Some Utilizations of Lambert W Function in Distribution Theory

We introduce a new class of positive infinitely divisible probability laws calling them 𝔏_γ distributions. Their cumulant-generating functions (cgf) are expressed in terms of the principal branch of the Lambert W function. The probability density functions (pdfs) of 𝔏_γ laws are bounded resembling pdf of a Lévy stable distribution. The exponential dispersion model constructed starting from an 𝔏_γ distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an 𝔏_γ distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with α = 1. We derive new results on 𝔏_γ laws and the related exponential dispersion models, including their convolution and scaling closure properties. We generate another exponential dispersion model starting from an exponentially compounded 𝔏_γ law. This distribution emerges in the Poisson mixture representation of a generalized Poisson law. We extend the Poisson approximation for the scaled Neyman type A exponential dispersion model. We derive saddlepoint-type approximations for some of these exponential dispersion models. The role of the Lambert W function is emphasized.     

The Community's Role in the Prevention and Study of Crime

Vladimir S. Malakhov outlines state-sponsored efforts to create a civic nation-state, which Gleb Pavlovskii and others have called "The Russian Project." The author points out that the multicultural, multiethnic, and multiconfessional legacies of "Rossiia"^* [Russia] make it extraordinarily difficult to change the meaning of the ethnonational term "Russian." Based on diverse United States and European histories, and despite their mixed experiences of immigrant incorporation, he advocates a return to the credibility of the term "multicultural." Toward this end, he suggests that the multiethnic meaning of "Rossiia" is being restored as the best defense against Russian chauvinist nationalism.     

The peculiar pattern of mortality of Jews in Moscow, 1993-95

Russian Jews, particularly men, have a large mortality advantage compared with the general Russian population. We consider possible explanations for this advantage using data on 445,000 deaths in Moscow, 1993-95. Log-linear analysis of the distribution of deaths by sex, age, ethnic group, and cause of death reveals a relatively high concentration of endogenous causes and a relatively low concentration of exogenous and behaviourally induced causes among Jews. There is also a significant concentration of deaths from breast cancer among Jewish women. Mortality estimates using the 1994 micro-census population as the denominator reveal an 11-year Russian-Jewish gap in the life expectancy of males at age 20, but only a 2-year life-expectancy gap for women. Only 40 per cent of the Russian-Jewish difference for men, but the entire difference for women, can be eliminated by adjustment for educational differences between the two ethnic groups. Similarities with other Jewish populations and possible explanations are discussed.     

The Travelling Salesman Problem on Permuted Monge Matrices

We consider traveling salesman problems (TSPs) with a permuted Monge matrix as cost matrix where the associated patching graph has a specially simple structure: a multistar, a multitree or a planar graph. In the case of multistars, we give a complete, concise and simplified presentation of Gaikov's theory. These results are then used for designing an O(m3 + mn) algorithm in the case of multitrees, where n is the number of cities and m is the number of subtours in an optimal assignment. Moreover we show that for planar patching graphs, the problem of finding an optimal subtour patching remains NP-complete.     

Linearizable special cases of the QAP

We consider special cases of the quadratic assignment problem (QAP) that are linearizable in the sense of Bookhold. We provide combinatorial characterizations of the linearizable instances of the weighted feedback arc set QAP, and of the linearizable instances of the traveling salesman QAP. As a by-product, this yields a new well-solvable special case of the weighted feedback arc set problem.     

Trends and geographic differentials in mortality under age 60 in India

The study examines overall and region-specific mortality changes and regional mortality variation in India since the 1970s, using data from the Sample Registration System (SRS). An evaluation of the quality of SRS data confirms their reliability for children and adults under age 60. The results suggest the convergence of mortality across the regions of India with important inter-state differences in the pace of health improvements over time. After spectacular progress during the 1970s and the 1980s, many Indian states have witnessed slower mortality improvements in both young and adult age groups. India faces difficulties in making further reductions in infant mortality and in the burden of chronic and man-made diseases at adult ages.     

Long-term trends in the longevity of scientific elites: evidence from the British and the Russian academies of science

National science academies represent intellectual elites and vanguard groups in the achievement of longevity. We estimated life expectancy (LE) at age 50 of members of the British Royal Society (RS) for the years 1670-2007 and of members of the Russian Academy of Sciences (RAS) for the years 1750-2006. The longevity of academicians was higher than that of their corresponding national populations, with the gap widening from the 1950s. Since the 1980s, LE in the RS has been higher than the maximum LE among all high-income countries. In each period, LE in the RS was greater than in the RAS, although since the 1950s it has risen in parallel in the two academies. This steep increase shared by academicians in Britain and Russia suggests that general populations have the potential for a substantial increase in survival to high ages.     

Steep increase in best-practice cohort life expectancy

We analyze trends in best-practice life expectancy among female cohorts born from 1870 to 1950. Cohorts experience declining rather than constant death rates, and cohort life expectancy usually exceeds period life expectancy. Unobserved mortality rates in non-extinct cohorts are estimated using the Lee-Carter model for mortality in 1960–2008. Best-practice cohort and period life expectancies increased nearly linearly. Across cohorts born from 1870 to 1920 the annual increase in cohort length of life was 0.43 years. Across calendar years from 1870 to 2008, the annual increase was 0.28 years. Cohort life expectancy increased from 53.7 years in the 1870 cohort to 83.8 years in the 1950 cohort. The corresponding cohort/period longevity gap increased from 1.2 to 10.3 years. Among younger cohorts, survival to advanced ages is substantially higher than could have been anticipated by period mortality regimes when these cohorts were young or middle-aged. A large proportion of the additional expected years of life are being lived at ages 65 and older. This substantially changes the balance between the stages of the life cycle.     

Research of injuries of passengers in city buses as a consequence of non-collision effects

In this study, a research of injuries of passengers in city buses is presented, which are not a consequence of collision of buses with other objects. The number of injured passengers in the territory of Belgrade was registered, during three consecutive years. Most frequently injured body part was the head, while women over the age of 60 are the most vulnerable population of passengers. The most often reason for the appearance of injuries was the effect of acceleration. Also, it is pointed out to the importance of consideration of the problem of multiple injuries. These injuries are more complex than the individual ones, and they may indicate to the existence of greater omissions in the design of the interior of the bus. Proposals for certain system solutions, as well as some recommendations for designing are given, which can improve the safety of passengers who use the city bus as a means of transportation.