An Existence Theory for Incomplete Designs

2016 ◽  
Vol 59 (2) ◽  
pp. 287-302 ◽  
Author(s):  
Peter Dukes ◽  
Esther R. Lamken ◽  
Alan C. H. Ling
Keyword(s):  
Existence Results ◽  
The Other ◽  
Existence Theory ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Incomplete Designs ◽  
Block Sizes

AbstractAn incomplete pairwise balanced design is equivalent to a pairwise balanced design with a distinguished block, viewed as a ‘hole’. If there are v points, a hole of size w, and all (other) block sizes equal k, this is denoted IPBD((v;w), k). In addition to congruence restrictions on v and w, there is also a necessary inequality: v > (k − 1)w. This article establishes two main existence results for IPBD((v;w), k): one in which w is fixed and v is large, and the other in the case v > (k −1+∊)w when w is large (depending on ∊). Several possible generalizations of the problemare also discussed.

scholarly journals Some Pairwise Balanced Designs

10.37236/1491 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
Malcolm Greig
Keyword(s):  
Block Design ◽  
Basis Sets ◽  
Combinatorial Designs ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Block Sizes

A pairwise balanced design, $B(K;v)$, is a block design on $v$ points, with block sizes taken from $K$, and with every pair of points occurring in a unique block; for a fixed $K$, $B(K)$ is the set of all $v$ for which a $B(K;v)$ exists. A set, $S$, is a PBD-basis for the set, $T$, if $T=B(S)$. Let $N_{a(m)}=\{n:n\equiv a\bmod m\}$, and $N_{\geq m}=\{n:n\geq m\}$; with $Q$ the corresponding restriction of $N$ to prime powers. This paper addresses the existence of three PBD-basis sets. 1. It is shown that $Q_{1(8)}$ is a basis for $N_{1(8)}\setminus E$, where $E$ is a set of 5 definite and 117 possible exceptions. 2. We construct a 78 element basis for $N_{1(8)}$ with, at most, 64 inessential elements. 3. Bennett and Zhu have shown that $Q_{\geq8}$ is a basis for $N_{\geq8}\setminus E'$, where $E'$ is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. (Constructions for 35 of these possible exceptions are known.) Finally, we provide brief details of some improvements and corrections to the generating/exception sets published in The CRC Handbook of Combinatorial Designs.

Pairwise Balanced Designs with Block Sizes Three and Four

1991 ◽  
Vol 43 (4) ◽  
pp. 673-704 ◽  
Author(s):  
Charles J. Colbourn ◽  
Alexander Rosa ◽  
Douglas R. Stinson
Keyword(s):  
Necessary Conditions ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Block Sizes

AbstractGiven integers ν, a and b, when does a pairwise balanced design on ν elements with a triples and b quadruples exist? Necessary conditions are developed, and shown to be sufficient for all v ≥ 96. An extensive set of constructions for pairwise balanced designs is used to obtain the result.

Block Sizes in Pairwise Balanced Designs

1984 ◽  
Vol 27 (3) ◽  
pp. 375-380 ◽  
Author(s):  
Charles J. Colbourn ◽  
Kevin T. Phelps ◽  
Vojtěch Rödl
Keyword(s):  
Block Size ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Good Characterization ◽  
Np Complete ◽  
Block Sizes

AbstractThe number of sets of integers which are realizable as block sizes of a pairwise balanced design of order n is between and ; in contrast, when the multiplicity of each block size is also specified, the number of multisets which can be realized is between and . Although this gives a reasonable bound on the number of multisets which can be realized, a good characterization is not likely to exist; deciding whether a multiset can be so realized is NP-complete.

Orthogonal Fractional Factorial Plans

1980 ◽  
Vol 29 (3-4) ◽  
pp. 143-160 ◽  
Author(s):  
Rahul Mukerjee
Keyword(s):  
Broad Class ◽  
Existence Results ◽  
Fractional Factorial ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Equal Frequency ◽  
Other Hand ◽  
Necessary And Sufficient ◽  
Generally True

This paper shows that the criterion of proportional frequency for (unblocked) orthogonal fractional factorial plans, as suggested by some previous authors, is not generally true. On the other hand, the criterion of equal frequency has been established as a necessary and sufficient condition in the general case. Some other properties of orthogonal fractional factorial plans have been investigated. A necessary and sufficient condition for designs involving two or more blocks has also been presented. A broad class of non-existence results follow.

scholarly journals Design for Affect: A Core Competency for the 21st Century

2015 ◽  
Vol 7 (2) ◽  
pp. 16-21
Author(s):  
Ravindra Chitturi
Keyword(s):  
21St Century ◽  
Core Competency ◽  
Optimal Combination ◽  
The Other ◽  
Functional Requirement ◽  
Emotional Experiences ◽  
Balanced Design ◽  
Product Aesthetics ◽  
Product Functionality

Abstract Consumers purchase products with the objective of reducing pain, increasing pleasure or both. Product aesthetics primarily contribute to enhancing consumer pleasure, and utilitarian attributes, such as product functionality, primarily help reduce consumer pain. So the question is how consumers choose between the goals of reducing pain and enhancing pleasure. In the case of functional dominance, consumers attach greater importance to fulfilling their minimum utilitarian needs over their minimum hedonic ones. By contrast, if consumers have to choose between two products, and one product meets their minimum functional requirement but exceeds their minimum aesthetic expectations, while the other meets their minimum aesthetic expectations but exceeds their minimum functional requirement, they select the product with superior aesthetics. A balanced design with an optimal combination of attributes and emotional experiences will reach a greater price on the market and insure higher profits.

scholarly journals On the existence of solutions for Volterra integral inclusions in Banach spaces

1993 ◽  
Vol 6 (3) ◽  
pp. 261-269
Author(s):  
Evgenios P. Avgerinos
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Of Solutions ◽  
Existence Results ◽  
The Other ◽  
Multivalued Map ◽  
Volterra Type ◽  
The Right ◽  
Condensing Maps

In this paper we examine a class of nonlinear integral inclusions defined in a separable Banach space. For this class of inclusions of Volterra type we establish two existence results, one for inclusions with a convex-valued orientor field and the other for inclusions with nonconvex-valued orientor field. We present conditions guaranteeing that the multivalued map that represents the right-hand side of the inclusion is α-condensing using for the proof of our results a known fixed point theorem for α-condensing maps.

scholarly journals Conformal scalar curvature equation on Sn: Functions with two close critical points (Twin Pseudo-Peaks)

2018 ◽  
Vol 20 (05) ◽  
pp. 1750051
Author(s):  
Man Chun Leung ◽  
Feng Zhou
Keyword(s):  
Scalar Curvature ◽  
Critical Points ◽  
Reduction Method ◽  
Existence Results ◽  
The Other ◽  
Curvature Equation ◽  
Two Bubbles

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on [Formula: see text] ([Formula: see text]) when the prescribed function (after being projected to [Formula: see text]) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness [Formula: see text]), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.

scholarly journals Existence theory for nonresonant singular boundary value problems

1995 ◽  
Vol 38 (3) ◽  
pp. 431-447 ◽  
Author(s):  
Donal O'Regan
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Trivial Solution ◽  
Boundary Value ◽  
Existence Results ◽  
Existence Theory ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Singular Boundary Value Problems

We present some existence results for the “nonresonant” singular boundary value problem a.e. on [0, 1] with Here μ is such that a.e. on [0, 1] with has only the trivial solution.

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