

Monotonic Refinements of a Ky Fan Inequality
Kwok K. Chong

Sub-super Solutions and the Existence of Extremal Solutions in Noncoercive Variational Inequalities
V.K. Le

Refinements of Carleman's Inequality
Bao-Quan Yuan 

Inequalities related to the Chebychev Functional Involving Integrals Over Different Intervals
I. Budimir, P. Cerone, and  J. Pecaric

Some Distortion Inequalities Associated with the Fractional Derivatives of Analytic and Univalent Functions
H.M. Srivastava, Yi Ling and Gejun Bao

Necessary and Sufficient Condition for Existence and Uniqueness of the Solution of Cauchy Problem for Holomorphic Fuchsian Operators
Mekki Terbeche

Bounds for Entropy and Divergence for Distributions over a Two-Element Set
Flemming Topsoe

A Pick Function Related to an Inequality for the Entropy Function
Christian Berg

Other issues

 

 

© 2000 School of Communications and Informatics, Victoria University of Technology. All rights reserved.
JIPAM is published by the School of Communications and Informatics which is part of the Faculty of Engineering and Science, located in Melbourne, Australia. All correspondence should be directed to the editorial office.

Copyright/Disclaimer