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Background: The relative contribution of environmental contaminants is an important, and frequently unanswered, question in human or ecological risk assessments. This interpretation of relative importance allows determination of the overall effect of a set of variables relative to other variables on an adverse health outcome. There are no underlying assumptions of independence of variables. The tool developed and used here is specifically designed for studying the effects of mixtures of chemicals on a particular function of the human body. Methods: We apply the approach to estimate the contributions of total exposure to six PFAS (perfluorodecanoic acid, perfluorohexane sulfonic acid, 2-(N-methyl-PFOSA) acetate, perfluorononanoic acid, perfluoroundecanoic acid and perfluoroundecanoic acid) to loss of bone mineral density relative to other factors related to risk of osteoporosis and bone fracture, using data from subjects who participated in the US National Health Examination and Nutrition Surveys (NHANES) of 2013–2014. Results: PFAS exposures contribute to bone mineral density changes relative to the following variables: age, weight, height, vitamin D2 and D3, gender, race, sex hormone binding globulin, testosterone, and estradiol. Conclusion: We note significant alterations to bone mineral density among more highly exposed adults and significant differences in effects between men and women.

PFASs have been detected in nearly every serum sample collected over the last two decades from US adults as part of the National Health and Nutrition Examination Survey (NHANES) and are commonly found in other data sets from around the world. However, less is known about infant PFAS exposures, primarily because the collection of infant serum samples is less common and frequently avoided. Cord blood samples are often preferred for chemical exposure assessments because this is thought to provide a good representation of infant serum concentrations, at least at the time of birth. In this paper, we will provide a statistical and probabilistic analysis of what can be expected for infants living in the US using NHANES from 2007 to 2008, which contains a rare subset of infant data. Regulatory efforts that require estimation of exposures among the very youth can be challenging, both because of a lack of data in general and because variability among this most vulnerable population can be uncertain. We report that US infant exposures are extremely common and that serum concentrations remain fairly constant, despite infant growth rates and relatively high caloric and fluid intake, with the possible exception of PFOS. Infant serum PFOS concentrations between months 1 and 3 are consistently higher than at less than one month, even though healthy infants at 1 and 2 months weigh more than they did at birth. This suggests that the babies are exposed to greater concentrations of PFOS after birth or that excretion kinetics differ for this PFAS.

Perfluoroalkyl and polyfluoroalkyl substances (PFAS) are a widely dispersed, broad class of synthetic chemicals with diverse biological effects, including effects on adipose and bone differentiation. PFAS most commonly occur as mixtures and only rarely, if ever, as single environmental contaminants. This poses significant regulatory questions and a pronounced need for chemical risk assessments, analytical methods, and technological solutions to reduce the risk to public and environmental health. The effects of PFAS on biological systems may be complex. Each may have several molecular targets initiating multiple biochemical events leading to a number of different adverse outcomes. An exposure to mixtures or coexposures of PFAS complicates the picture further. This review illustrates how PFAS target peroxisome proliferator–activated receptors. Additionally, we describe how such activation leads to changes in cell differentiation and bone development that contributes to metabolic disorder and bone weakness. This discussion sheds light on the importance of seemingly modest outcomes observed in test animals and highlights why the most sensitive end points identified in some chemical risk assessments are significant from a public health perspective.

We present a model for the transport of a single type of asbestos fibre through the human body. The model captures the transport modes that pertain particularly to the lungs and the mesothelium. Numerical solutions of the system follow observed movement in the body. We compare the accumulation of fibres in the lungs versus the mesothelium, and then we give analysis and results for various cases of exposure level and exposure time. Models, such as the one developed here, can give clues as to how asbestos fibres impact the body, and where to look for major impact.

Pseudo-random number generators of the form Xn+i = P{xn), Vn = h{xn) are ubiquitous in applications ranging from cryptology to statistics. Such systems have been studied extensively in the control theory literature when Xn € K. In this paper we make a detailed study of the invariants of such systems when the underlying field is the Galois field of two elements. We consider various groups that act on such system.

In the paper a theoretical analysis is given for the smallest ball that covers a finite number of points $p_1, p_2, \cdots, p_N \in \Bbb R^n$. Several fundamental properties of the smallest enclosing ball are described and proved. Particularly, it is proved that the $k$-circumscribing enclosing ball with smallest $k$ is the smallest enclosing ball, which dramatically reduces a possible large number of computations in the higher dimensional case. General formulas are deduced for calculating circumscribing balls. The difficulty of the closed-form description is discussed. Finally, as an application, the problem of finding a common quadratic Lyapunov function for a set of stable matrices is considered.

In this paper we give a basic derivation of smoothing and interpolating splines and through this derivation we show that the basic spline construction can be done through elementary Hilbert space techniques. Smoothing splines are shown to naturally separate into a filtering problem on the raw data and an interpolating spline construction. Both the filtering algorithm and the interpolating spline construction can be effectively implemented. We show that a variety of spline problems can be formulated into this common construction. By this construction we are also able to generalize the construction of smoothing splines to continuous data, a spline like filtering algorithm. Through the control theoretic approach it is natural to add multiple constraints and these techniques are developed in this paper. 1. Introduction. Estimation and smoothing for data sets that contain deter- ministic and random data present difficulties not present in purely random data sets. Yet such data sets are very common in practice and if the nature of the data is not respected conclusions may be drawn that have little relation to reality. In this pa- per we will present a unified treatment of such problems. We will extend the theory of smoothing splines to cover such situations. Some of the techniques that we will use have been developed in papers by Egerstedt and Martin, (4, 8, 14) and their col- leagues. The main technical contribution of this paper will be to show that many of these problems can be cast as minimum norm problems in suitable a Hilbert spaces. This approach unifies a series of problems that have been solved by Egerstedt, Zhou, Sun and Martin, (17, 18, 19). Furthermore, the approach of this paper gives a unified treatment of smoothing splines as developed by Wahba, (15), and the classical poly- nomial and exponential interpolating splines. The approach of this paper rests on the Hilbert space methods developed by Luenberger in (7).

The problem of fitting surfaces to data is a well studied problem in statistics. However when there is prior information the theory is not developed. It often happens in toxicology and in medicine that the effect of a single drug is well understood.. However if a pair of drugs is delivered in tandem or if two toxins are interacting the effect is not understood. In this paper we attack the problem of fitting a surface to a data set contained in a square when two of the boundaries are known. Our approach generalizes the concept of smoothing splines. Godkänd; 2004; 20070107 (ysko)

In this paper we give a new proof that for controllable and observable linear systems every L 2 [ 0 , T ] function can be approximated in the L 2 [ 0 , T ] sense with an output function generated by an L 2 [ 0 , T ] input function. We also give a new characterization of how continuous functions on [ 0 , T ] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.