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: Assessment of demand for food under concurrent PR and FR schedules in the brushtail possum (Trichosurus Vulpecula)

authors: Cameron, Kristie Elizabeth

abstract: The aim of this study was to compare the demand by possums for foods under different arrangements of concurrent progressive-ratio and fixed-ratio schedules of reinforcement. In Experiment 1, every possible food pair made up of berries, chicken, egg, foliage, insects, and mushroom was presented (30 pairs in total). The requirement on the progressive-ratio schedule increased within a session and the fixed-ratio was kept constant at 30. In Experiment 2, a subset of the foods from Experiment 1 were used (chicken, mushroom, egg, and berries) and in separate conditions the fixed-ratio was either 30 or 10 responses. In Experiment 3, the foods were the same as used in Experiment 2 and the progressive-ratio schedule increased every five sessions and the fixed-ratio was kept constant at 30. Exponential models of demand were applied to consumption rates to compare the parameters of initial demand, essential value and Pmax, and break point and cross point across foods. The models described the data well and consumption rates were similar when the incrementing schedules increased within- and across sessions. Demand was highest for berries, egg, and locust in Experiment 1 and egg and chicken in Experiments 2 and 3. This finding has practical implications for understanding possum food preferences in the wild as a function of other available food sources and for informing pest control strategies of potential lures.

text: Assessment of demand for food under concurrent PR and FR schedules in the brushtail possum (Trichosurus Vulpecula) The aim of this study was to compare the demand by possums for foods under different arrangements of concurrent progressive-ratio and fixed-ratio schedules of reinforcement. In Experiment 1, every possible food pair made up of berries, chicken, egg, foliage, insects, and mushroom was presented (30 pairs in total). The requirement on the progressive-ratio schedule increased within a session and the fixed-ratio was kept constant at 30. In Experiment 2, a subset of the foods from Experiment 1 were used (chicken, mushroom, egg, and berries) and in separate conditions the fixed-ratio was either 30 or 10 responses. In Experiment 3, the foods were the same as used in Experiment 2 and the progressive-ratio schedule increased every five sessions and the fixed-ratio was kept constant at 30. Exponential models of demand were applied to consumption rates to compare the parameters of initial demand, essential value and Pmax, and break point and cross point across foods. The models described the data well and consumption rates were similar when the incrementing schedules increased within- and across sessions. Demand was highest for berries, egg, and locust in Experiment 1 and egg and chicken in Experiments 2 and 3. This finding has practical implications for understanding possum food preferences in the wild as a function of other available food sources and for informing pest control strategies of potential lures.

year: 2015

title: The digital divide in New Zealand: The position of Maori and Pacific peoples

authors: Gibson, John

subjects: communication

abstract: Information and communication technology plays an ever-growing role in economic, political and social life and those who are unable to access such technologies are increasingly disadvantaged. Hence, what has come to be known as the 'digital divide' is of considerable significance. Here, indicators of that divide as it affects Maori and

text: The digital divide in New Zealand: The position of Maori and Pacific peoples Information and communication technology plays an ever-growing role in economic, political and social life and those who are unable to access such technologies are increasingly disadvantaged. Hence, what has come to be known as the 'digital divide' is of considerable significance. Here, indicators of that divide as it affects Maori and

year: 2002