Studies on some graph operators and related topics

Department of Mathematics, Cochin University of Science and Technology

‘Studies on some topics in product graphs’

For routing problems in interconnection networks it is important to find the shortest containers between any two vertices, since the w-wide diameter gives the maximum communication delay when there are up to w−1 faulty nodes in a network modeled by a graph. The concept of ‘wide diameter’ was introduced by Hsu [41] to unify the concepts of diameter and The concept of ‘domination’ has attracted interest due to its wide applications in many real world situations [38]. A connected dominating set serves as a virtual backbone of a network and it is a set of vertices that helps in routing. In this thesis, we make an earnest attempt to study some of these notions in graph products. This include, the diameter variability, the diameter vulnerability, the component factors and the domination criticality.connectivity

Sources of variation contributing to production and quality attributes of Kyrgyz cashmere in Osh and Naryn provinces : implications for industry development

We aimed to quantify the sources of variation contributing to the production and quality of cashmere produced in five districts in Osh and Naryn provinces of Kyrgyzstan. In early spring 2008 mid-side cashmere samples were taken from 719 cashmere adult females, and 41 cashmere adult males and castrates. Samples came from 53 villages and a total of 156 farmers’ flocks. For 91 goats from 33 farmers in 13 villages of two districts that had been sampled earlier, cashmere was combed from the goat at the time of a second visit (end of April 2008) when the cashmere would normally be harvested. Following standard cashmere objective measurement, data were examined using general linear modelling to quantify the effects of potential determinants. The mean fibre diameter (MFD) of cashmere differed between provinces (Osh 15.7 μm, Naryn 16.7 μm; P = 4.4 × 10−20). About 42% of the cashmere was <16 μm, 48% was 16.0–18.0 μm and 9.5% was >18.0 μm. Most of the cashmere samples were coloured (81%), with 63% black and 19% white. The percentage of cashmere samples that were white declined as MFD increased (26% < 14 μm to 11% of >18 μm). The primary determinants of cashmere MFD of individual goats were age of goat (range 1.46 μm, P = 1.8 × 10−12) and farm (range 6.5 μm, P = 1.7 × 10−14). The lesser effects detected for sex (range 0.9 μm, P = 0.026) and colour of cashmere (range 1.8 μm, P = 0.023) were based on small sample sizes and are unreliable. Age of goat had important affects on fibre diameter variation (up to 1.7% in coefficient of variation, P = 5.8 × 10−6) and fibre curvature (2.5–5°/mm, P = 2.1 × 10−4). By far the greatest effect on fibre curvature was cashmere MFD (P = 3.0 × 10−104) with a smaller effect of sex (about 5°/mm, P = 3.0 × 10−6). Village effects were detected on fibre diameter variability (range 4.5% in coefficient of variation, P = 0.027) and fibre curvature (range 15°/mm, P = 1.6 × 10−7). There was a strong negative association between increasing MFD and declining fibre curvature (−5.11 ± 0.181°/mm per 1 μm; P = 7.1 × 10−121; r2 = 0.51). Average combed cashmere weight was 164 g, the clean cashmere content was 0.661 and median clean cashmere production was 110 g per goat (range 60–351 g). Combed cashmere production increased with altitude of the village, probably related to different moulting times as spring temperatures warmed up later in higher altitude villages up to 3200 masl. Measurements of combed cashmere MFD were coarser than the mid-side samples taken earlier in the year. There are farmers and cashmere goats in the sampled districts of Kyrgyzstan which produce the finest qualities of commercial cashmere as the vast majority of cashmere is fine, has low variation in fibre diameter and has fibre crimping (curvature) typical of Chinese and Mongolian cashmere. There is substantial scope to increase the production and commercial value of cashmere produced by Kyrgyz goats. In particular, some villages and farmers need to change their buck selection practices if they wish to produce acceptable cashmere. Farmers should separate their finer and white cashmere prior to sale.

Weathering, fibre strength and colour properties of processed white cashmere

Weathering refers to the degradation of wool fibres that occur during growth from exposure of the fleece to sunlight, water and air. Weathering damage to Merino wool reduces quantities of fibre that are harvested, reduces length in both raw and processed wools, reduces spinning performance and dyeing outcomes. This work aimed to aimed to quantify if and to what extent weathering occurred in 38 lots of commercial dehaired white cashmere and cashmere top sourced from traditional and new origins of production and the extent of any association between weathering and tensile strength properties of the dehaired cashmere and cashmere top. The cashmere was tested for physical properties, bundle tenacity and extension, tristimulus values brightness (Y) and yellowness (Y-Z) and reflectance. Dye uptake was used as an index of weathering. Linear models, relating to weathering, bundle tenacity and Y-Z were fitted to origin and other objective measurements. Mean attributes (range) were: mean fibre diameter, 17.0 μm (13.5–21.3 μm); bundle tenacity of tops, 10.3 cN/tex (8.3–12.9 cN/tex), for dehaired fibre, 10.1 cN/tex (9.1–11.4 cN/tex). Stain uptake varied from 0.92 to 6.34 mg/g fibre indicating a six-fold variation in the extent of weathering. Both the extent of weathering and the bundle tenacity of commercial lots of cashmere were affected by the origin of the cashmere. Increased weathering reduced bundle tenacity, bundle extension, increased the yellowness and reduced reflectance of white cashmere. Bundle tenacity of cashmere declined as fibre diameter variability increased from 20 to 22.5%. For the samples tested, the cashmere from China, Mongolia, Afghanistan and Iran showed more weathering than cashmere from Australia, New Zealand and the USA. The differences in the extent of weathering and of bundle tenacity between cashmere from different origins were of commercial significance.

Variability of ascending aorta diameter measurements as assessed with electrocardiography-gated multidetector computerized tomography and computer assisted diagnosis software.

Recently, morphometric measurements of the ascending aorta have been done with ECG-gated multidector computerized tomography (MDCT) to help the development of future novel transcatheter therapies (TCT); nevertheless, the variability of such measurements remains unknown. Thirty patients referred for ECG-gated CT thoracic angiography were evaluated. Continuous reformations of the ascending aorta, perpendicular to the centerline, were obtained automatically with a commercially available computer aided diagnosis (CAD). Then measurements of the maximal diameter were done with the CAD and manually by two observers (separately). Measurements were repeated one month later. The Bland-Altman method, Spearman coefficients, and a Wilcoxon signed-rank test were used to evaluate the variability, the correlation, and the differences between observers. The interobserver variability for maximal diameter between the two observers was up to 1.2 mm with limits of agreement [-1.5, +0.9] mm; whereas the intraobserver limits were [-1.2, +1.0] mm for the first observer and [-0.8, +0.8] mm for the second observer. The intraobserver CAD variability was 0.8 mm. The correlation was good between observers and the CAD (0.980-0.986); however, significant differences do exist (P<0.001). The maximum variability observed was 1.2 mm and should be considered in reports of measurements of the ascending aorta. The CAD is as reproducible as an experienced reader.

Variability of input parameters related to pollutants build-up in stormwater quality modelling

The variability of input parameters is the most important source of overall model uncertainty. Therefore, an in-depth understanding of the variability is essential for uncertainty analysis of stormwater quality model outputs. This paper presents the outcomes of a research study which investigated the variability of pollutants build-up characteristics on road surfaces in residential, commercial and industrial land uses. It was found that build-up characteristics vary highly even within the same land use. Additionally, industrial land use showed relatively higher variability of maximum build-up, build-up rate and particle size distribution, whilst the commercial land use displayed a relatively higher variability of pollutant-solid ratio. Among the various build-up parameters analysed, D50 (volume-median-diameter) displayed the relatively highest variability for all three land uses.

Variability in Voluntary Intake of a Molasses-based Supplement by Cows and Calves

Supplements are often fed to ruminants in extensive grazing situations to provide minerals and nitrogen likely to be deficient in pasture. However a large proportion of animals offered such supplements may not consume any supplement, while among consumer animals the variability in supplement intake may be high (Wheeler et al., 1980; Dixon et al., 1998). An experiment examined the distribution of intake of a molasses-based supplement containing phosphorus and urea in a breeder herd. A herd of mixed-age breeder cows, calves, heifers and bulls were offered ad libitum a molasses-based supplement containing 13% urea and 17% phosphoric acid. After 2 weeks lithium-labelled supplement (2 mg Li/kg LW) was offered on one day to measure individual intakes of supplement. The molasses was offered in three 560 mm diameter feeders placed together near the water point.

Sources of variability in the free-ball micro-scale abrasion test

In the 'free-ball' version of the micro-scale abrasion or ball-cratering test the rotating ball rests against a tilted sample and a grooved drive shaft. Tests under nominally identical conditions with different apparatus commonly show small but significant differences in measured wear rate. An indirect method has been developed and demonstrated for continuous on-line measurement of the coefficient of friction in the free-ball test. Experimental investigation of the effects of sample tilt angle and drive shaft groove width shows that both these factors influence the stability of the rotation of the ball, and the shape of the abrasive slurry pool, which in turn affect the coefficient of friction in the wear scar area and the measured wear rate. It is suggested that in order to improve the reproducibility of this method the geometry of the apparatus should be specified. For the apparatus used in this work with a steel ball of 25 mm diameter, a sample tilt angle of 60-75° and a shaft groove width of about 10mm provided the most stable ball motion and a wear rate which showed least variability. © 2004 Elsevier B.V. All rights reserved.

Variability of particulate matter concentrations along roads and motorways determined by a moving measurement unit

The spatial variability of aerosol number and mass along roads was determined in different regions (urban, rural and coastal-marine) of the Netherlands. A condensation particle counter (CPC) and an optical aerosol spectrometer (LAS-X) were installed in a van along with a global positioning system (GPS). Concentrations were measured with high-time resolutions while driving allowing investigations not possible with stationary equipment. In particular, this approach proves to be useful to identify those locations where numbers and mass attain high levels ('hot spots'). In general, concentrations of number and mass of particulate matter increase along with the degree of urbanisation, with number concentration being the more sensitive indicator. The lowest particle numbers and PM1-concentrations are encountered in a coastal and rural area: <5000cm-3 and 6μgm-3, respectively. The presence of sea-salt material along the North-Sea coast enhances PM>1-concentrations compared to inland levels. High-particle numbers are encountered on motorways correlating with traffic intensity; the largest average number concentration is measured on the ring motorway around Amsterdam: about 160000cm-3 (traffic intensity 100000vehday-1). Peak values occur in tunnels where numbers exceed 106cm-3. Enhanced PM1 levels (i.e. larger than 9μgm-3) exist on motorways, major traffic roads and in tunnels. The concentrations of PM>1 appear rather uniformly distributed (below 6μgm-3 for most observations). On the urban scale, (large) spatial variations in concentration can be explained by varying intensities of traffic and driving patterns. The highest particle numbers are measured while being in traffic congestions or when behind a heavy diesel-driven vehicle (up to 600×103cm-3). Relatively high numbers are observed during the passages of crossings and, at a decreasing rate, on main roads with much traffic, quiet streets and residential areas with limited traffic. The number concentration exhibits a larger variability than mass: the mass concentration on city roads with much traffic is 12% higher than in a residential area at the edge of the same city while the number of particles changes by a factor of two (due to the presence of the ultrafine particles (aerodynamic diameter <100nm). It is further indicated that people residing at some 100m downwind a major traffic source are exposed to (still) 40% more particles than those living in the urban background areas. © 2004 Elsevier Ltd. All rights reserved.

Microbial spatial variability: An example from the Celtic Sea

In July 2004, dominant populations of microbial ultraplankton (<5 μm), in the surface of the Celtic Sea (between UK and Eire), were repeatedly mapped using flow cytometry, at 1.5 km resolution over a region of diameter 100 km. The numerically dominant representatives of all basic functional types were enumerated including one group of phototrophic bacteria (Syn), two groups of phytoplankton (PP, NP), three groups of heterotrophic bacterioplankton (HB) and the regionally dominant group of heterotrophic protists (HP). The distributions of all organisms showed strong spatial variability with little relation to variability in physical fields such as salinity and temperature. Furthermore, there was little agreement between distributions of different organisms. The only linear correlation consistently explaining more than 50% of the variance between any pairing of the organism groups enumerated is between two different groups of HB. Specifically, no linear, or non-linear, relationship is found between any pairings of SYB, PP or HB groups with their protist predators HP. Looking for multiple dependencies, factor analysis reveals three groupings: Syn, PP and low nucleic acid content HB (LNA); high nucleic acid content HB (HNA); HP and NP. Even the manner in which the spatial variability of Syn, PP and HB abundance varies as a function of lengthscale (represented by a semivariogram) differs significantly from that for HP. In summary, although all microbial planktonic groups enumerated are present and numerically dominant throughout the region studied, at face value the relationships between them seem weak. Nevertheless, the behaviour of a simple, illustrative ecological model, with strongly interacting phototrophs and heterotrophs, with stochastic forcing, is shown to be consistent with the observed poor correlations and differences in how spatial variability varies with lengthscale. Thus, our study suggests that a comparison of microbial abundances alone may not discern strong underlying trophic interactions. Specific knowledge of these processes, in particular grazing, will be required to explain the causes of the observed microbial spatial variability and its resulting consequences for the functioning of the ecosystem.

Variation of fibre diameter coefficient of variation and fibre curvature across mohair fleeces: Implications for animal selection, genetic selection and fleece evaluation

The presentwork aimed to determine howthe average fibre diameter coefficient of variation (CVD) and fibre curvature (FC) differences between nine sampling sites vary between sex and flock, to identify differences in variability between sampling sites as a result of between animal and between sire variability and to determine correlations between sampling sites in between animal and between sire variability. Australian Angoras (n = 313) from two farms in southern Australia were sampled at 12 and 18 months of age at nine sites (mid side, belly, brisket, hind flank, hip, hock, mid back, neck, shoulder). Staples were taken prior to shearing at skin level and CVD and FC determined. For each shearing, differences in CVD and FC between sampling sites, how these differences were affected by farm, sex, and sire, and the covariance between sites for sire and individual animal effects were investigated by restricted maximum likelihood (REML) analyses. The median mid side CVD at 12 and 18 months of age ranged from 23.6 to 25.1% but the actual range was 16.8–34.2%. The median mid side FC at 12 and 18 months of age ranged from 14.4 to 18.6◦/mm but the actual range was 10.5–26.3◦/mm. The general pattern for CVDwas for the mid back, hip and neck sites to have similar CVD, the brisket, hind flank and hock sites to have larger CVD and the belly to have smaller CVD than the mid side site. The between animal variation for CVD was lowest at the mid back site. This implies that the mid back would be the most effective site for between animal selection for CVD. Heritabilities for CVD (range at 18 months 0.18–0.30) were only about half the heritabilities for mean fibre diameter in the same study. There was a marked anterior–posterior increase in FC at both farms and with both ages. The results give no clear indication of the best site for between animal selection for FC, other than that the hock should be avoided. Heritabilities for FC are moderate to high (range at 18 months 0.44–0.77) and the genetic correlations are high except for the hock. Thus genetic selection for FC at any site, other than the hock, should be effective for changing FC over the entire fleece. There was more variability between animals than between sites and sires. These results are put into context with associated research on variation in mean fibre diameter and staple length.

Spatial variability of physical attributes of an alfisol under different hillslope curvatures

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Spatial variability of aggregate stability in latosols under sugarcane

Solos submetidos ao mesmo sistema de manejo manifestam variabilidade espacial diferenciada de seus atributos. A variabilidade espacial da estabilidade de agregados foi caracterizada em um Latossolo Vermelho distrófico e Latossolo Vermelho eutroférrico, sob cultivo de cana-de-açúcar. Foram realizadas amostragens de terra nos pontos de interseção de uma malha de 10 x 10 linhas, espaçadas de 10 m, totalizando 100 pontos de amostragem por área, coletadas nas camadas de 0,0-0,2 e 0,2-0,4 m de profundidade, para determinação de diâmetro médio geométrico (DMG), diâmetro médio ponderado (DMP), agregados na classe >2,0 mm e teor de matéria orgânica do solo. O Latossolo Vermelho eutroférrico apresenta maior estabilidade de agregados quando comparado com o Latossolo Vermelho distrófico, condizente com o maior teor de argila, matéria orgânica e mineralogia gibbsítica. A evolução diferenciada dos Latossolos estudados explica os maiores alcances, o menor coeficiente de variação (CV) e a menor variabilidade, observados no Latossolo Vermelho eutroférrico para todos os atributos estudados.

Spatial variability of infiltration and its relationship to some physical properties

Water infiltration into soil is one of the basic factors for estimating irrigation intensity according to the plants' requirements; this is aimed at avoiding problems of surface run-off and degradation. The purpose of the present investigation was to determine the spatial variation of infiltration and its relationship to some physical properties of soil by means of geostatistical techniques in Typic Plinthaquult soils having average texture and flat relief. A 113 point mesh was designned, having a regular distance of 10 m between points, samples being taken from 0 to 0.20 meters depth. Sand, silt and clay content, bulk density, macroporosity, microporosity and total porosity were determined. Infiltration tests were carried out in the field by means of a 15 cm diameter ring. Descriptive statistics and geostatistics were used for analysing the data. Infiltration, silt and microporosity data did not fit a normal distribution curve. Infiltration had high variability, having an average 36.03 mm h(-1). Total porosity was 56.73%, this being the only property that did not show spatial dependency. The smallest ranges were observed for bulk density, macroporosity and microporosity, having values of less than 40 m. The smallest degrees of spatial dependence were observed for infiltration, silt and clay, evidence also being shown of the influence of silt and clay on infiltration rate. Contour maps were constructed; fitting them to the semivariogram models, together with studying the correlations, led to establishing relationships between the properties.