1,721,050 research outputs found
Epsilon 1 - Primo Corso di Analisi Matematica
No full textEpsilon 1 offre una presentazione completa degli argomenti classici di un primo corso di Analisi Matematica nelle facoltà tecniche e scientifiche: funzioni e loro grafici, limiti, derivate e integrali di funzioni di una variabile reale, successioni e serie numeriche. Inoltre, accogliendo le esigenze dei vari ordinamenti didattici, contiene un’introduzione alle funzioni reali di più variabili reali e alle equazioni differenziali ordinarie. L’obiettivo generale è coniugare rigore metodologico con fruibilità e semplicità. La presentazione è graduale e dettagliata e non perde di vista le applicazioni teoriche e pratiche. Definizioni, enunciati e dimostrazioni sono accompagnati da numerosi esempi, figure, osservazioni ed esercizi di vari livelli e di difficoltà progressiva. Ciascuno di questi elementi è facilmente individuabile grazie a un'impostazione grafica modulare, che conferisce al testo una struttura particolarmente schematica e articolata. Dal punto di vista dei contenuti, le scelte più caratterizzanti del testo riguardano: un’esposizione simultanea di numeri, funzioni e loro grafici; una trattazione separata dei limiti di successioni (per facilitare l’assimilazione iniziale del concetto), che viene poi inquadrata nel contesto generale dei limiti di funzioni; una presentazione versatile dell’integrale di Riemann, introdotto subito anche come un limite. Ulteriori risorse sono disponibili sul sito web di McGraw-Hill Education, nella pagina dedicata al libro
Measure-Valued solutions to a nonlinear Fourth-Order regularization of forward-backward parabolic equations
Full text linkWe introduce and analyze a new, nonlinear fourth-order regularization of forwardbackward parabolic equations. In one space dimension, under general assumptions on the potentials, which include those of Perona--Malik type, we prove existence of Radon measure-valued solutions under both natural and essential boundary conditions. If the decay at infinity of the nonlinearities is sufficiently fast, we also exhibit examples of local solutions whose atomic part arises and/or persists (in contrast to the linear fourth-order regularization) and even disappears within finite time (in contrast to pseudoparabolic regularizations)
Macroscopic modelling of Alzheimer’s disease: difficulties and challenges
Full text linkIn the context of Alzheimer’s disease (AD), in silico research aims at giving complement- ary and better insight into the complex mechanisms which determine the development of AD. One of its important aspects is the construction of macroscopic mathematical models which are the basis for numerical simulations. In this paper we discuss some of the general and fundamental difficulties of macroscopic modelling of AD. In addition we formulate a mathem- atical model in the case of a specific problem in an early stage of AD, namely the propagation of pathological τ protein from the entorhinal cortex to the hippocampal region. The main feature of this model consists in the representation of the brain through two superposed finite graphs, which have the same vertices (that, roughly speaking, can be thought as parcels of a brain atlas), but different edges. We call these graphs “proximity graph” and “connectiv- ity graph”, respectively. The edges of the first graph take into account the distances of the vertices and the heterogeneity of the cerebral parenchyma, whereas the edges of the second graph represent the connections by white-matter fiber pathways between different structures. The diffusion of the proteins Aβ and τ are described through the Laplace operators on the graphs, whereas the phenomenon of aggregation of the proteins leading ultimately to senile plaques and neuro-fibrillar tangles (as already observed by A. Alzheimer in 1907) is modelled by means of the classical Smoluchowski aggregation system
Measure-valued solutions of scalar hyperbolic conservation laws, Part 1: Existence and time evolution of singular parts
Full text linkWe prove existence for a class of signed Radon measure-valued entropy solutions of the Cauchy
problem for a first order scalar hyperbolic conservation law in one space dimension. The initial
data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz
continuous. Existence is proven by a constructive procedure which makes use of a suitable
family of approximating problems. Relevant qualitative properties of such constructed solutions
are pointed out
The role of A and Tau proteins in Alzheimer’s disease: a mathematical model on graphs
Full text linkIn this Note we study amathematical model for the progression of Alzheimer'sDisease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as beta amyloid and Tau play in fact a crucial role in the development of Alzheimer's disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy
A sensitivity analysis of a mathematical model for the synergistic interplay of amyloid beta and tau on the dynamics of Alzheimer’s disease
Full text linkWe propose a mathematical model for the onset and progression of Alzheimer’s disease based on transport and diffusion equations. We treat brain neurons as a continuous medium and structure them by their degree of mal- functioning. Three different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble Amyloid beta, ii) effects of phosphorylated tau protein and iii) neuron-to- neuron prion-like transmission of the disease. We model these processes by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of tau protein and a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The latter equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. We are particularly interested in investigating the effects of the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer’s disease. The output of our numerical simulations, al- though in 2D with an over-simplified geometry, is in good qualitative agreement with clinical findings concerning both the disease distribution in the brain, which varies from early to advanced stages, and the effects of tau on the dynamics of the disease
A uniqueness criterion for measure-valued solutions of scalar hyperbolic conservation laws
Full text linkWe prove existence and uniqueness of Radon measure-valued solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension, the initial data being a finite superposition of Dirac masses and the flux being Lipschitz continuous, bounded and sufficiently smooth. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness
Well-Posedness of a Mathematical Model for Alzheimer's Disease
Full text linkWe consider the existence and uniqueness of solutions of an initial boundary value
problem for a coupled system of PDEs arising in a model for Alzheimer’s disease. Apart from
reaction diffusion equations, the system contains a transport equation in a bounded interval for a probability measure which is related to the malfunctioning of neurons. The main ingredients to prove existence are the method of characteristics for the transport equation, a priori estimates for solutions of the reaction diffusion equations, a variant of the classical contraction theorem, and the Wasserstein metric for the part concerning the probability measure. We stress that all hypotheses on the data are not suggested by mathematical artifacts, but are naturally imposed by modeling considerations. In particular the use of a probability measure is natural from a modeling point of view. The nontrivial part of the analysis is the suitable combination of the various mathematical tools, which is not quite routine and requires various technical adjustments
Radon measure-valued solutions of first order scalar conservation laws
Full text linkWe study nonnegative solutions of the Cauchy problem ∂ t u + ∂ x [ φ (u) ] = 0 in × (0, T), u = u 0 ≥ 0 in × 0 , \left\\beginaligned &\displaystyle\partial-tu+\partial-x[\varphi(u)]=0&% &\displaystyle\phantom\textin \mathbbR\times(0,T),\\ &\displaystyle u=u-0\geq 0&&\displaystyle\phantom\textin \mathbbR% \times\0\,\endaligned\right. where u 0 u-0 is a Radon measure and φ: [ 0, ∞) → \varphi\colon[0,\infty)\mapsto\mathbbR is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of u 0 u-0 is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case φ (u) = u \varphi(u)=u this happens for all times). In the latter case, we describe the evolution of the singular parts
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