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Summary 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.

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 simultaneous testing each gene against many features/vertices. Further gene/hyperedge interactions can be captured and explored using the line graph representations, a techniques that also reduces the complexity of dense hypergraphs. Such an approach provides access to graph centrality measures, which in turn identify 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 speciesSenecio lautusand results will be interpreted within this biological setting.

We give a framework for generalizing LDPC code constructions that use Transversal Designs or related structures such as mutually orthogonal Latin squares. Our construction offers 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. In contrast the LDPC codes constructed here are based on difference matrices and difference covering arrays, structures available for any order a. They satisfy the RC constraint and have, for a odd, length a2 and rate 1 − [(4a−3)/ a2 ], and for a even, length a2 − a and rate at least 1 − [(4a−6)/ (a2 −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. The simplicity of the construction allows us to theoretically verify these properties and analytically determine lower bounds for the minimum distance and stopping distance of the code. The BER and FER performance of our codes over AWGN (via simulation) is at the least equivalent to codes constructed previously, while in some cases significantly outperforming them.

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.

This paper takes a new approach to the modelling and visualisation of sequestered CO $$_2$$ in subsurface reservoirs and its migration pathways. We model the gas as discretised packets moving along paths within a graph or network. The movement of the packets will be governed by chip-firing. This new approach provides a new insight into the modelling of CO $$_2$$ migration in porous media with complex geological architectures. We compare and contrast this new graph theoretic approach with traditional methods and demonstrate that similar results are obtained, with the added advantage that the new methods are quick to implement and execute. In addition, the new methods are more flexible and can more accurately capture the variation in the stratification of the encasing rock.

In this paper, we study collections of mutually nearly orthogonal Latin squares (MNOLS), which come from a modification of the orthogonality condition for mutually orthogonal Latin squares. In particular, we find the maximum μ such that there exists a set of μ cyclic MNOLS of order n for n≤18, as well as providing a full enumeration of sets and lists of μ cyclic MNOLS of order n under a variety of equivalences with n≤18. This resolves in the negative a conjecture that proposed that the maximum μ for which a set of μ cyclic MNOLS of order n exists is [n/4]+1.

In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions is that, if N(n) denotes the size of the largest set of MOLS of order n, then N(n2)⩾N(n)+2. In particular, it follows that N(576)⩾9, improving the previously known lower bound N(576)⩾8., Scientific and Technological Research Council of Turkey (TÜBİTAK)

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$.

In this paper we will show the existence of a face $2$-colourable biembedding of the complete graph onto an orientable surface where each face is a cycle of a fixed length $k$, for infinitely many values of $k$. In particular, under certain conditions, we show that there exists at least $(n-2)[(p-2)!]^2/(e^2 kn)$ non-isomorphic face $2$-colourable biembeddings of $K_{2nk+1}$ in which all faces are cycles of length $k=4p+3$. These conditions are: $n\equiv 1\mod 4$, $k\equiv 3\mod 4$ and either $n$ is prime or $n\gg k$ and $n\equiv 0\mod 3$ implies $p\equiv 1\mod 3$. To achieve this result we begin by verifying the existence of $(n-2)[(p-2)!/e]^2$ non-equivalent 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 modulo $2nk+1$; (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 arrays $H(n;k)$ that satisfy condition (1) was established earlier in \cite{BCDY} and in this current paper we vary this construction and show that there are at least $(n-2)[(p-2)!/e]^2$ such non-equivalent $H(n;k)$ that satisfy condition (1) and then show that each of these Heffter arrays also satisfy condition (2) under certain conditions., Comment: arXiv admin note: text overlap with arXiv:1906.07366

Denote by A p k the Latin square of order n = p k formed by the Cayley table of the additive group ( Z p k , + ) , where p is an odd prime and k is a positive integer. It is shown that for each p there exists Q > 0 such that for all sufficiently large k , the number of transversals in A p k exceeds ( n Q ) n p ( p − 1 ) .

Cavenagh and Wanless recently proved that, for sufficiently large odd n , the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246) n . We adapt their proof to show that for sufficiently large t the number of additive permutations on [− t , t ] is greater than (3.246) 2 t + 1 and we go on to derive some much improved lower bounds on the numbers of Skolem-type sequences. For example, it is shown that for sufficiently large t ≡ 0 or 3 (mod 4) , the number of split Skolem sequences of order n = 7 t + 3 is greater than (3.246) 6 t + 3 . This compares with the previous best bound of 2 ⌊ n /3⌋ .

A surrogate model approximates a computationally expensive solver. Polynomial Chaos is a method used to construct surrogate models by summing combinations of carefully chosen polynomials. The polynomials are chosen to respect the probability distributions of the uncertain input variables (parameters); this allows for both uncertainty quantification and global sensitivity analysis. In this paper we apply these techniques to a commercial solver for the estimation of peak gas rate and cumulative gas extraction from a coal seam gas well. The polynomial expansion is shown to honour the underlying geophysics with low error when compared to a much more complex and computationally slower commercial solver. We make use of advanced numerical integration techniques to achieve this accuracy using relatively small amounts of training data.

It is shown that for v ≠ 6, 7, 10, 11, 12, 13, there exists an optimal packing with triples on v points that contains no Pasch configurations. Furthermore, for all v ≡ 5 (mod 6), there exists a pairwise balanced design of order v, whose blocks are all triples apart from a single quintuple, and that has no Pasch configurations amongst its triples.

This paper investigated the impact of geological and engineering factors on coal seam gas production in horizontal wells. The results were then used to compare the performance of proxy models based on linear and quadratic response surfaces and Universal Kriging (UK) to models based on Polynomial Chaos Expansion (PCE). A simple reservoir model was created using a commercial reservoir modelling software package which includes the capability to construct proxy models. The simple model was used for the dynamic modelling of cumulative gas production and peak gas rate under uncertainty in the input variables, e.g. the reservoir principal permeability and permeability in orthogonal directions varied as ratios of the principal permeability, porosity, gas content, coal saturation and the angle between the horizontal well and the principal permeability direction. The simulation results for cumulative gas production and peak gas rate were used to generate training data for proxy models, which were then used to predict the simulated output for other combinations of input parameters. Error analysis was conducted for each proxy model and used to compare and contrast the different modelling techniques. In addition, we investigated the sensitivity of the model to changes in the input variables. The results indicate that, for the given study, proxy models based on linear regression were not good estimators for cumulative gas production and peak gas rate. However, when enough training points were utilized, all other techniques provided good estimates. The best performing proxy model was a cubic PCE. In addition, the cubic PCE proxy model provided direct access to the sensitivity of the gas production to changes in the values of the input variables. Considering the main effects, the changes in the principle permeability and the porosity were predominant factors for the cumulative gas production and peak gas rate, respectively. As for the pairwise interactions, the combined effect of the principle permeability and the coal saturation had the most impact on the cumulative gas production, followed by the combined effect of the drilling angle and the ratio of the permeabilities in the y-direction and the x-direction, indicating that gas production can be improved by optimizing the orientation of the horizontal well. In addition, the combined effect of the coal saturation and the porosity had the most impact on the peak gas rate, an interesting result that warrants further investigation in future studies.

Sliced Sudoku-based space-filling designs and, more generally, quasi-sliced orthogonal array-based space-filling designs are useful experimental designs in several contexts, including computer experiments with categorical in addition to quantitative inputs and cross-validation. Here, we provide a straightforward construction of doubly orthogonal quasi-Sudoku Latin squares which can be used to generate quasi-sliced orthogonal arrays and, in turn, sliced space-filling designs which achieve uniformity in one- and two-dimensional projections for the full design and uniformity in two-dimensional projections for each slice. These constructions are very practical to implement and yield a spectrum of design sizes and numbers of factors not currently broadly available.

We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for asmany combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

We highlight some connections between graph labelling, combinatorial design theory and information theory. We survey results on the construction and enumeration of Skolem labellings and related structures. This includes discussion of two constructions of low density parity check codes from Skolem labellings. We raise several pertinent questions and suggestions for future research directions.

In this paper we use counting arguments to prove that the expected percentage coverage of a d dimensional parameter space of size n when performing k trials with either Latin Hypercube sampling or Orthogonal Array-based Latin Hypercube sampling is the same. We then extend these results to an experimental design setting by projecting onto a t < d dimensional subspace. These results are confirmed by simulations. The theory presented has both theoretical and practical significance in modelling and simulation science when sampling over high dimensional spaces., University of Queensland Centre for Coal Seam Gas; APLNG; Arrow Energy; Santos; QGC (Shell); Scientific and Technological Research Council of Turkey (TÜBİTAK)

Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.

This work provides an orthogonal trade for all possible volumes $$N \in \mathbb {Z}^+ \setminus \{1,2,3,4,5,7\}$$N?Z+\{1,2,3,4,5,7} for block size 4. All orthogonal trades of volume $$N\leqslant 15$$N?15 are classified up to isomorphism for this block size. The intersection problem for orthogonal arrays with block size 4 is solved for all but finitely many possible exceptions.

A Heffter array $$H(m,n;s,t)$$H(m,n?s,t) is an $$m \times n$$m×n matrix with nonzero entries from $$\mathbb {Z}_{2ms+1}$$Z2ms+1 such that (i) each row contains $$s$$s filled cells and each column contains $$t$$t filled cells, (ii) every row and column sum to 0, and (iii) no element from $$\{x,-x\}$${x,-x} appears twice. Heffter arrays are useful in embedding the complete graph $$K_{2ms+1}$$K2ms+1 on an orientable surface where the embedding has the property that each edge borders exactly one s-cycle and one t-cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be constructed in the case when $$s=m$$s=m, i.e every cell is filled. In this paper we concentrate on square arrays with empty cells where every row sum and every column sum is $$0$$0 in $${\mathbb {Z}}$$Z. We solve most of the instances of this case.

In this paper we have used simulations to make a conjecture about the coverage of a $t$ dimensional subspace of a $d$ dimensional parameter space of size $n$ when performing $k$ trials of Latin Hypercube sampling. This takes the form $P(k,n,d,t)=1-e^{-k/n^{t-1}}$. We suggest that this coverage formula is independent of $d$ and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the $t$ dimensional subspace at the sub-block size level., 9 pages, 5 figures

Let L be a Latin square of indeterminates. We explore the determinant and permanent of L and show that a number of properties of L can be recovered from the polynomials det(L) and per(L). For example, it is possible to tell how many transversals L has from per(L), and the number of 2 × 2 Latin subsquares in L can be determined from both det(L) and per(L). More generally, we can diagnose from det(L) or per(L) the lengths of all symbol cycles. These cycle lengths provide a formula for the coefficient of each monomial in det(L) and per(L) that involves only two different indeterminates. Latin squares A and B are trisotopic if B can be obtained from A by permuting rows, permuting columns, permuting symbols, and/or transposing. We show that nontrisotopic Latin squares with equal permanents and equal determinants exist for all orders n≥9 that are divisible by 3. We also show that the permanent, together with knowledge of the identity element, distinguishes Cayley tables of finite groups from each other. A similar result for determinants was already known.

The petroleum industry uses high level dynamic simulations applied to geocellular models to guide forecasts of oil, gas and water production. Uncertainty in model choice and input variable selection is often addressed through large numbers of computationally slow Monte Carlo simulations designed around physics based models. Here, an alternate approach is proposed, which uses a relatively small amount of data and a reduced number of simulations of the high level physics model to train a fast (to evaluate) proxy or surrogate model based on a Polynomial Chaos Expansion. We give details of the theory and incorporated techniques, which significantly increase flexibility. Input variables (e.g. cell-by-cell variations in porosity and permeability) are sampled from unknown probability distributions and sensitivity analysis is based on low level proxy models. The theory is tested by developing proxy models to predict total gas production from a five-spot well configuration in the Hermitage area that taps into the Walloon Coal Measures of the Surat Basin in Queensland. Synthetic training data is simulated using commercial dynamic simulation software based on a high level physics model.

In this paper, we will give intersection numbers for two simple 2-fold (3n, n, 3) group divisible designs. More precisely, we will develop constructions which show that there exists two simple 2-fold (3n, n, 3) group divisible designs which intersect in precisely $${k \in \{0, 1, 2, \ldots, 2n^2\}{\setminus} \{2n^2-1, 2n^2-2, 2n^2-3, 2n^2-5\}}$$ k ? { 0 , 1 , 2 , ? , 2 n 2 } \ { 2 n 2 - 1 , 2 n 2 - 2 , 2 n 2 - 3 , 2 n 2 - 5 } triples for $${n \geq 5}$$ n ? 5 . There are some exceptions for n = 2, 3, 4.

We construct face two-colourable triangulations of the graph 2Kn in an orientable surface; equivalently biembeddings of two twofold triple systems of order n, for all n ≡ 4 (mod 12). The biembeddings have a cyclic automorphism and the construction employs index 1 current graphs.

Bounds are obtained on the number of distinct transversal designs TD ( g , n ) (having g groups with n points in each group) for certain values of g and n. Amongst other results it is proved that, if 2 g ⩽ q + 1 where q is a prime power, then the number of nonisomorphic TD ( g , q r ) designs is at least q α r q 2 r ( 1 − o ( 1 ) ) as r → ∞ , where α = 1 / q 4 . The bounds obtained give equivalent bounds for the numbers of distinct and nonisomorphic sets of g − 2 mutually orthogonal Latin squares of order n in the corresponding cases. Applications to other combinatorial designs are also described.

In this note, we will give intersection numbers for two simple 2-fold ( 3 n , n , 3 ) group divisible designs (GDD). More precisely, we note that there exists two simple 2-fold ( 3 n , n , 3 ) GDDs which intersect in precisely k ∈ { 0 , 1 , 2 , … , 2 n 2 } \ { 2 n 2 − 1 , 2 n 2 − 2 , 2 n 2 − 3 , 2 n 2 − 5 } triples for n ⩾ 5 . There are some exceptions for n = 3 , 4 .

How can active learning, peer learning and prompt feedback be achieved in large first-year mathematics classes? Further, what technologies may support these aims? In this article, we assert that test revision sessions in first-year mathematics held in a technology-enhanced lecture theatre can be highly interactive with students solving problems, learning from each other and receiving immediate feedback. This is facilitated by pen-enabled screens and synchronization software. We argue that the educational benefits achievable through the technology do outweigh the technological distractions, and that these benefits can be achieved by focused, targeted one-off sessions and not only by a semester-long, regular approach. Repeat mid-semester test revision sessions were offered on a non-compulsory basis using pen-enabled screens for all students. Students worked practice test questions and marked solutions to mathematical problems on the screens. Students’ work was then displayed anonymously for their peers to s...

Let $B_p$ be the Latin square given by the addition table for the integers modulo an odd prime $p$. Here we consider the properties of Latin trades in $B_p$ which preserve orthogonality with one of the $p-1$ MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in $p$ for the number of times each symbol occurs in such a trade, with an overall lower bound of $(\log{p})^2/\log\log{p}$ for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in $B_p$ hits the main diagonal either $p$ or at most $p-\log_2{p}-1$ times. Finally, if $p\equiv 1\pmod{6}$ we show the existence of a Latin square which is orthogonal to $B_p$ and which contains a $2\times 2$ subsquare.

A µ -way Latin trade of volume s is a collection of µ partial Latin squares T 1 , T 2 , ? , T µ , containing exactly the same s filled cells, such that, if cell ( i , j ) is filled, it contains a different entry in each of the µ partial Latin squares, and such that row i in each of the µ partial Latin squares contains, set-wise, the same symbols, and column j likewise. It is called a µ -way k -homogeneous Latin trade if, in each row and each column, T r , for 1 ? r ? µ , contains exactly k elements, and each element appears in T r exactly k times. It is also denoted as a ( µ , k , m ) Latin trade, where m is the size of the partial Latin squares.We introduce some general constructions for µ -way k -homogeneous Latin trades, and specifically show that, for all k ? m , 6 ? k ? 13 , and k = 15 , and for all k ? m , k = 4 , 5 (except for four specific values), a 3 -way k -homogeneous Latin trade of volume k m exists. We also show that there is no ( 3 , 4 , 6 ) Latin trade and there is no ( 3 , 4 , 7 ) Latin trade. Finally, we present general results on the existence of 3 -way k -homogeneous Latin trades for some modulo classes of m .

Due to the changing nature of learning and teaching in universities, there is a growing need for professional development for lecturers and tutors teaching in disciplines in the mathematical sciences. Mathematics teaching staff receive some training in learning and teaching but many of the courses running at university level are not tailored to the mathematical sciences. This article reports on a collaborative research project aimed at investigating the type of professional development that Australian tertiary mathematics teachers need and their preference for delivery modes. Effective teaching promotes effective learning in our students and discipline-specific professional development will enhance outcomes for teachers, students, and mathematics.

A construction is described that yields improved lower bounds for the numbers of 2-designs and resolvable 2-designs with the parameters of AG d (n, q).

It is shown that for every admissible order v for which a cyclic Steiner triple system exists, there exists a biembedding of a cyclic Steiner quasigroup of order v with a copy of itself. Furthermore, it is shown that for each n >= 2 the projective Steiner quasigroup of order 2(n) - 1 has a biembedding with a copy of itself. (C) 2010 Wiley Periodicals, Inc. J Combin Designs 19: 16-27, 2010

Two constructions are described that yield an improved lower bound for the number of 2-designs with the parameters of PG d (n, q), and a lower bound for the number of resolved 2-designs with the parameters of AG d (n, q).

For each positive integer n ≥ 2, there is a well-known regular orientable Hamiltonian embedding of Kn, n, and this generates a regular face 2-colourable triangular embedding of Kn, n, n. In the case n ≡ 0 (mod 8), and only in this case, there is a second regular orientable Hamiltonian embedding of Kn, n. This paper presents an analysis of the face 2-colourable triangular embedding of Kn, n, n that results from this. The corresponding Latin squares of side n are determined, together with the full automorphism group of the embedding.

There has been much interest in mutually unbiased bases (MUBs) and their connections with various other discrete structures, such as projective planes, mutually orthogonal Latin squares (MOLS) etc. It has been conjectured by Saniga et al. (J Opt B Quantum Semiclass Opt 6:L19---L20, 2004) that the existence of a complete set of MUBs in ? d is linked to the existence of a complete set of MOLS of side length d. Since more is known about MOLS than about MUBs, most research has concentrated on constructing MUBs from MOLS (Roy and Scott, J Math Phys 48:072110, 2007; Paterek et al., Phys Rev A 70:012109, 2009). This paper gives a simple algebraic construction of MOLS from two known constructions of MUBs in the odd prime power case.

The parity vectors of two Latin squares of the same side n provide a necessary condition for the two squares to be biembeddable in an orientable surface. We investigate constraints on the parity vector of a Latin square resulting from structural properties of the square, and show how the parity vector of a direct product may be obtained from the parity vectors of the constituent factors. Parity vectors for Cayley tables of all Abelian groups, some non-Abelian groups, Steiner quasigroups and Steiner loops are determined. Finally, we give a lower bound on the number of main classes of Latin squares of side n that admit no self-embeddings.

We apply a recursive construction for biembeddings of Latin squares to produce a new infinite family of biembeddings of cyclic Latin squares of even side having a high degree of symmetry. Reapplication of the construction yields two further classes of biembeddings.

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d D of D there exists a minimal defining set d F of F such that $${d_D = d_F\cap D}$$. The unique simple design with parameters $${{\left(v,k, {v-2\choose k-2}\right)}}$$is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.

We give a necessary condition for the biembedding of two Latin squares in an orientable surface. As a consequence, it is shown that for n≥2, there is no biembedding of two Latin squares both lying in the same main class as the square obtained from the Cayley table of the Abelian 2-group C2n.

Let P and Q be uniquely completable partial Latin squares. It is an open problem to determine necessary and sufficient conditions so that the completable product [email protected]?Q is also uniquely completable. So far, only a few specific examples of P have been given such that the completable product of P with itself ([email protected]?P) does not have a unique completion. In this paper, we find a whole class of such partial Latin squares.

A Steiner 2-(v,3) trade is a pair (T"1,T"2) of disjoint partial Steiner triple systems, each on the same set of v points, such that each pair of points occurs in T"1 if and only if it occurs in T"2. A Steiner 2-(v,3) trade is called d-homogeneous if each point occurs in exactly d blocks of T"1 (or T"2). In this paper we construct minimal d-homogeneous Steiner 2-(v,3) trades of foundation v and volume dv/3 for sufficiently large values of v. (Specifically, v>3(1.75d^2+3) if v is divisible by 3 and v>d(4^d^/^3^+^1+1) otherwise.)

Suppose that L is a latin square of order m and P subset of L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2(n). The back circulant latin square of even order m has a well-known critical set of size m(2)/4, and this is the smallest known critical set for a latin square of order m. The abelian 2-group of order 2(n) has a critical set of size 4(n)-3(n), and this is the largest known critical set for a latin square of order 2(n). We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2(n) and the abelian 2-group of order 2(n). (C) 2007 Wiley Periodicals, Inc.

We construct face two-colourable triangulations of the graph ?$2K_n$? in an orientable surface; equivalently biembeddings of two twofold triple systems of order ?$n$?, for all ?$n \equiv 16$? or 28 (mod 48). The biembeddings come from index 1 current graphs lifted under a group ?$\mathbb{Z}_{n /4} \times \mathbb{K}_4$?. Konstruiramo triangulacije grafa ?$2K_n$? v orientabilno ploskev, ki dopuščajo dvo-barvno barvanje lic oziroma dvojne vložitve dveh dvojnih sistemov trojic reda n za vse ?$n \equiv 16$? ali 28 (mod 48). Dvojne vložitve izhajajo iz tokovnih grafov indeksa 1, dvignjenih z grupo ?$\mathbb{Z}_{n /4} \times \mathbb{K}_4$?.

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A d-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or d times. In this paper we give a construction for minimal d-homogeneous latin trades of size dm, for every integer d>=3, and m>=1.75d^2+3. We also improve this bound for small values of d. Our proof relies on the construction of cyclic sequences whose adjacent sums are distinct.

This paper discusses existence results for latin trades and provides a Glueing Construction which is subsequently used to construct all latin trades of finite order greater than three.