ABSTRACT A weak ‐colouring of a design is an assignment of colours to its points from a set of available colours, such that there are no monochromatic blocks. A colouring of a design is block‐equitable, if for each block, the number of points coloured with any available pair of colours differ by at most one. Weak and block‐equitable colourings of balanced incomplete block designs have been previously considered. In this paper, we extend these concepts to group divisible designs (GDDs) and packing designs. We first determine when a ‐GDD of type can have a block‐equitable ‐colouring. We then give a direct construction of maximum block‐equitable 2‐colourable packings with block size 4; a recursive construction has previously appeared in the literature. We also generalise a bound given in the literature for the maximum size of block‐equitably 2‐colourable packings to . Furthermore, we establish the asymptotic existence of uniform ‐GDDs with arbitrarily many groups and arbitrary chromatic numbers (with the exception of and ). A structural analysis of 2‐ and 3‐uniform 3‐GDDs obtained from 4‐chromatic STS , where is given. We briefly discuss weak colourings of packings, and finish by considering some further constraints on weak colourings of GDDs, namely, requiring all groups to be either monochromatic or equitably coloured.

Does higher clustering shorten attractor periods? We examine whether the global clustering coefficient $ \symbf{C} $, a direct measure of triangle density, predicts attractor lengths in synchronous, signed-threshold Boolean networks on Watts-Strogatz (WS) graphs. We generate 330 directed, signed WS networks spanning sizes $ \symbf{N\,=\,10-100} $ and mean degrees $ \symbf{\bar{k}=2-10} $, simulate dynamics from $ \symbf{M\,=\,100} $ random initial states per graph with exact attractor detection, and summarize each graph by the average log attractor period (equivalently, the geometric mean period). Our primary analysis relates this log-period summary to $ \symbf{C} $ while adjusting for $ \symbf{N} $, $ \symbf{\bar{k}} $, the mean directed shortest path, and including nonlinear size-degree and clustering-degree interactions. Higher clustering robustly shortens attractor periods: a 0.10 increase in $ \symbf{C} $ ($ \symbf{C\in[0,1]} $) corresponds to an $ \symbf{{\approx 13\%-14\%}} $ lower expected geometric mean period, and moving from $ \symbf{C\,=\,0.000} $ to $ \symbf{C\,=\,0.460} $ yields an $ \symbf{\approx 50\%} $ reduction, holding other properties fixed. The effect persists when the linear $ \symbf{C} $ term is replaced by a nonlinear function of $ \symbf{C} $, and it replicates in held-out graph instances (graphs not used to fit the model). Shorter periods are not explained by an increase in fixed points under the strict comparator ($ \symbf{\gt} $); rather, higher triangle density shifts mass from long periods to medium-length periods. In threshold-like logic, settling speed and oscillatory stability are central to computation and control. Our results provide a direct, quantitative link between triangle density and these long-run behaviours, showing that $ \symbf{C} $ acts as a structural lever on temporal complexity.

A square integer relative Heffter array is an n × n array whose rows and columns sum to zero, each row and each column has exactly k entries and either x or -x appears in the array for every x ∈ℤ_2nk+t∖ J, where J is a subgroup of size t. There are many open problems regarding the existence of these arrays. In this paper we construct two new infinite families of these arrays with the additional property that they are strippable. These constructions complete the existence theory for square integer relative Heffter arrays in the case where k=3 and n is prime.

For an integer partition h1+…+hn=N, a 2-realization of this partition is a latin square of order N with disjoint subsquares of orders h1,…,hn. The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to m-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.

Multiple myeloma (MM) is a plasma cell cancer that occurs in the bone marrow. A leading treatment for MM is the monoclonal antibody Daratumumab, targeting the CD38 receptor, which is highly overexpressed in myeloma cells. In this work we model drug resistance via loss of CD38 expression, which is a proposed mechanism of resistance to Daratumumab treatment. We develop an ODE model that includes drug resistance via two mechanisms: a direct effect in which CD38 expression is lost without cell death in response to Daratumumab, and an indirect effect in which CD38 expression switches on and off in the cancer cells; myeloma cells that do not express CD38 have lower fitness but are shielded from the drug action. The model also incorporates competition with healthy cells, death of healthy cells due to off-target drug effects, and a Michaelis-Menten type immune response. Using optimal control theory, we study the effect of the drug resistance mechanisms and the off-target drug effect on the optimal treatment regime. We identify a general increase in the duration and costs of optimal treatment, as a result of these added mechanisms. Several distinct optimal treatment regimes are identified within the parameter space.

The objective of this research is to demonstrate hypergraph versatility and applicability for modeling diverse biological systems. The inherent structure of hypergraphs allows for encoding of higher-order feature interactions, providing a flexible framework for efficient models that can enhance our understanding of physical phenomena and one that can be generalized across various datasets. By adopting innovative methods including centrality measure and populations of models rather than singular instances, biases and overfitting tendencies are mitigated, again presenting promise for application across a broad spectrum of biological systems. Furthermore, emphasis is placed on the significance of probabilistic distribution analysis in elucidating threshold selection and feature relevance while maintaining high levels of accuracy. Our results demonstrate the advantages of hypergraph models on two different datasets; with the first on gene expression and the identification of outlier genes and the second on classifying starch grains. There is significant scope in the application of the hypergraph to a wider class of biological systems, with the potential to improve understanding of the biological processes.

The use of graph centrality measures applied to biological networks, such as protein interaction networks, underpins much research into identifying key players within biological processes. This approach however is restricted to dyadic interactions and it is well-known that in many instances interactions are polyadic. In this study we illustrate the merit of using hypergraph centrality applied to a hypernetwork as an alternative. Specifically, we review and propose an extension to a recently introduced node and edge nonlinear hypergraph centrality model which provides mutually dependent node and edge centralities. A Saccharomyces Cerevisiae protein complex hypernetwork is used as an example application with nodes representing proteins and hyperedges representing protein complexes. The resulting rankings of the nodes and edges are considered to see if they provide insight into the essentiality of the proteins and complexes. We find that certain variations of the model predict essentiality more accurately and that the degree-based variation illustrates that the centrality-lethality rule extends to a hypergraph setting. In particular, through exploitation of the models flexibility, we identify small sets of proteins densely populated with essential proteins. One of the key advantages of applying this model to a protein complex hypernetwork is that it also provides a classification method for protein complexes, unlike previous approaches which are only concerned with classifying proteins.

This paper introduces a novel hypergraph classification algorithm. The use of hypergraphs in this framework has been widely studied. In previous work, hypergraph models are typically constructed using distance or attribute based methods. That is, hyperedges are generated by connecting a set of samples which are within a certain distance or have a common attribute. These methods however, do not often focus on multi-way interactions directly. The algorithm provided in this paper looks to address this problem by constructing hypergraphs which explore multi-way interactions of any order. We also increase the performance and robustness of the algorithm by using a population of hypergraphs. The algorithm is evaluated on two datasets, demonstrating promising performance compared to a generic random forest classification algorithm.

Plants adapt to their local environment through complex interactions between genes, gene networks and hormones. Although the impact of gene expression on trait regulation and evolution has been recognised for many decades, its role in the evolution of adaptation is still a subject of intense exploration. We used a Multi-parent Advanced Generation Inter-Cross (MAGIC) population, which we derived from crossing multiple parents from two distinct coastal ecotypes of an Australia wildflower, Senecio lautus . We focused on studying the contrasting gravitropic behaviours of these ecotypes, which have evolved independently multiple times and show strong responses to natural selection in field experiments, emphasising the role of natural selection in their evolution. Here, we investigated how gene expression differences have contributed to the adaptive evolution of gravitropism. We studied gene expression in 60 pools at five time points (30, 60, 120, 240 and 480 min) after rotating half of the pools 90°. We found 428 genes with differential expression in response to the 90° rotation treatment. Of these, 81 genes (~19%) have predicted functions related to the plant hormones auxin and ethylene, which are crucial for the gravitropic response. By combining insights from Arabidopsis mutant studies and analysing our gene networks, we propose a preliminary model to explain the differences in gravitropism between ecotypes. This model suggests that the differences arise from changes in the transport and availability of the two hormones auxin and ethylene. Our findings indicate that the genetic basis of adaptation involves interconnected signalling pathways that work together to give rise to new ecotypes.

Unravelling the interplay among genes, networks, and signalling molecules is key to understanding how many natural populations adapt. Although the impact of gene expression on trait regulation and evolution has been recognised for many decades, its role in the evolution of adaptations is still a subject of intense exploration. Using a hybrid population derived from two contrasting ecotypes of an Australian wildflower, Senecio lautus, we investigated the role of gene expression divergence in their origins. Coastal ecotypes of S. lautus have contrasting vegetative heights and gravitropic behaviours that evolved independently many times, highlighting the role of natural selection in their evolution. We examined gene expression in 10 gravitropic and 10 agravitropic hybrid families from the hybrid population of Senecio at Lennox Head, NSW. We found 428 genes that showed differential expression between the gravitropic control and treatment groups when we rotated the hybrids 90 degrees. Of these, 81 genes (~19%) had predicted functions linked to several plant hormones. Using knowledge from Arabidopsis mutant screens and assessing our gene networks, we construct a model for differences in gravitropism between ecotypes that relies on modulating the movement and accessibility of the hormone auxin, known to control the gravitropic response across plants. Our findings suggest a role for the hormonal control of gravitropism in plant adaptation to coastal environments, where ecotypes are known to differ from their counterparts in other habitats. More generally, we posit that the genetics of adaptation encompasses the evolution of intertwined signalling pathways that ultimately contribute to the origin of new ecotypes.

A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for d=2 (Latin squares) and d=3 (Latin cubes). Explicit constructions show that this bound is near-optimal for large n> d . For d>n , a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.

We give a framework that generalizes LDPC code constructions using transversal designs or related structures such as mutually orthogonal Latin squares. Our constructions offer a broader range of code lengths and codes rates. Similar earlier constructions rely on the existence of finite fields of order a power of a prime, which significantly restricts the functionality of the resulting codes. In contrast, the LDPC codes constructed here are based on difference matrices and difference covering arrays, structures that are available for any order $a$ , resulting in LDPC codes across a broader class of parameters, notably length $a(a-1)$ , for all even $a$ . Such values are not possible with earlier constructions, thus establishing the novelty of these new constructions. Specifically the codes constructed here satisfy the RC constraint and for $a$ odd, have length $a^{2}$ and rate $1-(4a-3)/a^{2}$ , and for $a$ even, length $a^{2}-a$ and rate at least $1-(4a-6)/(a^{2}-a)$ . When 3 does not divide $a$ , these LDPC codes have stopping distance at least 8. When $a$ is odd and both 3 and 5 do not divide $a$ , our construction delivers an infinite family of QC-LDPC codes with minimum distance at least 10. We also determine lower bounds for the stopping distance of the code. Further we include simulation results illustrating the performance of our codes. The BER and FER performance of our codes over AWGN (via simulation) is at least equivalent to codes constructed previously.

Multidisciplinary approaches can significantly advance our understanding of complex systems. For instance, gene co-expression networks align prior knowledge of biological systems with studies in graph theory, emphasising pairwise gene to gene interactions. In this paper, we extend these ideas, promoting hypergraphs as an investigative tool for studying multi-way interactions in gene expression data. Additional freedoms are achieved by representing individual genes with hyperedges, and simultaneously testing each gene against many features/vertices. Further gene/hyperedge interactions can be captured and explored using the line graph representations, a technique that reduces the complexity of dense hypergraphs. Such an approach provides access to graph centrality measures, which identifies salient features within a data set. For instance dominant or hub-like hyperedges, leading to key knowledge on gene expression. The validity of this approach is established through the study of gene expression data for the plant species Senecio lautus and results will be interpreted within this biological setting.

This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from partially balanced incomplete block designs. Since Gallager’s construction of LDPC codes by randomly allocating bits in a sparse parity-check matrix, many researchers have used a variety of more structured combinatorial approaches. Many of these constructions start with the Galois field; however, this limits the choice of parameters of the constructed codes. Here we present a construction of LDPC codes of length 4n^2 - 2n for all n using the cyclic group of order 2 n . These codes achieve high information rate (greater than 0.8) for n ≥ 8 , have girth at least 6 and have minimum distance 6 for n odd. The results provide proof of concept and lay the groundwork for potential high performing codes

In the last decade, Australia has experienced an overall decline in red cell demand, but there has been an increased need for phenotyped matched red cells. Lifeblood and mathematicians from Queensland universities have developed a probabilistic model to determine the percentage of the donor panel that would need extended antigen typing to meet this increasing demand, and an estimated timeline to achieve the optimum required phenotyped (genotyped) panel. Mathematical modelling, based on Multinomial distributions, was used to provide guidance on the percentage of typed donor panel needed, based on recent historical blood request data and the current donor panel size. Only antigen combinations determined to be uncommon, but not rare, were considered. Simulations were run to attain at least 95% success percentage. Modelling predicted a target of 38% of the donor panel, or 205,000 donors, would need to be genotyped to meet the current demand. If 5% of weekly returning donors were genotyped, this target would be reached within 12 years. For phenotyping, 35% or 188,000 donors would need to be phenotyped to meet Lifeblood's demand. With the current level of testing, this would take eight years but could be performed within three years if testing was increased to 9% of weekly returning donors. An additional 26,140 returning donors need to be phenotyped annually to maintain this panel. This mathematical model will inform business decisions and assist Lifeblood in determining the level of investment required to meet the desired timeline to achieve the optimum donor panel size.

In this paper, we forecast cumulative production for stimulated gas wells using a combination of fast-to-implement modeling methodologies, including polynomial chaos expansion (PCE) and Gaussian processes (GP) proxy models coupled with populations of phenomenological models (POMs). These modeling techniques allow for a reduction in forecast uncertainty and are shown to be effective techniques for extrapolating early time data for stimulated well production from a field of wells in the Surat Basin, Queensland, Australia. The proposed techniques strategically capture and capitalize on production trends across an entire gas field, even in the presence of early production transients. We demonstrate that learning cycles can be shortened, leading to reasonable forecasts, as well as meaningful and actionable insights.

It is shown that for v not equal 7 , 8 , 11 , 12, or 13, there exists an optimal covering with triples on v points that contains no Pasch configurations.

Square Heffter arrays are n×n arrays such that each row and each column contains k filled cells, each row and column sum is divisible by 2nk+1 and either x or −x appears in the array for each integer 1⩽x⩽nk. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face 2-colourable embedding of the complete graph K2nk+1 on an orientable surface, where for each colour, the faces give a k-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo 2nk+1; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) k≡0(mod4); or (b) n≡1(mod4) and k≡3(mod4); or (c) n≡0(mod4), k≡3(mod4) and n≫k. As corollaries to the above we obtain pairs of orthogonal k-cycle decompositions of K2nk+1.

It is shown that, up to isomorphism, there are 35 810 097 maximum partial triple systems on 17 points and 47 744 568 maximal partial triple systems on 16 points. It is also established that there are 157 151 non-isomorphic pairwise balanced designs, PBD(17, {3, 5})s, having a single block of size 5. Structural properties of all these systems are determined, including their automorphism groups, and the numbers of Pasch configurations, mitres and Fano planes contained in them. The systems themselves are available from the authors.

In this paper, we use constructions of Heffter arrays to verify the existence of face 2-colorable embeddings of cycle decompositions of the complete graph. Specifically, forn degrees 1(mod 4)andk degrees 3(mod4),n >> k > 7and whenn degrees 0(mod 3)thenk degrees 7(mod 12), there exist face 2-colorable embeddings of the complete graphK2nk+1onto an orientable surface where each face is a cycle of a fixed lengthk. In these embeddings the vertices ofK2nk+1will be labeled with the elements ofZ2nk+1in such a way that the group,(Z2nk+1,+)acts sharply transitively on the vertices of the embedding. This result is achieved by verifying the existence of nonequivalent Heffter arrays,H(n;k), which satisfy the conditions: (1) for each row and each column the sequential partial sums determined by the natural ordering must be distinct modulo2nk+1; (2) the composition of the natural orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. The existence of Heffter arraysH(n;k)that satisfy condition (1) was established earlier in Burrage et al. and in this current paper, we vary this construction and show, fork > 11, that there are at least(n-2)[((k-11)/4)!/e]2such nonequivalentH(n;k)that satisfy both conditions (1) and (2).