Experimental Designs Balanced for the Estimation of Residual Effects of Treatments

1949 ◽  
Vol 2 (2) ◽  
pp. 149 ◽  
Author(s):  
EJ Williams
Keyword(s):  
Residual Effect ◽  
Latin Squares ◽  
Experimental Designs ◽  
Residual Effects ◽  
Direct Effects ◽  
Balanced Design ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Wide Applicability ◽  
Balanced Designs

Where an experiment can be carried out by applying different treatments in succession to the same unit of experimental material, accurate comparisons can be made between the effects of different treatments. To allow for the residual effect of previous treatments on the result obtained for any given treatment, it is desirable to adjust the results for such effects. Methods of constructing balanced designs for the estimation of these residual effects are described in this paper, and are summarized as follows. Designs balanced for effect of single preceding treatment: When n, the number of treatments, is even, a balanced design is possible with n replications ; when n is odd, 2n replications are required.Designs balanced for the effects of any number of preceding treatments, ignoring the interaction of residual effects: When n is a prime or a power of a prime, a balanced design is possible in n(n-1) replications, which may be set out as a set of n-1 mutually orthogonal Latin squares, with the same first columns. Designs which are not expressible as mutually orthogonal Latin squares are also possible. Designs balanced for the effect of the two preceding treatments and their interactions : A design can be developed from a set of n-l mutually orthogonal Latin squares obeying certain restrictions. The method of analysis of designs of this type is set out in detail, together with a numerical example. Direct effects of treatments are shown to be only slightly confounded, the maximum confounding being 4 per cent., when there are three treatments. These designs have wide applicability wherever successive treatments can be applied to the same unit of experimental material.

Experimental Designs Balanced for Pairs of Residual Effect

1950 ◽  
Vol 3 (3) ◽  
pp. 351 ◽  
Author(s):  
EJ Williams
Keyword(s):  
Experimental Data ◽  
Residual Effect ◽  
Combinatorial Problem ◽  
Round Table ◽  
Experimental Designs ◽  
Residual Effects ◽  
Present Treatment ◽  
Balanced Designs

When different treatments are applied In succession to the same unit of experimental material, it is necessary to consider the residual effects of preceding treatments on the present treatment. To facilitate the interpretation of the results of such experiments, balanced designs have been developed. In the designs discussed in this paper, each treatment is preceded by each pair of other treatments, so that the designs are balanced for the effects of pairs of residual effects. It is shown that balanced designs are possible for n treatments, using n(n-1) replications. Solutions of the combinatorial problem involved can be derived from solutions of the Round Table problem, discussed m many books of mathematical puzzles. The method of analysis of designs of this type is described, and exemplified by application to a set of experimental data

Projective planes of infinite but isolic order

1976 ◽  
Vol 41 (2) ◽  
pp. 391-404 ◽  
Author(s):  
J. C. E. Dekker
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Latin Squares ◽  
Combinatorial Theory ◽  
Ternary Ring ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Planar Ternary Ring ◽  
Partial Recursive Functions

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

New bounds of Mutually unbiased maximally entangled bases in C^dxC^(kd)

2018 ◽  
Vol 18 (13&14) ◽  
pp. 1152-1164
Author(s):  
Xiaoya Cheng ◽  
Yun Shang
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Power ◽  
Latin Squares ◽  
Mutually Unbiased Bases ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Bipartite System

Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(p^a-1) MUMEBs in \cd by properties of Guss sums for arbitrary odd d. It improves the known lower bound p^a-1 for odd d. Certainly, it also generalizes the lower bound 2(p^a-1) for d being a single prime power. Furthermore, we construct MUMEBs in \ckd for general k\geq 2 and odd d. We get the similar lower bounds as k,b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in \ckd, and obtain greater lower bounds than reducing the problem into prime power dimension in some cases.

A Partial Generalization of Mann's Theorem Concerning Orthogonal Latin Squares

1988 ◽  
Vol 31 (4) ◽  
pp. 409-413 ◽  
Author(s):  
E. T. Parker ◽  
Lawrence Somer
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Necessary Condition ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set

AbstractLetn = 4t+- 2, where the integert ≧ 2. A necessary condition is given for a particular Latin squareLof ordernto have a complete set ofn — 2mutually orthogonal Latin squares, each orthogonal toL.This condition extends constraints due to Mann concerning the existence of a Latin square orthogonal to a given Latin square.

scholarly journals On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1895 ◽  
Author(s):  
M. Higazy ◽  
A. El-Mesady ◽  
M. S. Mohamed
Keyword(s):  
Computer Science ◽  
Finite Fields ◽  
Orthogonal Array ◽  
Orthogonal Arrays ◽  
Latin Squares ◽  
Error Correcting Codes ◽  
Recursive Construction ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Definition Of

During the last two centuries, after the question asked by Euler concerning mutually orthogonal Latin squares (MOLS), essential advances have been made. MOLS are considered as a construction tool for orthogonal arrays. Although Latin squares have numerous helpful properties, for some factual applications these structures are excessively prohibitive. The more general concepts of graph squares and mutually orthogonal graph squares (MOGS) offer more flexibility. MOGS generalize MOLS in an interesting way. As such, the topic is attractive. Orthogonal arrays are essential in statistics and are related to finite fields, geometry, combinatorics and error-correcting codes. Furthermore, they are used in cryptography and computer science. In this paper, our current efforts have concentrated on the definition of the graph-orthogonal arrays and on proving that if there are k MOGS of order n, then there is a graph-orthogonal array, and we denote this array by G-OA(n2,k,n,2). In addition, several new results for the orthogonal arrays obtained from the MOGS are given. Furthermore, we introduce a recursive construction method for constructing the graph-orthogonal arrays.

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