Closed form summation of C-finite sequences

We consider sums of the form $$\sum_{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)... F_k(a_kn+b_kj+c_k),$$ in which each $\{F_i(n)\}$ is a sequence that satisfies a linear recurrence of degree $D(i)<\infty$, with constant coefficients. We assume further that the $a_i$'s and the $a_i+b_i$'s are all nonnegative integers. We prove that such a sum always has a closed form, in the sense that it evaluates to a linear combination of a finite set of monomials in the values of the sequences $\{F_i(n)\}$ with coefficients that are polynomials in $n$. We explicitly describe two different sets of monomials that will form such a linear combination, and give an algorithm for finding these closed forms, thereby completely automating the solution of this class of summation problems. We exhibit tools for determining when these explicit evaluations are unique of their type, and prove that in a number of interesting cases they are indeed unique. We also discuss some special features of the case of ``indefinite summation," in which $a_1=a_2=... = a_k = 0$

Iron Behaving Badly: Inappropriate Iron Chelation as a Major Contributor to the Aetiology of Vascular and Other Progressive Inflammatory and Degenerative Diseases

The production of peroxide and superoxide is an inevitable consequence of aerobic metabolism, and while these particular "reactive oxygen species" (ROSs) can exhibit a number of biological effects, they are not of themselves excessively reactive and thus they are not especially damaging at physiological concentrations. However, their reactions with poorly liganded iron species can lead to the catalytic production of the very reactive and dangerous hydroxyl radical, which is exceptionally damaging, and a major cause of chronic inflammation. We review the considerable and wide-ranging evidence for the involvement of this combination of (su)peroxide and poorly liganded iron in a large number of physiological and indeed pathological processes and inflammatory disorders, especially those involving the progressive degradation of cellular and organismal performance. These diseases share a great many similarities and thus might be considered to have a common cause (i.e. iron-catalysed free radical and especially hydroxyl radical generation). The studies reviewed include those focused on a series of cardiovascular, metabolic and neurological diseases, where iron can be found at the sites of plaques and lesions, as well as studies showing the significance of iron to aging and longevity. The effective chelation of iron by natural or synthetic ligands is thus of major physiological (and potentially therapeutic) importance. As systems properties, we need to recognise that physiological observables have multiple molecular causes, and studying them in isolation leads to inconsistent patterns of apparent causality when it is the simultaneous combination of multiple factors that is responsible. This explains, for instance, the decidedly mixed effects of antioxidants that have been observed, etc...Comment: 159 pages, including 9 Figs and 2184 reference

Health Disparities Based on Socioeconomic Inequities: Implications for Urban Health Care

Health is unevenly distributed across socioeconomic status. Persons of lower income, education, or occupational status experience worse health and die earlier than do their better-off counterparts. This article discusses these disparities in the context of urban medical practice. The article begins with a discussion of the complex relationship among socioeconomic status, race, and health in the United States. It highlights the effects of institutional, individual, and internalized racism on the health of African Americans, including the insidious consequences of residential segregation and concentrated poverty. Next, the article reviews health disparities based on socioeconomic status across the life cycle, beginning in fetal health and ending with disparities among the elderly. Potential explanations for these socioeconomic-based disparities are addressed, including reverse causality (e.g., being poor causes lower socioeconomic status) and confounding by genetic factors. The article underscores social causation as the primary explanation for health disparities and highlights the cumulative effects of social disadvantage across stages of the life cycle and across environments (e.g., fetal, family, educational, occupational, and neighborhood). The article concludes with a discussion of the implications of health disparities for the practice of urban medicine, including the role that concentration of disadvantage plays among patients and practice sites and the need for quality improvement to mitigate these disparities.http://pt.wkhealth.com/pt/re/lwwgateway/landingpage.htm;jsessionid=NzLDMW4qhdTyJ3yM5YbtyKKvnkXM1GLG67L6KpFfnFGKgcHNkgnJ!-488008196!181195628!8091!-1?issn=1040-2446&volume=79&issue=12&spage=113