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Alzheimer's disease (AD) is a multifactorial disorder characterized by the progressive deterioration of neuronal networks. The primary cause and sequence of its progression are only partially understood but abnormalities in folding and accumulation of insoluble proteins such as ß-amyloid and Tau-protein are both associated with the pathogenesis of AD. Mitochondria play a crucial role in cell survival and death, and changes in mitochondrial structure and/or function are related to many human diseases. Increasing evidence suggests that compromised mitochondrial function contributes to the aging process and thus may increase the risk of AD. Dysfunctional mitochondria contribute to reactive oxygen species which can lead to extensive macromolecule oxidative damage and the progression of amyloid pathology. Oxidative stress and amyloid toxicity leave neurons chemically vulnerable. The mitochondrial toxicity induced by ß-amyloid is still not clear but may include numerous mechanisms, such as the increased permeability of mitochondrial membranes, the disruption of calcium homeostasis, the alteration of oxidative phosphorylation with a consequent overproduction of reactive oxygen species. Other mechanisms have been associated with the pathophysiology of AD. Inflammatory changes are observed in AD brain overall, particularly at the amyloid deposits, which are rich in activated microglia. Once stimulated, the microglia release a wide variety of pro-inflammatory mediators including cytokines, complement components and free radicals, all of which potentially contribute to further neuronal dysfunction and eventually death. Clinically, novel approaches to visualize early neuroinflammation in the human brain are needed to improve the monitoring and control of therapeutic strategies that target inflammatory and other pathological mechanisms. Similarly, there is growing interest in developing agents that modulate mitochondrial function.

Abstract: The aim of the present paper is to show how the Lagrange Inversion Formula (LIF) can be applied in a straight-forward way i) to find the generating function of many combinatorial sequences, ii) to extract the coefficients of a formal power series, iii) to compute combinatorial sums, and iv) to perform the inversion of combinatorial identities. Particular forms of the LIF are studied, in order to simplify the computation steps. Some examples are taken from the literature, but their proof is different from the usual, and others are new.

Abstract.: In this paper we treat the static dictionary problem , very well known in computer science. It consists in storing a set S of m elements in the range [1 . . . n ] so that membership queries on S 's elements can be handled in O(1) time. It can be approached as a table compression problem in which a size n table has m ones and the other elements are zeros. We focus our attention on sparse case (m n ). We use a simple algorithm to solve the problem and make an average-case analysis of the total space required when the input derives from uniform probability distribution. We also find some conditions able to minimize storage requirements. We then propose and analyze a new algorithm able to reduce storage requirements drastically to O(m 4/3 ) .