Single machine scheduling and due date assignment with positionally dependent processing times

We consider single machine scheduling and due date assignment problems in which the processing time of a job depends on its position in a processing sequence. The objective functions include the cost of changing the due dates, the total cost of discarded jobs that cannot be completed by their due dates and, possibly, the total earliness of the scheduled jobs. We present polynomial-time dynamic programming algorithms in the case of two popular due date assignment methods: CON and SLK. The considered problems are related to mathematical models of cooperation between the manufacturer and the customer in supply chain scheduling.

Emotions and the habitus: Young people with socio-emotional differences (re)producing social, emotional and cultural capital in family and leisure space-times

In this paper we explore the importance of emotionally inter-dependent relationships to the functioning of embodied social capital and habitus. Drawing upon the experiences of young people with socio-emotional differences, we demonstrate how emotionally inter-dependent and relatively nurturing relationships are integral to the acquisition of social capital and to the co-construction and embodiment of habitus. The young people presented in this paper often had difficulties in forging social relationships and in acquiring symbolic and cultural capital in school spaces. However, we outline how these young people (re)produce and embody alternative kinds of habitus, based on emotionally reciprocal relationships forged through formal and informal leisure activities and familial and fraternal social relationships. These alternative forms of habitus provide sites of subjection, scope for acquiring social and cultural capital and a positive sense of identity in the face of problematic relations and experiences in school spaces.

Production lot sizing and scheduling with non-triangular sequence-dependent setup times

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Analysis of recurrent failure times: A time-dependent Yule process approach

Analysis of recurrent events has been widely discussed in medical, health services, insurance, and engineering areas in recent years. This research proposes to use a nonhomogeneous Yule process with the proportional intensity assumption to model the hazard function on recurrent events data and the associated risk factors. This method assumes that repeated events occur for each individual, with given covariates, according to a nonhomogeneous Yule process with intensity function λx(t) = λ 0(t) · exp( x′β). One of the advantages of using a non-homogeneous Yule process for recurrent events is that it assumes that the recurrent rate is proportional to the number of events that occur up to time t. Maximum likelihood estimation is used to provide estimates of the parameters in the model, and a generalized scoring iterative procedure is applied in numerical computation. ^ Model comparisons between the proposed method and other existing recurrent models are addressed by simulation. One example concerning recurrent myocardial infarction events compared between two distinct populations, Mexican-American and Non-Hispanic Whites in the Corpus Christi Heart Project is examined. ^

Growth and shrinkage in Antarctic krill Euphausia superba is sex-dependent

ABSTRACT: The ability of Antarctic krill Euphausia superba Dana to withstand the overwintering period is critical to their success. Laboratory evidence suggests that krill may shrink in body length during this time in response to the low availability of food. Nevertheless, verification that krill can shrink in the natural environment is lacking because winter data are difficult to obtain. One of the few sources of winter krill population data is from commercial vessels. We examined length-frequency data of adult krill (>35 mm total body length) obtained from commercial vessels in the Scotia-Weddell region and compared our results with those obtained from a combination of science and commercial sampling operations carried out in this region at other times of the year. Our analyses revealed body-length shrinkage in adult females but not males during overwinter, based on both the tracking of modal size classes over seasons and sex-ratio patterns. Other explanatory factors, such as differential mortality, immigration and emigration, could not explain the observed differences. The same pattern was also observed at South Georgia and in the Western Antarctic Peninsula. Fitted seasonally modulated von Bertalanffy growth functions predicted a pattern of overwintering shrinkage in all body-length classes of females, but only stagnation in growth in males. This shrinkage most likely reflects morphometric changes resulting from the contraction of the ovaries and is not necessarily an outcome of winter hardship. The sex-dependent changes that we observed need to be incorporated into life cycle and population dynamic models of this species, particularly those used in managing the fishery. KEY WORDS: Southern Ocean · Population dynamics · Production · Life cycle · Fishery

Growth and shrinkage in Antarctic krill Euphausia superba is sex-dependent

ABSTRACT: The ability of Antarctic krill Euphausia superba Dana to withstand the overwintering period is critical to their success. Laboratory evidence suggests that krill may shrink in body length during this time in response to the low availability of food. Nevertheless, verification that krill can shrink in the natural environment is lacking because winter data are difficult to obtain. One of the few sources of winter krill population data is from commercial vessels. We examined length-frequency data of adult krill (>35 mm total body length) obtained from commercial vessels in the Scotia-Weddell region and compared our results with those obtained from a combination of science and commercial sampling operations carried out in this region at other times of the year. Our analyses revealed body-length shrinkage in adult females but not males during overwinter, based on both the tracking of modal size classes over seasons and sex-ratio patterns. Other explanatory factors, such as differential mortality, immigration and emigration, could not explain the observed differences. The same pattern was also observed at South Georgia and in the Western Antarctic Peninsula. Fitted seasonally modulated von Bertalanffy growth functions predicted a pattern of overwintering shrinkage in all body-length classes of females, but only stagnation in growth in males. This shrinkage most likely reflects morphometric changes resulting from the contraction of the ovaries and is not necessarily an outcome of winter hardship. The sex-dependent changes that we observed need to be incorporated into life cycle and population dynamic models of this species, particularly those used in managing the fishery. KEY WORDS: Southern Ocean · Population dynamics · Production · Life cycle · Fishery

ENZYME-FREE UBIQUITIN LIGATION. FROM NATIVE CHEMICAL LIGATION AND SPIN LABELING TO DIMERIZATION BY INTER-UBIQUITIN MIMICKING LINKAGE AND PH-DEPENDENT CONFORMATIONAL SWITCH

The delicate balance between the production and disposal of proteins is vital for the changes required in the cell to respond to given stimulus. Ubiquitination is a protein modification with a range of signaling outcomes when ubiquitin is attached to a protein through a highly ordered enzymatic cascade process. Understanding ubiquitination is a growing field and nowadays the application of chemical reactions allows the isolation of quantitative materials for structural studies. Therefore, in this dissertation it is described some of these suitable chemical methodologies to produce an isopeptide bond toward the polymerization of ubiquitin bypassing the enzymatic control with the purpose of showing if these chemical modifications have a direct impact on the structure of ubiquitin. First, the possibility of incorporating non-natural lysine analogs known as mercaptolysines into the polypeptide chain of Ubiquitin was explored when they were attached to ubiquitin by native chemical ligation at its C terminus. The sulfhydryl group was used for the attachment of a paramagnetic label to map the surface of ubiquitin. Second, the condensation catalyzed by silver nitrate was used for the dimer assembly. In particular, the main focus was on examining whether orthogonal protection and deprotection of each monomer have an impact on the reaction yield, since the synthetic strategy has been previously attempted successfully. Third, the formation of ubiquitin dimers was approached by building an inter-ubiquitin linkage mimicking the isopeptide bond with two approaches, the classic disulfide exchange as well as the thiol-ene click reaction by thermal initiation in aqueous conditions. After assembling the dimeric units, they were studied by Nuclear Magnetic Resonance, in order to establish a conformational state profile which depends on the pH conditions. The latter is a very important concept since some ligands have a preferred affinity when the protein-protein hydrophobic patches are in close proximity.

Inter-agency working : good intentions and interaction dynamics

This paper explores inter-agency working and examines the implications of inter-agency operations for delivering multi-domain service outcomes. Cross-agency collaborative approaches to service delivery are suggested to provide the vehicle for achieving integrated service and policy goals. However, it is argued these need to be crafted ‘fit’ for purpose’ and may not be the requisite approach for all joint purposes. Moreover, some commentators suggest that the optimism about these partnership arrangements and cross-agency actions to resolve complex multi-dimensional problems may be misplaced and propose that further research into the actual rather than desired consequences of these arrangements may find that, at times, partnership working creates negative effects. While collaboration and partnerships are often framed as the way to achieve real breakthroughs in service delivery across agencies, there remain key challenges to interagency working. As more and insistent calls for agencies and other community actors to work together in resolving complex social problems are heeded, the implications of working across organizational boundaries need to be further investigated. This paper investigates cases of inter-agency programmes to understand the dimensions and limitations of inter-agency working. The paper concludes by offering a framework for better inter-agency working that has applicability across all sectors.

Kinematics of intracellular chlamydiae provide evidence for contact-dependent development

A crucial process of chlamydial development involves differentiation of the replicative reticulate body (RB) into the infectious elementary body (EB). We present experimental evidence to provide support for a contact-dependent hypothesis for explaining the trigger involved in differentiation. We recorded live-imaging of Chlamydia trachomatis-infected McCoy cells at key times during development and tracked the temporospatial trajectories of individual chlamydial particles. We found that movement of the particles is related to development. Early to mid-developmental stages involved slight wobbling of RBs. The average speed of particles increased sharply at 24 h postinfection (after the estimated onset of RB to EB differentiation). We also investigated a penicillin-supplemented culture containing EBs, RBs, and aberrantly enlarged, stressed chlamydiae. Near-immobile enlarged particles are consistent with their continued tethering to the chlamydial inclusion membrane (CIM). We found a significantly negative, nonlinear association between speed and size/type of particles, providing further support for the hypothesis that particles become untethered near the onset of RB to EB differentiation. This study establishes the relationship between the motion properties of the chlamydiae and developmental stages, whereby wobbling RBs gradually lose contact with the CIM, and RB detachment from the CIM is coincidental with the onset of late differentiation.

Vibration of ship structures and its control

Noise and vibration in complex ship structures are becoming a prominent issue for ship building industry and ship companies due to the constant demand of building faster ships of lighter weight, and the stringent noise and libration regulation of the industry. In order to retain the full benefit of building faster ships without compromising too much on ride comfort and safety, noise and vibration control needs to be implemented. Due to the complexity of ship structures, the coupling of different wave types and multiple wave propagation paths, active control of global hull modes is difficult to implement and very expensive. Traditional passive control such as adding damping materials is only effective in the high frequency range. However, most severe damage to ship structures is caused by large structural deformation of hull structures and high dynamic stress concentration at low frequencies. The most discomfort and fatigue of passengers and the crew onboard ships is also due to the low frequency noise and vibration. Innovative approaches are therefore, required to attenuate the noise and vibration at low frequencies. This book was developed from several specialized research topics on vibration and vibration control of ship structures, mostly from the author's own PhD work at the University of Western Australia. The book aims to provide a better understanding of vibration characteristics of ribbed plate structures, plate/plate coupled structures and the mechanism governing wave propagation and attenuation in periodic and irregular ribbed structures as well as in complex ship structures. The book is designed to be a reference book for ship builders, vibro-acoustic engineers and researchers. The author also hopes that the book can stimulate more exciting future work in this area of research. It is the author's humble desire that the book can be some use for those who purchase it. This book is divided into eight chapters. Each chapter focuses on providing solution to address a particular issue on vibration problems of ship structures. A brief summary of each chapter is given in the general introduction. All chapters are inter-dependent to each other to form an integration volume on the subject of vibration and vibration control of ship structures and alike. I am in debt to many people in completing this work. In particular, I would like to thank Professor J. Pan, Dr N.H. Farag, Dr K. Sum and many others from the University of Western Australia for useful advices and helps during my times at the University and beyond. I would also like to thank my wife, Miaoling Wang, my children, Anita, Sophia and Angela Lin, for their sacrifice and continuing supports to make this work possible. Financial supports from Australian Research Council, Australian Defense Science and Technology Organization and Strategic Marine Pty Ltd at Western Australia for this work is gratefully acknowledged.

Time-dependent rates of molecular evolution

For over half a century, it has been known that the rate of morphological evolution appears to vary with the time frame of measurement. Rates of microevolutionary change, measured between successive generations, were found to be far higher than rates of macroevolutionary change inferred from the fossil record. More recently, it has been suggested that rates of molecular evolution are also time dependent, with the estimated rate depending on the timescale of measurement. This followed surprising observations that estimates of mutation rates, obtained in studies of pedigrees and laboratory mutation-accumulation lines, exceeded long-term substitution rates by an order of magnitude or more. Although a range of studies have provided evidence for such a pattern, the hypothesis remains relatively contentious. Furthermore, there is ongoing discussion about the factors that can cause molecular rate estimates to be dependent on time. Here we present an overview of our current understanding of time-dependent rates. We provide a summary of the evidence for time-dependent rates in animals, bacteria and viruses. We review the various biological and methodological factors that can cause rates to be time dependent, including the effects of natural selection, calibration errors, model misspecification and other artefacts. We also describe the challenges in calibrating estimates of molecular rates, particularly on the intermediate timescales that are critical for an accurate characterization of time-dependent rates. This has important consequences for the use of molecular-clock methods to estimate timescales of recent evolutionary events.

Empirical analysis on relationship between traffic conditions and crash occurrences

This paper investigates relationship between traffic conditions and the crash occurrence likelihood (COL) using the I-880 data. To remedy the data limitations and the methodological shortcomings suffered by previous studies, a multiresolution data processing method is proposed and implemented, upon which binary logistic models were developed. The major findings of this paper are: 1) traffic conditions have significant impacts on COL at the study site; Specifically, COL in a congested (transitioning) traffic flow is about 6 (1.6) times of that in a free flow condition; 2)Speed variance alone is not sufficient to capture traffic dynamics’ impact on COL; a traffic chaos indicator that integrates speed, speed variance, and flow is proposed and shows a promising performance; 3) Models based on aggregated data shall be interpreted with caution. Generally, conclusions obtained from such models shall not be generalized to individual vehicles (drivers) without further evidences using high-resolution data and it is dubious to either claim or disclaim speed kills based on aggregated data.

Volume-dependent response of precooling for intermittent-sprint exercise in the heat

Purpose: To assess the effects of pre-cooling volume on neuromuscular function and performance in free-paced intermittent-sprint exercise in the heat. Methods: Ten male, teamsport athletes completed four randomized trials involving an 85-min free-paced intermittentsprint exercise protocol in 33°C±33% relative humidity. Pre-cooling sessions included whole body (WB), head+hand (HH), head (H) and no cooling (CONT), applied for 20-min pre-exercise and 5-min mid exercise. Maximal voluntary contractions (MVC) were assessed pre- and postintervention and mid- and post-exercise. Exercise performance was assessed with sprint times, % decline and distances covered during free-paced bouts. Measures of core(Tc) and skin (Tsk) temperatures, heart rate, perceptual exertion and thermal stress were monitored throughout. Venous and capillary blood was analyzed for metabolite, muscle damage and inflammatory markers. Results: WB pre-cooling facilitated the maintenance of sprint times during the exercise protocol with reduced % decline (P=0.04). Mean and total hard running distances increased with pre cooling 12% compared to CONT (P<0.05), specifically, WB was 6-7% greater than HH (P=0.02) and H (P=0.001) respectively. No change was evident in mean voluntary or evoked force pre- to post-exercise with WB and HH cooling (P>0.05). WB and HH cooling reduced Tc by 0.1-0.3°C compared to other conditions (P<0.05). WB Tsk was suppressed for the entire session(P=0.001). HR responses following WB cooling were reduced(P=0.05; d=1.07) compared to CONT conditions during exercise. Conclusion: A relationship between pre-cooling volume and exercise performance seems apparent, as larger surface area coverage augmented subsequent free-paced exercise capacity, in conjunction with greater suppression of physiological load. Maintenance of MVC with pre-cooling, despite increased work output suggests the role of centrally-mediated mechanisms in exercise pacing regulation and subsequent performance.

Meal induced thermogenesis and appetite : methodological issues and responses to energy restriction

Diet Induced Thermogenesis (DIT) is the energy expended consequent to meal consumption, and reflects the energy required for the processing and digestion of food consumed throughout each day. Although DIT is the total energy expended across a day in digestive processes to a number of meals, most studies measure thermogenesis in response to a single meal (Meal Induced Thermogenesis: MIT) as a representation of an individual’s thermogenic response to acute food ingestion. As a component of energy expenditure, DIT may have a contributing role in weight gain and weight loss. While the evidence is inconsistent, research has tended to reveal a suppressed MIT response in obese compared to lean individuals, which identifies individuals with an efficient storage of food energy, hence a greater tendency for weight gain. Appetite is another factor regulating body weight through its influence on energy intake. Preliminary research has shown a potential link between MIT and postprandial appetite as both are responses to food ingestion and have a similar response dependent upon the macronutrient content of food. There is a growing interest in understanding how both MIT and appetite are modified with changes in diet, activity levels and body size. However, the findings from MIT research have been highly inconsistent, potentially due to the vastly divergent protocols used for its measurement. Therefore, the main theme of this thesis was firstly, to address some of the methodological issues associated with measuring MIT. Additionally this thesis aimed to measure postprandial appetite simultaneously to MIT to test for any relationships between these meal-induced variables and to assess changes that occur in MIT and postprandial appetite during periods of energy restriction (ER) and following weight loss. Two separate studies were conducted to achieve these aims. Based on the increasing prevalence of obesity, it is important to develop accurate methodologies for measuring the components potentially contributing to its development and to understand the variability within these variables. Therefore, the aim of Study One was to establish a protocol for measuring the thermogenic response to a single test meal (MIT), as a representation of DIT across a day. This was done by determining the reproducibility of MIT with a continuous measurement protocol and determining the effect of measurement duration. The benefit of a fixed resting metabolic rate (RMR), which is a single measure of RMR used to calculate each subsequent measure of MIT, compared to separate baseline RMRs, which are separate measures of RMR measured immediately prior to each MIT test meal to calculate each measure of MIT, was also assessed to determine the method with greater reproducibility. Subsidiary aims were to measure postprandial appetite simultaneously to MIT, to determine its reproducibility between days and to assess potential relationships between these two variables. Ten healthy individuals (5 males, 5 females, age = 30.2 ± 7.6 years, BMI = 22.3 ± 1.9 kg/m2, %Fat Mass = 27.6 ± 5.9%) undertook three testing sessions within a 1-4 week time period. During the first visit, participants had their body composition measured using DXA for descriptive purposes, then had an initial 30-minute measure of RMR to familiarise them with the testing and to be used as a fixed baseline for calculating MIT. During the second and third testing sessions, MIT was measured. Measures of RMR and MIT were undertaken using a metabolic cart with a ventilated hood to measure energy expenditure via indirect calorimetry with participants in a semi-reclined position. The procedure on each MIT test day was: 1) a baseline RMR measured for 30 minutes, 2) a 15-minute break in the measure to consume a standard 576 kcal breakfast (54.3% CHO, 14.3% PRO, 31.4% FAT), comprising muesli, milk toast, butter, jam and juice, and 3) six hours of measuring MIT with two, ten-minute breaks at 3 and 4.5 hours for participants to visit the bathroom. On the MIT test days, pre and post breakfast then at 45-minute intervals, participants rated their subjective appetite, alertness and comfort on visual analogue scales (VAS). Prior to each test, participants were required to be fasted for 12 hours, and have undertaken no high intensity physical activity for the previous 48 hours. Despite no significant group changes in the MIT response between days, individual variability was high with an average between-day CV of 33%, which was not significantly improved by the use of a fixed RMR to 31%. The 95% limits of agreements which ranged from 9.9% of energy intake (%EI) to -10.7%EI with the baseline RMRs and between 9.6%EI to -12.4%EI with the fixed RMR, indicated very large changes relative to the size of the average MIT response (MIT 1: 8.4%EI, 13.3%EI; MIT 2: 8.8%EI, 14.7%EI; baseline and fixed RMRs respectively). After just three hours, the between-day CV with the baseline RMR was 26%, which may indicate an enhanced MIT reproducibility with shorter measurement durations. On average, 76, 89, and 96% of the six-hour MIT response was completed within three, four and five hours, respectively. Strong correlations were found between MIT at each of these time points and the total six-hour MIT (range for correlations r = 0.990 to 0.998; P < 0.01). The reproducibility of the proportion of the six-hour MIT completed at 3, 4 and 5 hours was reproducible (between-day CVs ≤ 8.5%). This indicated the suitability to use shorter durations on repeated occasions and a similar percent of the total response to be completed. There was a lack of strong evidence of any relationship between the magnitude of the MIT response and subjective postprandial appetite. Given a six-hour protocol places a considerable burden on participants, these results suggests that a post-meal measurement period of only three hours is sufficient to produce valid information on the metabolic response to a meal. However while there was no mean change in MIT between test days, individual variability was large. Further research is required to better understand which factors best explain the between-day variability in this physiological measure. With such a high prevalence of obesity, dieting has become a necessity to reduce body weight. However, during periods of ER, metabolic and appetite adaptations can occur which may impede weight loss. Understanding how metabolic and appetite factors change during ER and weight loss is important for designing optimal weight loss protocols. The purpose of Study Two was to measure the changes in the MIT response and subjective postprandial appetite during either continuous (CONT) or intermittent (INT) ER and following post diet energy balance (post-diet EB). Thirty-six obese male participants were randomly assigned to either the CONT (Age = 38.6 ± 7.0 years, weight = 109.8 ± 9.2 kg, % fat mass = 38.2 ± 5.2%) or INT diet groups (Age = 39.1 ± 9.1 years, weight = 107.1 ± 12.5 kg, % fat mass = 39.6 ± 6.8%). The study was divided into three phases: a four-week baseline (BL) phase where participants were provided with a diet to maintain body weight, an ER phase lasting either 16 (CONT) or 30 (INT) weeks, where participants were provided with a diet which supplied 67% of their energy balance requirements to induce weight loss and an eight-week post-diet EB phase, providing a diet to maintain body weight post weight loss. The INT ER phase was delivered as eight, two-week blocks of ER interspersed with two-week blocks designed to achieve weight maintenance. Energy requirements for each phase were predicted based on measured RMR, and adjusted throughout the study to account for changes in RMR. All participants completed MIT and appetite tests during BL and the ER phase. Nine CONT and 15 INT participants completed the post-diet EB MIT and 14 INT and 15 CONT participants completed the post-diet EB appetite tests. The MIT test day protocol was as follows: 1) a baseline RMR measured for 30 minutes, 2) a 15-minute break in the measure to consume a standard breakfast meal (874 kcal, 53.3% CHO, 14.5% PRO, 32.2% FAT), and 3) three hours of measuring MIT. MIT was calculated as the energy expenditure above the pre-meal RMR. Appetite test days were undertaken on a separate day using the same 576 kcal breakfast used in Study One. VAS were used to assess appetite pre and post breakfast, at one hour post breakfast then a further three times at 45-minute intervals. Appetite ratings were calculated for hunger and fullness as both the intra-meal change in appetite and the AUC. The three-hour MIT response at BL, ER and post-diet EB respectively were 5.4 ± 1.4%EI, 5.1 ± 1.3%EI and 5.0 ± 0.8%EI for the CONT group and 4.4 ± 1.0%EI, 4.7 ± 1.0%EI and 4.8 ± 0.8%EI for the INT group. Compared to BL, neither group had significant changes in their MIT response during ER or post-diet EB. There were no significant time by group interactions (p = 0.17) indicating a similar response to ER and post-diet EB in both groups. Contrary to what was hypothesised, there was a significant increase in postprandial AUC fullness in response to ER in both groups (p < 0.05). However, there were no significant changes in any of the other postprandial hunger or fullness variables. Despite no changes in MIT in both the CONT or INT group in response to ER or post-diet EB and only a minor increase in postprandial AUC fullness, the individual changes in MIT and postprandial appetite in response to ER were large. However those with the greatest MIT changes did not have the greatest changes in postprandial appetite. This study shows that postprandial appetite and MIT are unlikely to be altered during ER and are unlikely to hinder weight loss. Additionally, there were no changes in MIT in response to weight loss, indicating that body weight did not influence the magnitude of the MIT response. There were large individual changes in both variables, however further research is required to determine whether these changes were real compensatory changes to ER or simply between-day variation. Overall, the results of this thesis add to the current literature by showing the large variability of continuous MIT measurements, which make it difficult to compare MIT between groups and in response to diet interventions. This thesis was able to provide evidence to suggest that shorter measures may provide equally valid information about the total MIT response and can therefore be utilised in future research in order to reduce the burden of long measurements durations. This thesis indicates that MIT and postprandial subjective appetite are most likely independent of each other. This thesis also shows that, on average, energy restriction was not associated with compensatory changes in MIT and postprandial appetite that would have impeded weight loss. However, the large inter-individual variability supports the need to examine individual responses in more detail.

Comment on "Local accumulation times for source, diffusion, and degradation models in two and three dimensions" [J. Chem. Phys. 138, 104121 (2013)]

In a recent paper, Gordon, Muratov, and Shvartsman studied a partial differential equation (PDE) model describing radially symmetric diffusion and degradation in two and three dimensions. They paid particular attention to the local accumulation time (LAT), also known in the literature as the mean action time, which is a spatially dependent timescale that can be used to provide an estimate of the time required for the transient solution to effectively reach steady state. They presented exact results for three-dimensional applications and gave approximate results for the two-dimensional analogue. Here we make two generalizations of Gordon, Muratov, and Shvartsman’s work: (i) we present an exact expression for the LAT in any dimension and (ii) we present an exact expression for the variance of the distribution. The variance provides useful information regarding the spread about the mean that is not captured by the LAT. We conclude by describing further extensions of the model that were not considered by Gordon,Muratov, and Shvartsman. We have found that exact expressions for the LAT can also be derived for these important extensions...