Mapping quantitative physiological traits in apple ( Malus × domestica Borkh.)

Efficient breeding and selection of high-quality apple cultivars requires knowledge and understanding of the underlying genetics. The availability of genetic linkage maps constructed with molecular markers enables the detection and analysis of major genes and quantitative trait loci contributing to the quality traits of a genotype. A segregating population of the cross between the apple varieties 'sFiesta' (syn. 'sRed Pippin') and 'sDiscovery' has been observed over three years at three different sites in Switzerland and data on growth habit, blooming behaviour, juvenile period and fruit quality has been recorded. QTL analyses were performed, based on a genetic linkage map consisting of 804 molecular markers and covering all 17 apple chromosomes. With the maximum likelihood based interval mapping method, the investigated complex traits could be dissected into a number of QTLs affecting the observed characters. Genomic regions participating in the genetic control of stem diameter, plant height increment, leaf size, blooming time, blooming intensity, juvenile phase length, time of fruit maturity, number of fruit, fruit size and weight, fruit flesh firmness, sugar content and fruit acidity were identified and compared with previously mapped QTLs in apple. Although 'sDiscovery' fruit displayed a higher acid content, both acidity QTLs were attributed to the sweeter parent 'sFiesta'. This indicated homozygosity at the acidity loci in 'sDiscovery' preventing their detection in the progeny due to the lack of segregatio

Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types

The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\mathbb{H}^3$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.Comment: 26 pages, 10 figure

Molecular markers linked to the apple scab resistance gene Vbj derived from Malus baccata jackii

Breeding for scab-resistant apple cultivars by pyramiding several resistance genes in the same genetic background is a promising way to control apple scab caused by the fungus Venturia inaequalis. To achieve this goal, DNA markers linked to the genes of interest are required in order to select seedlings with the desired resistance allele combinations. For several apple scab resistance genes, molecular markers are already available; but until now, none existed for the apple scab resistance gene Vbj originating from the crab apple Malus baccata jackii. Using bulk segregant analysis, three RAPD markers linked to Vbj were first identified. These markers were transformed into more reliable sequence-characterised amplified region (SCAR) markers that proved to be co-dominant. In addition, three SSR markers and one SCAR were identified by comparing homologous linkage groups of existing genetic maps. Discarding plants showing genotype-phenotype incongruence (GPI plants) plants, a linkage map was calculated. Vbj mapped between the markers CH05e03 (SSR) and T6-SCAR, at 0.6cM from CH05e03 and at 3.9cM from T6-SCAR. Without the removal of the GPI plants, Vbj was placed 15cM away from the closest markers. Problems and pitfalls due to GPI plants and the consequences for mapping the resistance gene accurately are discussed. Finally, the usefulness of co-dominant markers for pedigree analysis is also demonstrate

On commensurable hyperbolic Coxeter groups

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space Hn, new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions

Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra

We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume bounds follow from techniques related to the proof of Thurston's Orbifold Theorem, Schl\"afli's formula, and previous results of the author giving volume bounds for right-angled hyperbolic polyhedra.Comment: 36 pages, 19 figure

Bed-load Transport of Mixed-size Sediment: Fractional Transport Rates, Bed Forms, and the Development of a Coarse Bed-surface Layer

Fractional transport rates, bed surface texture, and bed configuration were measured after a mixed size sediment had reached an equilibrium transport state for seven different flow strengths in a recirculating laboratory flume. Fractional transport rates were also measured at the beginning of each run when the bed was well mixed and planar. The start-up observations allow us to describe the variation of fractional transport rates with bed shear stress for a constant bed surface texture and bed configuration. The start-up and equilibrium observations together allow, for the first time, an unambiguous description of the mutual adjustment among the transport, the bed configuration, and the bed surface, as the transport system moves toward equilibrium. We find that a substantial interaction exists among the transport, bed surface, and bed configuration. Bed forms and a coarse surface layer coexist over a range of bed shear stress. Under some flow conditions the size and shape of the bed forms are controlled by the presence of the coarse surface layer. At higher flows the coarse surface layer is eliminated by scour in the lee of the bed forms. If the bed surface is defined as that over which the bed forms move, a coherent relation between the bed surface texture and the transport grain size distribution may be defined. At equilibrium the transport rates of all fractions were not equally mobile, defined as identical transport and bed grain size distributions, although equal mobility was approached for runs in which the bed shear stress was more than twice that for initial motion of the mixture. Under some flow conditions the transport was observed to adjust away from equal mobility as the bed adjusted from a well-mixed start-up condition to an equilibrium state. Development of a partial static armor, wherein some individual grains become essentially immobile even though other grains in the same fraction remain in transport, is suggested to explain these adjustments between the transport and bed surface grain size distributions. Constraints on equilibrium mixed size sediment transport are defined. The special conditions for which equal mobility must hold and the relevance to natural conditions of flume results and the equal mobility concept are discussed

Molecular markers linked to the apple scab resistance gene Vbj derived from Malus baccata jackii

Breeding for scab-resistant apple cultivars by pyramiding several resistance genes in the same genetic background is a promising way to control apple scab caused by the fungus Venturia inaequalis. To achieve this goal, DNA markers linked to the genes of interest are required in order to select seedlings with the desired resistance allele combinations. For several apple scab resistance genes, molecular markers are already available; but until now, none existed for the apple scab resistance gene Vbj originating from the crab apple Malus baccata jackii. Using bulk segregant analysis, three RAPD markers linked to Vbj were first identified. These markers were transformed into more reliable sequence-characterised amplified region (SCAR) markers that proved to be co-dominant. In addition, three SSR markers and one SCAR were identified by comparing homologous linkage groups of existing genetic maps. Discarding plants showing genotype\u2013phenotype incongruence (GPI plants) plants, a linkage map was calculated. Vbj mapped between the markers CH05e03 (SSR) and T6-SCAR, at 0.6 cM from CH05e03 and at 3.9 cM from T6-SCAR. Without the removal of the GPI plants, Vbj was placed 15 cM away from the closest markers. Problems and pitfalls due to GPI plants and the consequences for mapping the resistance gene accurately are discussed. Finally, the usefulness of co-dominant markers for pedigree analysis is also demonstrated