scholarly journals Highly D‑efficient Weighing Design and Its Construction

2018 ◽  
Vol 5 (331) ◽  
pp. 143-151
Author(s):  
Bronisław Ceranka ◽  
Małgorzata Graczyk
Keyword(s):  
Sufficient Conditions ◽  
Block Designs ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Construction Methods ◽  
Design Optimality ◽  
Spring Balance ◽  
New Construction ◽  
Balanced Incomplete Block ◽  
Necessary And Sufficient

In this paper, some aspects of design optimality on the basis of spring balance weighing designs are considered. The properties of D‑optimal and D‑efficiency designs are studied. The necessary and sufficient conditions determining the mentioned designs and some new construction methods are introduced. The methods of determining designs that have the required properties are based on a set of incidence matrices of balanced incomplete block designs and group divisible designs.

scholarly journals Some new construction methods of variance balanced block designs with repeated blocks

2008 ◽  
Vol 5 (1) ◽  
Author(s):  
Bronisław Ceranka ◽  
Małgorzata Graczyk
Keyword(s):  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Construction Methods ◽  
New Construction ◽  
Balanced Incomplete Block

Some new construction methods of the variance balanced block designs with repeated blocks are given. They are based on the specialized product of incidence matrices of the balanced incomplete block designs.

scholarly journals A Regular D‑optimal Weighing Design with Negative Correlations of Errors

2019 ◽  
Vol 5 (344) ◽  
pp. 7-16
Author(s):  
Małgorzata Graczyk ◽  
Bronisław Ceranka
Keyword(s):  
Sufficient Conditions ◽  
Optimal Estimation ◽  
Block Designs ◽  
Correlated Errors ◽  
Unknown Parameters ◽  
Chemical Balance ◽  
Construction Methods ◽  
New Construction ◽  
Balanced Incomplete Block ◽  
Necessary And Sufficient

The issues concerning optimal estimation of unknown parameters in the model of chemical balance weighing designs with negative correlated errors are considered. The necessary and sufficient conditions determining the regular D‑optimal design and some new construction methods are presented. They are based on the incidence matrices of balanced incomplete block designs and balanced bipartite weighing designs.  

scholarly journals A-optimal chemical balance weighing design under certain condition

2007 ◽  
Vol 4 (1) ◽  
Author(s):  
Bronisław Ceranka ◽  
Małgorzata Graczyk
Keyword(s):  
Lower Bound ◽  
Sufficient Conditions ◽  
Design Matrix ◽  
Block Designs ◽  
Chemical Balance ◽  
Incidence Matrices ◽  
Construction Methods ◽  
New Construction ◽  
The Right ◽  
Necessary And Sufficient

The paper is studying the estimation problem of individual weights of \(p\) objects using the design matrix \(\mathbf{X}\) of the A-optimal chemical balance weighing design under the restriction \(p_1 + p_2 = q  \leq p\), where \(p_1\) and \(p_2\) represent the number of objects placed on the left pan and on the right pan, respectively, in each of the measurement operations. The lower bound of \(\mathrm{Tr}(\mathbf{X}^{\prime}\mathbf{X})^{-1}\) is attained and the necessary and sufficient conditions for this lower bound to be obtained are given. There are given new construction methods of the A-optimal chemical balance weighing designs based on incidence matrices of the balanced bipartite weighing designs and the ternary balanced block designs.

scholarly journals ON D-OPTIMAL CHEMICAL BALANCE WEIGHING DESIGNS

2015 ◽  
Vol 1 (311) ◽  
Author(s):  
Bronisław Ceranka ◽  
Małgorzata Graczyk
Keyword(s):  
Optimal Design ◽  
Measurement Errors ◽  
Block Designs ◽  
Incomplete Block ◽  
Chemical Balance ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Construction Methods ◽  
Existence Conditions ◽  
Balanced Incomplete Block

The paper deals with the problem of determining the chemical balance weighing designs satisfying the criterion of D-optimality under assumption that the measurement errors are equal correlated and they have the same variances. The existence conditions and the form of the optimal design are given. Moreover, some construction methods of the design matrices based on the incidence matrices of the balanced incomplete block designs and ternary balanced block designs are presented. Any example of construction is given.

On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, III: Rectangular Matrices

1966 ◽  
Vol 18 ◽  
pp. 9-17
Author(s):  
Kulendra N. Majindar
Keyword(s):  
Necessary Conditions ◽  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block ◽  
Definition Of ◽  
Combinatorial Configurations

In this paper, we give a connection between incidence matrices of affine resolvable balanced incomplete block designs and rectangular integer matrices subject to certain arithmetical conditions. The definition of these terms can be found in paper II of this series or in (2). For some necessary conditions on the parameters of affine resolvable balanced incomplete block designs and their properties see (2).

On Integer Matrices and Incidence Matrices of Certain Combinatorial Configurations, II: Rectangular Matrices

1966 ◽  
Vol 18 ◽  
pp. 6-8
Author(s):  
Kulendra N. Majindar
Keyword(s):  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block ◽  
Definition Of ◽  
Image Position ◽  
Combinatorial Configurations

In this paper we establish a connection between rectangular integer matrices and incidence matrices of resolvable balanced incomplete block designs. The definition of these terms has been given in paper I of this series.Our theorem can be stated as follows:THEOREM 2. Let A be a v X b matrix with integer elements such that2.1

Trend‐Free Nested Balanced Incomplete Block Designs and Designs for Diallel Cross Experiments

2007 ◽  
Vol 59 (3-4) ◽  
pp. 203-221
Author(s):  
Kishan Lal ◽  
Rajender Prasad ◽  
V. K. Gupta
Keyword(s):  
Block Design ◽  
Block Size ◽  
Diallel Cross ◽  
Block Designs ◽  
Incomplete Block ◽  
Diallel Crosses ◽  
Balanced Incomplete Block Designs ◽  
Block Level ◽  
Balanced Incomplete Block ◽  
Necessary And Sufficient

Abstract: Nested balanced incomplete block (NBIB) designs are useful when the experiments are conducted to deal with experimental situations when one nuisance factor is nested within the blocking factor. Similar to block designs, trend may exist in experimental units within sub‐blocks or within blocks in NBIB designs over time or space. A necessary and sufficient condition, for a nested block design to be trend‐free at sub‐block level, is derived. Families and catalogues of NBIB designs that can be converted into trend‐free NBIB designs at sub‐block and block levels have been obtained. A NBIB design with sub‐block size 2 has a one to one correspondence with designs for diallel crosses experiments. Therefore, optimal block designs for dialled cross experiments have been identified to check if these can be converted in to trend‐free optimal block designs for diallel cross experiments. A catalogue of such designs is also obtained. Trend‐free design is illustrated with example for a NBIB design and a design for diallel crosses experiments. AMS (2000) Subject Classification: 62K05, 62K10.

Combinatorial Problems

1950 ◽  
Vol 2 ◽  
pp. 93-99 ◽  
Author(s):  
S. Chowla ◽  
H. J. Ryser
Keyword(s):  
Combinatorial Problem ◽  
Difference Sets ◽  
Projective Planes ◽  
Combinatorial Problems ◽  
Block Designs ◽  
Incomplete Block ◽  
Incidence Matrices ◽  
Finite Projective Planes ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].

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