title: Row-column factorial designs and mutually orthogonal frequency rectangles

authors: Rahim, Fahim

subjects: Row-column factorial design

abstract: A (full) qᵏ factorial design with replication λ is the multi-set containing all possible q-ary sequences of length k, each occurring exactly λ times. An m × n row-column factorial design is any arrangement of λ replicates of the qᵏ  factorial design in an m × n array. We say that the design has strength t if each row and column is an orthogonal array of strength t. We denote such a design by Iₖ (m,n,q,t).
A frequency rectangle of type FR(m,n;q) is an m × n array based on a symbol set S of size q, such that each element of S appears exactly n/q times in each row and m/q times in each column. Two frequency rectangles of the same type are said to be orthogonal if each possible pair of symbols appears the same number of times when the two arrays are superimposed. By k–MOFR(m,n;q) we mean a set of k frequency rectangles of type FR(m,n;q) in which every pair is orthogonal.
In Chapter 4, we give the necessary and sufficient conditions when a row-column factorial design of strength 1 exists. We show that an array of type Iₖ (m,n,q,1) exists if and only if (a) q|m, q|n and qᵏ|mn; (b) (k,q,m,n) ≠ (2,6,6,6) and (c) if (k,q,m) = (2,2,2) then 4 divides n. In Chapter 5, we discuss designs of strength 2 and above. We solve the case completely when t = 2 and q is a prime power: we show that there exists an array of type Iₖ(qᴹ,qᴺ,q,2) if and only if k ≤ M + N, k ≤ (qᴹ - 1)/(q - 1) and (k,M,q) ≠ (3,2,2). We also show that Iₖ+α(2αb,2ᵏ,2,2) exists whenever α ≥ 2 and 2α + α + 1 ≤ k < 2αb - α, assuming there exists a Hadamard matrix of order 4b. For strength 3 we restrict ourselves to the binary case, solving it completely when q is a power of 2.
In Chapter 6, our focus is on mutually orthogonal frequency rectangles (MOFR). We use orthogonal arrays and Hadamard matrices to construct sets of MOFR. We also describe a new form of orthogonality for a set of frequency rectangles. We say that a k–MOFR(m,n;q) is t–orthogonal if each subset of size t, when superimposed, forms a qᵗ  factorial design with replication mn/qᵗ. A set of vectors over a finite field is said to be t-independent if each subset of size t is linearly independent. We describe a relationship between a set of t–orthogonal MOFR and a set of t-independent vectors. We use known results from coding theory and related literature to formulate a table for the size of a set of t-independent vectors of length N ≤ 16, over F₂. We also describe a method to construct a set of (p - 1)–MOFR(2p,2p;2) where p is an odd prime, improving known lower bounds for all p ≥ 19.

text: Row-column factorial designs and mutually orthogonal frequency rectangles A (full) qᵏ factorial design with replication λ is the multi-set containing all possible q-ary sequences of length k, each occurring exactly λ times. An m × n row-column factorial design is any arrangement of λ replicates of the qᵏ  factorial design in an m × n array. We say that the design has strength t if each row and column is an orthogonal array of strength t. We denote such a design by Iₖ (m,n,q,t).
A frequency rectangle of type FR(m,n;q) is an m × n array based on a symbol set S of size q, such that each element of S appears exactly n/q times in each row and m/q times in each column. Two frequency rectangles of the same type are said to be orthogonal if each possible pair of symbols appears the same number of times when the two arrays are superimposed. By k–MOFR(m,n;q) we mean a set of k frequency rectangles of type FR(m,n;q) in which every pair is orthogonal.
In Chapter 4, we give the necessary and sufficient conditions when a row-column factorial design of strength 1 exists. We show that an array of type Iₖ (m,n,q,1) exists if and only if (a) q|m, q|n and qᵏ|mn; (b) (k,q,m,n) ≠ (2,6,6,6) and (c) if (k,q,m) = (2,2,2) then 4 divides n. In Chapter 5, we discuss designs of strength 2 and above. We solve the case completely when t = 2 and q is a prime power: we show that there exists an array of type Iₖ(qᴹ,qᴺ,q,2) if and only if k ≤ M + N, k ≤ (qᴹ - 1)/(q - 1) and (k,M,q) ≠ (3,2,2). We also show that Iₖ+α(2αb,2ᵏ,2,2) exists whenever α ≥ 2 and 2α + α + 1 ≤ k < 2αb - α, assuming there exists a Hadamard matrix of order 4b. For strength 3 we restrict ourselves to the binary case, solving it completely when q is a power of 2.
In Chapter 6, our focus is on mutually orthogonal frequency rectangles (MOFR). We use orthogonal arrays and Hadamard matrices to construct sets of MOFR. We also describe a new form of orthogonality for a set of frequency rectangles. We say that a k–MOFR(m,n;q) is t–orthogonal if each subset of size t, when superimposed, forms a qᵗ  factorial design with replication mn/qᵗ. A set of vectors over a finite field is said to be t-independent if each subset of size t is linearly independent. We describe a relationship between a set of t–orthogonal MOFR and a set of t-independent vectors. We use known results from coding theory and related literature to formulate a table for the size of a set of t-independent vectors of length N ≤ 16, over F₂. We also describe a method to construct a set of (p - 1)–MOFR(2p,2p;2) where p is an odd prime, improving known lower bounds for all p ≥ 19.

year: 2023

title: Separation of variables and contractions on two-dimensional hyperboloid

authors: Kalnins, Ernie G.

abstract: In this paper analytic contractions have been established in the R → ∞ con- traction limit for exactly solvable basis functions of the Helmholtz equation on the two- dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae.

text: Separation of variables and contractions on two-dimensional hyperboloid In this paper analytic contractions have been established in the R → ∞ con- traction limit for exactly solvable basis functions of the Helmholtz equation on the two- dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae.

year: 2012

title: Translated application interfaces - issues of engagement

authors: Mato, Paora James

abstract: In New Zealand, English is the language that dominates contemporary technologies. Usability testing was completed on a range of applications, available with a Māori-language interface, to gauge levels of awareness, engagement and perception. Nearly all of the respondents were unaware of the availability of these interfaces but most indicated they would prefer to use the Māori-language versions. In terms of engagement and usability, users initially engaged using Māori but switched to English when they wanted to quickly complete the task at hand. Few remained fully engaged with the Māori-language interfaces. High levels of language switching were reported and some frustration as the participants encountered new and unfamiliar uses of words. At face value the feedback suggests the translated interfaces contained unnecessary complications and that better design and content might have enhanced the user experience. However, there is evidence that extended use would enable users to become more familiar with the interfaces alluding to initial barriers created by a previous competence in another language -- in this case English. With this previous competence in mind it might be more useful to employ design concepts that would alleviate initial difficulties and serve to keep the user engaged in the target language for longer periods of time.

text: Translated application interfaces - issues of engagement In New Zealand, English is the language that dominates contemporary technologies. Usability testing was completed on a range of applications, available with a Māori-language interface, to gauge levels of awareness, engagement and perception. Nearly all of the respondents were unaware of the availability of these interfaces but most indicated they would prefer to use the Māori-language versions. In terms of engagement and usability, users initially engaged using Māori but switched to English when they wanted to quickly complete the task at hand. Few remained fully engaged with the Māori-language interfaces. High levels of language switching were reported and some frustration as the participants encountered new and unfamiliar uses of words. At face value the feedback suggests the translated interfaces contained unnecessary complications and that better design and content might have enhanced the user experience. However, there is evidence that extended use would enable users to become more familiar with the interfaces alluding to initial barriers created by a previous competence in another language -- in this case English. With this previous competence in mind it might be more useful to employ design concepts that would alleviate initial difficulties and serve to keep the user engaged in the target language for longer periods of time.

year: 2015