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L. Matteucci and M. C. Nucci

Abstract

Rheumatoid arthritis is an autoimmune disease of unknown etiology that manifests as a persistent inammatory syn- ovitis and eventually destroys the joints. The immune system recognizes synovial cells as not self and consequently causes lymphocyte and antibody proliferation that is promoted by the pro-inammatory cytokines, the most significant being tumor necrosis factor TNF-. In the treatment of rheumatoid arthritis either monoclonal antibodies or soluble receptors are used to neutralize the TNF- bioactivity, such as sTNFR2, Etanercept and Iniximab. In [M. Jit et al. Rheumatology 2005;44:323- 331] a mathematical model that represents the TNF-dynamics in the inamed synovial joint within which locally produced TNF-_ can bind to cell-surface receptors was proposed. It consists of four coupled ordinary differential equations, that were integrated numerically assuming a range of estimates of the key parameters. In this paper we complement the previous work by determining the general solution of those equations for speci_c conditions on the parameters. Then we characterize the behavior of TNF- in the presence of different inhibitors and also evaluate the inhibitors effectiveness in the treatment of rheumatoid arthritis.

Open access

Annalisa Pascarella and Francesca Pitolli

Abstract

The MagnetoEncephaloGraphy (MEG) has gained great interest in neurorehabilitation training due to its high temporal resolution. The challenge is to localize the active regions of the brain in a fast and accurate way. In this paper we use an inversion method based on random spatial sampling to solve the real-time MEG inverse problem. Several numerical tests on synthetic but realistic data show that the method takes just a few hundredths of a second on a laptop to produce an accurate map of the electric activity inside the brain. Moreover, it requires very little memory storage. For these reasons the random sampling method is particularly attractive in real-time MEG applications.

Open access

Giacomo Aletti, Davide Lonardoni, Giovanni Naldi and Thierry Nieus

Abstract

One major challenge in neuroscience is the identifcation of interrelations between signals reecting neural activity and how information processing occurs in the neural circuits. At the cellular and molecular level, mechanisms of signal transduction have been studied intensively and a better knowledge and understanding of some basic processes of information handling by neurons has been achieved. In contrast, little is known about the organization and function of complex neuronal networks. Experimental methods are now available to simultaneously monitor electrical activity of a large number of neurons in real time. Then, the qualitative and quantitative analysis of the spiking activity of individual neurons is a very valuable tool for the study of the dynamics and architecture of the neural networks. Such activity is not due to the sole intrinsic properties of the individual neural cells but it is mostly the consequence of the direct inuence of other neurons. The deduction of the effective connectivity between neurons, whose experimental spike trains are observed, is of crucial importance in neuroscience: first for the correct interpretation of the electro-physiological activity of the involved neurons and neural networks, and, for correctly relating the electrophysiological activity to the functional tasks accomplished by the network. In this work, we propose a novel method for the identification of connectivity of neural networks using recorded voltages. Our approach is based on the assumption that the network has a topology with sparse connections. After a brief description of our method, we will report the performances and compare it to the cross-correlation computed on the spike trains, which represents a gold standard method in the field.

Open access

Marco Scianna and Annachiara Colombi

Abstract

The invasive capability is fundamental in determining the malignancy of a solid tumor. In particular, tumor invasion fronts are characterized by different morphologies, which result both from cell-based processes (such as cell elasticity, adhesive properties and motility) and from subcellular molecular dynamics (such as growth factor internalization, ECM protein digestion and MMP secretion). Of particular relevance is the development of tumors with unstable fingered morphologies: they are in fact more aggressive and hard to be treated than smoother ones as, even if their invasive depth is limited, they are dificult to be surgically removed. The phenomenon of malignant fingering has been reproduced with several mathematical approaches. In this respect, we here present a qualitative comparison between the results obtained by an individual cell-based model (an extended version of the cellular Potts model) and by a measure-based theoretic method. In particular, we show that in both cases a fundamental role in finger extension is played by intercellular adhesive forces and taxis-like migration.

Open access

Orazio Muscato and Vincenza Di Stefano

Abstract

The Wigner transport equation can be solved stochastically by Monte Carlo techniques based on the theory of piecewise deterministic Markov processes. A new stochastic algorithm, without time discretization error, has been implemented and studied in the case of the quantum transport through a rectangular potential barrier.

Open access

Clemente Cesarano

Abstract

Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

Open access

Clemente Cesarano

Abstract

Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

Open access

Yasushige Watase

Summary

We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1].

The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : AB, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h −1(𝔭) where 𝔭 2 Spec B.

Open access

Charyyar Ashyralyyev

Abstract

Reverse parabolic equation with integral condition is considered. Well-posedness of reverse parabolic problem in the Hölder space is proved. Coercive stability estimates for solution of three boundary value problems (BVPs) to reverse parabolic equation with integral condition are established.

Open access

D. Brunetto, C. Andrà, N. Parolini and M. Verani

Abstract

This paper aims at bridging Mathematical Modelling and Mathematics Education by studying the opinion dynamics of students who work in small groups during mathematics classrooms. In particular, we propose a model which hinges upon the pioneering work of Hegselmann and Krause on opinion dynamics and integrates recent results of interactionist research in Mathematical Education. More precisely, the proposed model incorporates the following features: 1) the feelings of each student towards the classmates (building upon the so-called \I can" -\you can" framework); 2) the different levels of preparation of the students; 3) the presence of the teacher, who may or may not intervene to drive the students towards the correct solution of the problem. Several numerical experiments are presented to assess the capability of the model in reproducing typical realistic scenarios.