Decompositions of complete graphs into triangles and Hamilton cycles

For all odd integers n greater than or equal to 1, let G(n) denote the complete graph of order n, and for all even integers n greater than or equal to 2 let G,, denote the complete graph of order n with the edges of a 1-factor removed. It is shown that for all non-negative integers h and t and all positive integers n, G, can be decomposed into h Hamilton cycles and t triangles if and only if nh + 3t is the number of edges in G(n). (C) 2004 Wiley Periodicals, Inc.

Atomic Latin squares of order eleven

A Latin square is pan-Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i j. A Latin square is atomic if all of its conjugates are pan-Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1-factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan-Hamiltonian Latin square of order n describes a perfect 1-factorization of Kn,n, and vice versa. Perfect 1-factorizations of Kn,n can be constructed from a perfect 1-factorization of Kn+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn-square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self-orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self-orthogonal Latin squares in the same main class as a given Latin square.

Common multiples of complete graphs

A graph H is said to divide a graph G if there exists a set S of subgraphs of G, all isomorphic to H, such that the edge set of G is partitioned by the edge sets of the subgraphs in S. Thus, a graph G is a common multiple of two graphs if each of the two graphs divides G.

The Fornax spectroscopic survey I. Survey strategy and preliminary results on the redshift distribution of a complete sample of stars and galaxies

The Fornax Spectroscopic Survey will use the Two degree Field spectrograph (2dF) of the Angle-Australian Telescope to obtain spectra for a complete sample of all 14000 objects with 16.5 less than or equal to b(j) less than or equal to 19.7 in a 12 square degree area centred on the Fornax Cluster. The aims of this project include the study of dwarf galaxies in the cluster (both known low surface brightness objects and putative normal surface brightness dwarfs) and a comparison sample of background field galaxies. We will also measure quasars and other active galaxies, any previously unrecognised compact galaxies and a large sample of Galactic stars. By selecting all objects-both stars and galaxies-independent of morphology, we cover a much larger range of surface brightness and scale size than previous surveys. In this paper we first describe the design of the survey. Our targets are selected from UK Schmidt Telescope sky survey plates digitised by the Automated Plate Measuring (APM) facility. We then describe the photometric and astrometric calibration of these data and show that the APM astrometry is accurate enough for use with the 2dF. We also describe a general approach to object identification using cross-correlations which allows us to identify and classify both stellar and galaxy spectra. We present results from the first 2dF field. Redshift distributions and velocity structures are shown for all observed objects in the direction of Fornax, including Galactic stars? galaxies in and around the Fornax Cluster, and for the background galaxy population. The velocity data for the stars show the contributions from the different Galactic components, plus a small tail to high velocities. We find no galaxies in the foreground to the cluster in our 2dF field. The Fornax Cluster is clearly defined kinematically. The mean velocity from the 26 cluster members having reliable redshifts is 1560 +/- 80 km s(-1). They show a velocity dispersion of 380 +/- 50 km s(-1). Large-scale structure can be traced behind the cluster to a redshift beyond z = 0.3. Background compact galaxies and low surface brightness galaxies are found to follow the general galaxy distribution.

Synchronous oogenesis during the semilunar spawning cycle of the tropical abalone Haliotis asinina

On the southern Great Barrier Reef, Haliotis asinina (Vetigastropoda: Pleurotomarioidea) synchronously spawn every 2 wk in a predictable fashion. allowing detailed analysis of reproduction, gametogenesis, and gonad development. Histological examination of the ovaries of members of the Heron Reef population during this semilunar cycle reveals that oogenesis is also synchronous and predictable, and requires more than two spawning cycles (i.e. >28 days) to complete. Shortly after a spawning event the ovary comprises two cohorts of primary oocytes, one of which will be released at the next spawning event, and clusters of oogonia. At this time there is a rapid proliferation and expansion of trabeculae, germinal epithelial, and oogonia, and a dramatic increase in the size of the vitellogenic oocytes to be: spawned at the next spawning event. Within 4 days these oocytes have filled the ovary. On the day of the next spawning a lumen forms in the ovary as a result of localized degradation of trabeculae. The large primary oocytes dissociate from the receding trabeculae. initiate maturation, and accumulate in the lumen; these oocytes become embedded in a jelly coat layer. The next cohort of oocytes remain attached to the trabeculae. The jelly coat appears to be completely dissolved within 30 min of spawning. Comparison of the oogenesis and ovary development in II. asinina with other abalone species indicates that these processes are very similar in tropical and temperate abalone. This suggests that insights into the regulation of reproduction and spawning in H. asinina are likely to be applicable to other haliotids.

Quasars from a complete spectroscopic survey

With the advent of multi-fibre spectrographs such as the 'Two-Degree Field' (2dF) instrument at the Angle-Australian Telescope, quasar surveys that are free of any preselection of candidates and any biases this implies have become possible for the first time. The first of these is that which is being undertaken as part of the Fornax Spectroscopic Survey, a survey of the area around the Fornax Cluster of galaxies, and aims to obtain the spectra of all objects in the magnitude range 16.5 < b(j) < 19.7. To date, 3679 objects in the central pi -deg(2) area have been successfully identified from their spectral characteristics. Of these, 71 are found to be quasars, 61 with redshifts 0.3 < z < 2.2 and 10 with redshifts z > 2.2. Using this complete quasar sample, a new determination of quasar number counts is made, enabling an independent check of existing quasars surveys. Cumulative counts per square degree at a magnitude limit of b(j) < 19.5 are found to be 11.5 +/- 2.2 for 0.3 < z < 2.2, 2.22 +/- 0.93 for z > 2.2 and 13.7 +/- 3.1 for z > 0.3. Given the likely detection of extra quasars in the Fornax survey, we make a more detailed examination of existing quasar selection techniques. First, looking at the use of a stellar criterion, four of the 71 quasars are 'non-stellar' on the basis of the automated plate measuring facility (APM) b(j) classification, however inspection shows all are consistent with stellar, but misclassified due to image confusion. Examining the ultraviolet excess and multicolour selection techniques, for the selection criteria investigated, ultraviolet excess would find 69 +/- 6 per cent of our 0.3 < z < 2.2 quasars and only 50(-18)(+14), per cent of our z > 2.2 quasars, while the completeness level for multicolour selection is found to be 90(-4)(+3) per cent for 0.3 < z < 2.2 quasars and 80(-12)(+14) per cent for z > 2.2 quasars. The extra quasars detected by our all-object survey thus have unusually red star-like colours, and this appears to be a result of the continuum shape rather than any emission features. An intrinsic dust extinction model may, at least partly, account for the red colours.

The crystal structure of free human hypoxanthineguanine phosphoribosyltransferase reveals extensive conformational plasticity throughout the catalytic cycle

Human hypoxanthine-guanine phosphoribosyltransferase (HGPRT) catalyses the synthesis of the purine nucleoside monophosphates, IMP and GMP, by the addition of a 6-oxopurine base, either hypoxanthine or guanine, to the 1-beta-position of 5-phospho-U-D-ribosyl-1-pyrophosphate (PRib-PP). The mechanism is sequential, with PRib-PP binding to the free enzyme prior to the base. After the covalent reaction, pyrophosphate is released followed by the nucleoside monophosphate. A number of snapshots of the structure of this enzyme along the reaction pathway have been captured. These include the structure in the presence of the inactive purine base analogue, 7-hydroxy [4,3-d] pyrazolo pyrimidine (HPP) and PRib-PP. Mg2+, and in complex with IMP or GMP. The third structure is that of the immucillinHP.Mg2+.PPi complex, a transition-state analogue. Here, the first crystal structure of free human HGPRT is reported to 1.9 angstrom resolution, showing that significant conformational changes have to occur for the substrate(s) to bind and for catalysis to proceed. Included in these changes are relative movement of subunits within the tetramer, rotation and extension of an active-site alpha-helix (D137-D153), reorientation of key active-site residues K68, D137 and K165, and the rearrangement of three active-site loops (100-128, 165-173 and 186-196). Toxoplasina gondii HGXPRT is the only other 6-oxopurine phosphoribosyltransferase structure solved in the absence of ligands. Comparison of this structure with human HGPRT reveals significant differences in the two active sites, including the structure of the flexible loop containing K68 (human) or K79 (T gondii). (c) 2005 Elsevier Ltd. All rights reserved.

The intersection problem for K_4-e designs

A G-design of order n is a pair (P,B) where P is the vertex set of the complete graph K-n and B is an edge-disjoint decomposition of K-n into copies of the simple graph G. Following design terminology, we call these copies ''blocks''. Here K-4 - e denotes the complete graph K-4 with one edge removed. It is well-known that a K-4 - e design of order n exists if and only if n = 0 or 1 (mod 5), n greater than or equal to 6. The intersection problem here asks for which k is it possible to find two K-4 - e designs (P,B-1) and (P,B-2) of order n, with \B-1 boolean AND B-2\ = k, that is, with precisely k common blocks. Here we completely solve this intersection problem for K-4 - e designs.

Strongly 2-perfect cycle systems and their quasigroups

A recent result of Bryant and Lindner shows that the quasigroups arising from 2-perfect m-cycle systems form a variety only when m = 3, 5 and 7. Here we investigate the situation in the case where the distance two cycles are required to be in the original system.

A comparison of dissected follicle numbers and follicle counts on the ovarian surface for the evaluation of ovarian follicular populations in Bos indicus cows

Ovaries (n = 140) from 70 mixed-age multiparous, lactating Brahman cross (3/4-7/8 Bos indicus) cows were used to examine the hypothesis that counts of follicles visible on the surface of the ovaries of Bos indicus cows and their classification into diameter size classes, are closely correlated with numbers of follicles in those size classes found by complete dissection of the ovary. immediately after ovariectomy, mean diameters (long and short axes averaged) of all follicles greater than or equal to 2 mm visible on the surface of each ovary were measured. All follicles greater than or equal to 2 mm were dissected from the ovaries, excess stroma removed and follicle diameters measured under a stereomicroscope using an ocular graticule. For each ovary, follicles were classified in either small (8 mm) categories based on either diameters of surface or dissected follicles. Data for numbers of surface and dissected follicles (mean +/- SE) in small, medium, large categories and total follicle numbers, respectively, were 24.4 +/- 1.6 vs. 28.0 +/- 1.9, 1.6 +/- + 0.2 vs. 11.6 +/- 1.0, 0.5 +/- 0.1 vs. 0.7 +/- 0.1 and 26.4 +/- 1.6 vs. 40.4 +/- 2.5. Correlation coefficients (r) for counts of surface and dissected follicles in small, medium, large and total follicle numbers were 0.76, 0.40, 0.69 and 0.79, respectively. Medium size follicles presented only a small translucent area on the surface of the ovary, leading to an underestimate of numbers when categorised by surface evaluation. Counts of follicles visible on the surface of the ovaries of Bos indicus cows and their classification into size classes based on estimated diameter, are closely correlated with numbers of follicles in those size classes found at dissection of the ovary for small (8 mm) and total follicles but not for medium sized (4-8 mm) follicles. (C) 1997 Elsevier Science B.V.

On single laws for varieties of quasigroups associated with extended cycle systems

It has been previously shown by Lindner and Rodger that quasigroups associated with 2-perfect extended m-cycle systems can be equationally defined if and only if m is an element of {3, 5, 7}. In this paper we present a single identity for each such m which is equivalent to the identities given for these varieties.

A novel variant genotype C of hepatitis B virus identified in isolates from Australian Aborigines: complete genome sequence and phylogenetic relatedness

There have been no reports of DNA sequences of hepatitis B virus (HBV) strains from Australian Aborigines, although the hepatitis B surface antigen (HBsAg) was discovered among them. To investigate the characteristics of DNA sequences of HBV strains from Australian Aborigines, the complete nucleotide sequences of HBV strains were determined and subjected to molecular evolutionary analysis. Serum samples positive for HBsAg were collected from five Australian Aborigines. Phylogenetic analysis of the five complete nucleotide sequences compared with DNA sequences of 54 global HBV isolates from international databases revealed that three of the five were classified into genotype D and were most closely related in terms of evolutionary distance to a strain isolated from a healthy blood donor in Papua New Guinea. Two of the five were classified into a novel variant genotype C, which has not been reported previously, and were closely related to a strain isolated from Polynesians, particularly in the X and Core genes. These two strains of variant genotype C differed from known genotype C strains by 5.9-7.4% over the complete nucleotide sequence and 4.0-5.6 % in the small-S gene, and had residues Arg(122), Thr(127) and Lys(160) characteristic of serotype ayw3, which have not been reported previously in genotype C. In conclusion, this is the first report of the characteristics of complete nucleotide sequences of HBV from Australian Aborigines. These results contribute to the investigation of the worldwide spread of HBV, the relationship between serotype and genotype and the ancient common origin of Australian Aborigines.

Rate and timing of vegetative growth, flowering and fruit development of Persoonia virgata (Proteaceae)

Persoonia virgata R. Br. is harvested from the wild in both its vegetative and flowering stages. There has been no systematic study published on the annual growth cycle and anecdotal reports are conflicting. The growth pattern, flowering and fruit development of P. virgata in its natural habitat was recorded monthly for two consecutive years. The main growth period occurred in late spring-mid-autumn (November-May) when the shrubs were producing little or no fruit. Very few open flowers were observed at the site over the 2 years, with only 6.7 and 12.7% of stems bearing open flowers in January and February 1996, respectively. A second study of flowering on container-grown shrubs showed that individual flowers were open for only 2-5 days, with individual stems taking 3-8.5 weeks to complete flowering. The main fruit growth period occurred from May to September, and in June and July 1996 the total fruit set per stem was 41.6 and 36.1%, respectively. The fruit took at least 6 months to develop during which vegetative growth was minimal. The harvesting of plants in the flowering or fruiting stages removes the annual seed crop, which may reduce regeneration of this obligate seed regenerator and threaten its survival after fire.