An asymptotic existence of balanced incomplete block (BIB) designs and pairwise balanced designs (PBD) has been discussed in [1]-[3]. On the other hand, the existence of additive BIB designs and pairwise additive BIB ...An asymptotic existence of balanced incomplete block (BIB) designs and pairwise balanced designs (PBD) has been discussed in [1]-[3]. On the other hand, the existence of additive BIB designs and pairwise additive BIB designs with k = 2?and?λ = 1?has been discussed with direct and recursive constructions in [4]-[8]. In this paper, an asymptotic existence of pairwise additive BIB designs is proved by use of Wilson’s theorem on PBD, and?also for some l?and k the exact existence of l?pairwise additive BIB designs with block size k and?λ = 1?is discussed.展开更多
A strong partially balanced design SPBD（v, b, k; λ,0） whose b is the maximum number of blocks in all SPBD（v, b, k; λ, 0）, as an optimal strong partially balanced design, briefly OSPBD（v, k, λ） is studied. In ...A strong partially balanced design SPBD（v, b, k; λ,0） whose b is the maximum number of blocks in all SPBD（v, b, k; λ, 0）, as an optimal strong partially balanced design, briefly OSPBD（v, k, λ） is studied. In investigation of authentication codes it has been found that the strong partially balanced design can be used to construct authentication codes. This note investigates the existence of optimal strong partially balanced design OSPBD（v, k, 1） for k = 3 and 4, and shows that there exists an OSPBD（v, k, 1） for any v ≥ k.展开更多
The technique of fitting a response surface design is useful in modelling of experimental designs.Response surface is used in situations where the response of interest is influenced by several experimental variables.T...The technique of fitting a response surface design is useful in modelling of experimental designs.Response surface is used in situations where the response of interest is influenced by several experimental variables.The objective of fitting a response surface design is to reduce cost of experimentation and to obtain optimal designs.The property of rotatability is a desirable quantity of experimental design and requires the variance of the fitted design to be constant on circles or spheres about the centre of the design.In this article,a construction technique of fitting modified non-sequential third order rotatable design(TORD)using Pairwise Balanced Design(PBD)is presented.The variance function of a third order response surface design and the properties of Pairwise Balanced Design are utilised for the construction.展开更多
A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) o...A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.展开更多
Given any positive integers k3 and λ,let c（k,λ）denote the smallest integer such that u ∈ B（k,λ）for every integer uc（k,λ）that satisfies the congruences λv（v-1）≡0（mod k（k-1））and λ（u-1）≡0（mod k-1...Given any positive integers k3 and λ,let c（k,λ）denote the smallest integer such that u ∈ B（k,λ）for every integer uc（k,λ）that satisfies the congruences λv（v-1）≡0（mod k（k-1））and λ（u-1）≡0（mod k-1）.In this article we make an improvement on the bound of c（k,λ）provided by Chang in[4]and prove that c（k,λ）exp{k<sup>3k<sup>6</sup></sup>}.In particular,c（k,1）exp{k<sup>k<sup>2</sup></sup>}.展开更多
Given any set K of positive integers and positive integer λ, let c( K, λ) denote the smallest integer such that v∈B( K, λ) for every integer v】】c( K, λ) that satisfies the congruences λv( v-1)≡0 (modβ (K)) a...Given any set K of positive integers and positive integer λ, let c( K, λ) denote the smallest integer such that v∈B( K, λ) for every integer v】】c( K, λ) that satisfies the congruences λv( v-1)≡0 (modβ (K)) and λ( v- 1)≡0 (modα(K)). Let K0 be an equivalent set of K, k and k* be the smallest and the largest integers in K0. We prove that c( K,λ)≤exp展开更多
Let k be any integer and k≥3. In this article it is proved that the necessary condition υ=k (mod k(k-1)) for the existence of an RB(v,k,1) is sufficient whenever u>exp{exp{k12k2}}.
文摘An asymptotic existence of balanced incomplete block (BIB) designs and pairwise balanced designs (PBD) has been discussed in [1]-[3]. On the other hand, the existence of additive BIB designs and pairwise additive BIB designs with k = 2?and?λ = 1?has been discussed with direct and recursive constructions in [4]-[8]. In this paper, an asymptotic existence of pairwise additive BIB designs is proved by use of Wilson’s theorem on PBD, and?also for some l?and k the exact existence of l?pairwise additive BIB designs with block size k and?λ = 1?is discussed.
文摘A strong partially balanced design SPBD（v, b, k; λ,0） whose b is the maximum number of blocks in all SPBD（v, b, k; λ, 0）, as an optimal strong partially balanced design, briefly OSPBD（v, k, λ） is studied. In investigation of authentication codes it has been found that the strong partially balanced design can be used to construct authentication codes. This note investigates the existence of optimal strong partially balanced design OSPBD（v, k, 1） for k = 3 and 4, and shows that there exists an OSPBD（v, k, 1） for any v ≥ k.
文摘The technique of fitting a response surface design is useful in modelling of experimental designs.Response surface is used in situations where the response of interest is influenced by several experimental variables.The objective of fitting a response surface design is to reduce cost of experimentation and to obtain optimal designs.The property of rotatability is a desirable quantity of experimental design and requires the variance of the fitted design to be constant on circles or spheres about the centre of the design.In this article,a construction technique of fitting modified non-sequential third order rotatable design(TORD)using Pairwise Balanced Design(PBD)is presented.The variance function of a third order response surface design and the properties of Pairwise Balanced Design are utilised for the construction.
基金Supported by NSFC grant No. 10371002 (Y. Chang) and No.19901008 (J. Lei)
文摘A idempotent quasigroup (Q, o) of order n is equivalent to an n(n-1)×3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n+1)-set. Denote by T(n+1) the set of (n+1)n(n-1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n+1) into n+1 n(n-1)×3 partial orthogonal arrays A_x, x∈X based on X\{x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.
基金Supported by NSFC Grant No.19701002 and Huo Yingdong Foundation
文摘Given any positive integers k3 and λ,let c（k,λ）denote the smallest integer such that u ∈ B（k,λ）for every integer uc（k,λ）that satisfies the congruences λv（v-1）≡0（mod k（k-1））and λ（u-1）≡0（mod k-1）.In this article we make an improvement on the bound of c（k,λ）provided by Chang in[4]and prove that c（k,λ）exp{k<sup>3k<sup>6</sup></sup>}.In particular,c（k,1）exp{k<sup>k<sup>2</sup></sup>}.
基金This work was supported by the National Natural Science Foundation of China (Grant No.19701002)and Hou Yingdong Foundation.It was also partially supported by Climbing Foundation of Northern Jiaotong University.
文摘Given any set K of positive integers and positive integer λ, let c( K, λ) denote the smallest integer such that v∈B( K, λ) for every integer v】】c( K, λ) that satisfies the congruences λv( v-1)≡0 (modβ (K)) and λ( v- 1)≡0 (modα(K)). Let K0 be an equivalent set of K, k and k* be the smallest and the largest integers in K0. We prove that c( K,λ)≤exp
基金the National Natural Science Foundation of China (No.19701002) HuoYingdong Foundation.
文摘Let k be any integer and k≥3. In this article it is proved that the necessary condition υ=k (mod k(k-1)) for the existence of an RB(v,k,1) is sufficient whenever u>exp{exp{k12k2}}.
