On existence and uniqueness of positive solutions to a class of fractional boundary value problems

[EN] The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fractional boundary value problem D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = 0 , where 2 < α ≤ 3 and D 0 + α is the Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem in partially ordered metric spaces. The autonomous case of this problem was studied in the paper [Zhao et al., Abs. Appl. Anal., to appear], but in Zhao et al. (to appear), the question of uniqueness of the solution is not treated. We also present some examples where we compare our results with the ones obtained in Zhao et al. (to appear). 2010 Mathematics Subject Classification: 34B15

Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems

[EN] We establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Our analysis relies on a fixed point theorem in partially ordered sets.

Tuzla City (BiH): an example of geohazard induced by salt extraction

Salt deposits characterize the subsurface of Tuzla (BiH) and made it famous since the ancient times. Archeological discoveries demonstrate the presence of a Neolithic pile-dwelling settlement related to the existence of saltwater springs that contributed to make the most of the area a swampy ground. Since the Roman times, the town is reported as “the City of Salt deposits and Springs”; "tuz" is the Turkish word for salt, as the Ottomans renamed the settlement in the 15th century following their conquest of the medieval Bosnia (Donia and Fine, 1994). Natural brine springs were located everywhere and salt has been evaporated by means of hot charcoals since pre-Roman times. The ancient use of salt was just a small exploitation compared to the massive salt production carried out during the 20th century by means of classical mine methodologies and especially wild brine pumping. In the past salt extraction was practised tapping natural brine springs, while the modern technique consists in about 100 boreholes with pumps tapped to the natural underground brine runs, at an average depth of 400-500 m. The mining operation changed the hydrogeological conditions enabling the downward flow of fresh water causing additional salt dissolution. This process induced severe ground subsidence during the last 60 years reaching up to 10 meters of sinking in the most affected area. Stress and strain of the overlying rocks induced the formation of numerous fractures over a conspicuous area (3 Km2). Consequently serious damages occurred to buildings and infrastructures such as water supply system, sewage networks and power lines. Downtown urban life was compromised by the destruction of more than 2000 buildings that collapsed or needed to be demolished causing the resettlement of about 15000 inhabitants (Tatić, 1979). Recently salt extraction activities have been strongly reduced, but the underground water system is returning to his natural conditions, threatening the flooding of the most collapsed area. During the last 60 years local government developed a monitoring system of the phenomenon, collecting several data about geodetic measurements, amount of brine pumped, piezometry, lithostratigraphy, extension of the salt body and geotechnical parameters. A database was created within a scientific cooperation between the municipality of Tuzla and the city of Rotterdam (D.O.O. Mining Institute Tuzla, 2000). The scientific investigation presented in this dissertation has been financially supported by a cooperation project between the Municipality of Tuzla, The University of Bologna (CIRSA) and the Province of Ravenna. The University of Tuzla (RGGF) gave an important scientific support in particular about the geological and hydrogeological features. Subsidence damage resulting from evaporite dissolution generates substantial losses throughout the world, but the causes are only well understood in a few areas (Gutierrez et al., 2008). The subject of this study is the collapsing phenomenon occurring in Tuzla area with the aim to identify and quantify the several factors involved in the system and their correlations. Tuzla subsidence phenomenon can be defined as geohazard, which represents the consequence of an adverse combination of geological processes and ground conditions precipitated by human activity with the potential to cause harm (Rosenbaum and Culshaw, 2003). Where an hazard induces a risk to a vulnerable element, a risk management process is required. The single factors involved in the subsidence of Tuzla can be considered as hazards. The final objective of this dissertation represents a preliminary risk assessment procedure and guidelines, developed in order to quantify the buildings vulnerability in relation to the overall geohazard that affect the town. The historical available database, never fully processed, have been analyzed by means of geographic information systems and mathematical interpolators (PART I). Modern geomatic applications have been implemented to deeply investigate the most relevant hazards (PART II). In order to monitor and quantify the actual subsidence rates, geodetic GPS technologies have been implemented and 4 survey campaigns have been carried out once a year. Subsidence related fractures system has been identified by means of field surveys and mathematical interpretations of the sinking surface, called curvature analysis. The comparison of mapped and predicted fractures leaded to a better comprehension of the problem. Results confirmed the reliability of fractures identification using curvature analysis applied to sinking data instead of topographic or seismic data. Urban changes evolution has been reconstructed analyzing topographic maps and satellite imageries, identifying the most damaged areas. This part of the investigation was very important for the quantification of buildings vulnerability.

Bioimaging of animal embryogenesis: mathematical methods and computational algorithms

Some fundamental biological processes such as embryonic development have been preserved during evolution and are common to species belonging to different phylogenetic positions, but are nowadays largely unknown. The understanding of cell morphodynamics leading to the formation of organized spatial distribution of cells such as tissues and organs can be achieved through the reconstruction of cells shape and position during the development of a live animal embryo. We design in this work a chain of image processing methods to automatically segment and track cells nuclei and membranes during the development of a zebrafish embryo, which has been largely validates as model organism to understand vertebrate development, gene function and healingrepair mechanisms in vertebrates. The embryo is previously labeled through the ubiquitous expression of fluorescent proteins addressed to cells nuclei and membranes, and temporal sequences of volumetric images are acquired with laser scanning microscopy. Cells position is detected by processing nuclei images either through the generalized form of the Hough transform or identifying nuclei position with local maxima after a smoothing preprocessing step. Membranes and nuclei shapes are reconstructed by using PDEs based variational techniques such as the Subjective Surfaces and the Chan Vese method. Cells tracking is performed by combining informations previously detected on cells shape and position with biological regularization constraints. Our results are manually validated and reconstruct the formation of zebrafish brain at 7-8 somite stage with all the cells tracked starting from late sphere stage with less than 2% error for at least 6 hours. Our reconstruction opens the way to a systematic investigation of cellular behaviors, of clonal origin and clonal complexity of brain organs, as well as the contribution of cell proliferation modes and cell movements to the formation of local patterns and morphogenetic fields.

Mathematical models of cognitive processes

The research activity carried out during the PhD course was focused on the development of mathematical models of some cognitive processes and their validation by means of data present in literature, with a double aim: i) to achieve a better interpretation and explanation of the great amount of data obtained on these processes from different methodologies (electrophysiological recordings on animals, neuropsychological, psychophysical and neuroimaging studies in humans), ii) to exploit model predictions and results to guide future research and experiments. In particular, the research activity has been focused on two different projects: 1) the first one concerns the development of neural oscillators networks, in order to investigate the mechanisms of synchronization of the neural oscillatory activity during cognitive processes, such as object recognition, memory, language, attention; 2) the second one concerns the mathematical modelling of multisensory integration processes (e.g. visual-acoustic), which occur in several cortical and subcortical regions (in particular in a subcortical structure named Superior Colliculus (SC)), and which are fundamental for orienting motor and attentive responses to external world stimuli. This activity has been realized in collaboration with the Center for Studies and Researches in Cognitive Neuroscience of the University of Bologna (in Cesena) and the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA). PART 1. Objects representation in a number of cognitive functions, like perception and recognition, foresees distribute processes in different cortical areas. One of the main neurophysiological question concerns how the correlation between these disparate areas is realized, in order to succeed in grouping together the characteristics of the same object (binding problem) and in maintaining segregated the properties belonging to different objects simultaneously present (segmentation problem). Different theories have been proposed to address these questions (Barlow, 1972). One of the most influential theory is the so called “assembly coding”, postulated by Singer (2003), according to which 1) an object is well described by a few fundamental properties, processing in different and distributed cortical areas; 2) the recognition of the object would be realized by means of the simultaneously activation of the cortical areas representing its different features; 3) groups of properties belonging to different objects would be kept separated in the time domain. In Chapter 1.1 and in Chapter 1.2 we present two neural network models for object recognition, based on the “assembly coding” hypothesis. These models are networks of Wilson-Cowan oscillators which exploit: i) two high-level “Gestalt Rules” (the similarity and previous knowledge rules), to realize the functional link between elements of different cortical areas representing properties of the same object (binding problem); 2) the synchronization of the neural oscillatory activity in the γ-band (30-100Hz), to segregate in time the representations of different objects simultaneously present (segmentation problem). These models are able to recognize and reconstruct multiple simultaneous external objects, even in difficult case (some wrong or lacking features, shared features, superimposed noise). In Chapter 1.3 the previous models are extended to realize a semantic memory, in which sensory-motor representations of objects are linked with words. To this aim, the network, previously developed, devoted to the representation of objects as a collection of sensory-motor features, is reciprocally linked with a second network devoted to the representation of words (lexical network) Synapses linking the two networks are trained via a time-dependent Hebbian rule, during a training period in which individual objects are presented together with the corresponding words. Simulation results demonstrate that, during the retrieval phase, the network can deal with the simultaneous presence of objects (from sensory-motor inputs) and words (from linguistic inputs), can correctly associate objects with words and segment objects even in the presence of incomplete information. Moreover, the network can realize some semantic links among words representing objects with some shared features. These results support the idea that semantic memory can be described as an integrated process, whose content is retrieved by the co-activation of different multimodal regions. In perspective, extended versions of this model may be used to test conceptual theories, and to provide a quantitative assessment of existing data (for instance concerning patients with neural deficits). PART 2. The ability of the brain to integrate information from different sensory channels is fundamental to perception of the external world (Stein et al, 1993). It is well documented that a number of extraprimary areas have neurons capable of such a task; one of the best known of these is the superior colliculus (SC). This midbrain structure receives auditory, visual and somatosensory inputs from different subcortical and cortical areas, and is involved in the control of orientation to external events (Wallace et al, 1993). SC neurons respond to each of these sensory inputs separately, but is also capable of integrating them (Stein et al, 1993) so that the response to the combined multisensory stimuli is greater than that to the individual component stimuli (enhancement). This enhancement is proportionately greater if the modality-specific paired stimuli are weaker (the principle of inverse effectiveness). Several studies have shown that the capability of SC neurons to engage in multisensory integration requires inputs from cortex; primarily the anterior ectosylvian sulcus (AES), but also the rostral lateral suprasylvian sulcus (rLS). If these cortical inputs are deactivated the response of SC neurons to cross-modal stimulation is no different from that evoked by the most effective of its individual component stimuli (Jiang et al 2001). This phenomenon can be better understood through mathematical models. The use of mathematical models and neural networks can place the mass of data that has been accumulated about this phenomenon and its underlying circuitry into a coherent theoretical structure. In Chapter 2.1 a simple neural network model of this structure is presented; this model is able to reproduce a large number of SC behaviours like multisensory enhancement, multisensory and unisensory depression, inverse effectiveness. In Chapter 2.2 this model was improved by incorporating more neurophysiological knowledge about the neural circuitry underlying SC multisensory integration, in order to suggest possible physiological mechanisms through which it is effected. This endeavour was realized in collaboration with Professor B.E. Stein and Doctor B. Rowland during the 6 months-period spent at the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA), within the Marco Polo Project. The model includes four distinct unisensory areas that are devoted to a topological representation of external stimuli. Two of them represent subregions of the AES (i.e., FAES, an auditory area, and AEV, a visual area) and send descending inputs to the ipsilateral SC; the other two represent subcortical areas (one auditory and one visual) projecting ascending inputs to the same SC. Different competitive mechanisms, realized by means of population of interneurons, are used in the model to reproduce the different behaviour of SC neurons in conditions of cortical activation and deactivation. The model, with a single set of parameters, is able to mimic the behaviour of SC multisensory neurons in response to very different stimulus conditions (multisensory enhancement, inverse effectiveness, within- and cross-modal suppression of spatially disparate stimuli), with cortex functional and cortex deactivated, and with a particular type of membrane receptors (NMDA receptors) active or inhibited. All these results agree with the data reported in Jiang et al. (2001) and in Binns and Salt (1996). The model suggests that non-linearities in neural responses and synaptic (excitatory and inhibitory) connections can explain the fundamental aspects of multisensory integration, and provides a biologically plausible hypothesis about the underlying circuitry.

Gli assetti proprietari e di controllo nell'agroalimentare italiano

The recent default of important Italian agri-business companies provides a challenging issue to be investigated through an appropriate scientific approach. The events involving CIRIO, FERRUZZI or PARMALAT rise an important research question: what are the determinants of performance for Italian companies in the Italian agri – food sector? My aim is not to investigate all the factors that are relevant in explaining performance. Performance depends on a wide set of political, social, economic variables that are strongly interconnected and that are often very difficult to express by formal or mathematical tools. Rather, in my thesis I mainly focus on those aspects that are strictly related to the governance and ownership structure of agri – food companies representing a strand of research that has been quite neglected by previous scholars. The conceptual framework from which I move to justify the existence of a relationship between the ownership structure of a company, governance and performance is the model set up by Airoldi and Zattoni (2005). In particular the authors investigate the existence of complex relationships arising within the company and between the company and the environment that can bring different strategies and performances. They do not try to find the “best” ownership structure, rather they outline what variables are connected and how they could vary endogenously within the whole economic system. In spite of the fact that the Airoldi and Zattoni’s model highlights the existence of a relationship between ownership and structure that is crucial for the set up of the thesis the authors fail to apply quantitative analyses in order to verify the magnitude, sign and the causal direction of the impact. In order to fill this gap we start from the literature trying to investigate the determinants of performance. Even in this strand of research studies analysing the relationship between different forms of ownership and performance are still lacking. In this thesis, after a brief description of the Italian agri – food sector and after an introduction including a short explanation of the definitions of performance and ownership structure, I implement a model in which the performance level (interpreted here as Return on Investments and Return on Sales) is related to variables that have been previously identified by the literature as important such as the financial variables (cash and leverage indices), the firm location (North Italy, Centre Italy, South Italy), the power concentration (lower than 25%, between 25% and 50% and between 50% and 100% of ownership control) and the specific agri – food sector (agriculture, food and beverage). Moreover we add a categorical variable representing different forms of ownership structure (public limited company, limited liability company, cooperative) that is the core of our study. All those variables are fully analysed by a preliminary descriptive analysis. As in many previous contributions we apply a panel least squares analysis for 199 Italian firms in the period 1998 – 2007 with data taken from the Bureau Van Dijck Dataset. We apply two different models in which the dependant variables are respectively the Return on Investments (ROI) and the Return on Sales (ROS) indicators. Not surprisingly we find that companies located in the North Italy representing the richest area in Italy perform better than the ones located in the Centre and South of Italy. In contrast with the Modigliani - Miller theorem financial variables could be significant and the specific sector within the agri – food market could play a relevant role. As the power concentration, we find that a strong property control (higher than 50%) or a fragmented concentration (lower than 25%) perform better. This result apparently could suggest that “hybrid” forms of concentrations could create bad functioning in the decision process. As our key variables representing the ownership structure we find that public limited companies and limited liability companies perform better than cooperatives. This is easily explainable by the fact that law establishes that cooperatives are less profit – oriented. Beyond cooperatives public limited companies perform better than limited liability companies and show a more stable path over time. Results are quite consistent when we consider both ROI and ROS as dependant variables. These results should not lead us to claim that public limited company is the “best” among all possible governance structures. First, every governance solution should be considered according to specific situations. Second more robustness analyses are needed to confirm our results. At this stage we deem these findings, the model set up and our approach represent original contributions that could stimulate fruitful future studies aimed at investigating the intriguing issue concerning the effect of ownership structure on the performance levels.

The Libor Market Model: from theory to calibration

This thesis is focused on the financial model for interest rates called the LIBOR Market Model. In the appendixes, we provide the necessary mathematical theory. In the inner chapters, firstly, we define the main interest rates and financial instruments concerning with the interest rate models, then, we set the LIBOR market model, demonstrate its existence, derive the dynamics of forward LIBOR rates and justify the pricing of caps according to the Black’s formula. Then, we also present the Swap Market Model, which models the forward swap rates instead of the LIBOR ones. Even this model is justified by a theoretical demonstration and the resulting formula to price the swaptions coincides with the Black’s one. However, the two models are not compatible from a theoretical point. Therefore, we derive various analytical approximating formulae to price the swaptions in the LIBOR market model and we explain how to perform a Monte Carlo simulation. Finally, we present the calibration of the LIBOR market model to the markets of both caps and swaptions, together with various examples of application to the historical correlation matrix and the cascade calibration of the forward volatilities to the matrix of implied swaption volatilities provided by the market.

Sostanzialismo e relazionalismo dello spaziotempo alla luce della relatività generale

Until recently the debate on the ontology of spacetime had only a philosophical significance, since, from a physical point of view, General Relativity has been made "immune" to the consequences of the "Hole Argument" simply by reducing the subject to the assertion that solutions of Einstein equations which are mathematically different and related by an active diffeomorfism are physically equivalent. From a technical point of view, the natural reading of the consequences of the "Hole Argument” has always been to go further and say that the mathematical representation of spacetime in General Relativity inevitably contains a “superfluous structure” brought to light by the gauge freedom of the theory. This position of apparent split between the philosophical outcome and the physical one has been corrected thanks to a meticulous and complicated formal analysis of the theory in a fundamental and recent (2006) work by Luca Lusanna and Massimo Pauri entitled “Explaining Leibniz equivalence as difference of non-inertial appearances: dis-solution of the Hole Argument and physical individuation of point-events”. The main result of this article is that of having shown how, from a physical point of view, point-events of Einstein empty spacetime, in a particular class of models considered by them, are literally identifiable with the autonomous degrees of freedom of the gravitational field (the Dirac observables, DO). In the light of philosophical considerations based on realism assumptions of the theories and entities, the two authors then conclude by saying that spacetime point-events have a degree of "weak objectivity", since they, depending on a NIF (non-inertial frame), unlike the points of the homogeneous newtonian space, are plunged in a rich and complex non-local holistic structure provided by the “ontic part” of the metric field. Therefore according to the complex structure of spacetime that General Relativity highlights and within the declared limits of a methodology based on a Galilean scientific representation, we can certainly assert that spacetime has got "elements of reality", but the inevitably relational elements that are in the physical detection of point-events in the vacuum of matter (highlighted by the “ontic part” of the metric field, the DO) are closely dependent on the choice of the global spatiotemporal laboratory where the dynamics is expressed (NIF). According to the two authors, a peculiar kind of structuralism takes shape: the point structuralism, with common features both of the absolutist and substantival tradition and of the relationalist one. The intention of this thesis is that of proposing a method of approaching the problem that is, at least at the beginning, independent from the previous ones, that is to propose an approach based on the possibility of describing the gravitational field at three distinct levels. In other words, keeping the results achieved by the work of Lusanna and Pauri in mind and following their underlying philosophical assumptions, we intend to partially converge to their structuralist approach, but starting from what we believe is the "foundational peculiarity" of General Relativity, which is that characteristic inherent in the elements that constitute its formal structure: its essentially geometric nature as a theory considered regardless of the empirical necessity of the measure theory. Observing the theory of General Relativity from this perspective, we can find a "triple modality" for describing the gravitational field that is essentially based on a geometric interpretation of the spacetime structure. The gravitational field is now "visible" no longer in terms of its autonomous degrees of freedom (the DO), which, in fact, do not have a tensorial and, therefore, nor geometric nature, but it is analyzable through three levels: a first one, called the potential level (which the theory identifies with the components of the metric tensor), a second one, known as the connections level (which in the theory determine the forces acting on the mass and, as such, offer a level of description related to the one that the newtonian gravitation provides in terms of components of the gravitational field) and, finally, a third level, that of the Riemann tensor, which is peculiar to General Relativity only. Focusing from the beginning on what is called the "third level" seems to present immediately a first advantage: to lead directly to a description of spacetime properties in terms of gauge-invariant quantites, which allows to "short circuit" the long path that, in the treatises analyzed, leads to identify the "ontic part” of the metric field. It is then shown how to this last level it is possible to establish a “primitive level of objectivity” of spacetime in terms of the effects that matter exercises in extended domains of spacetime geometrical structure; these effects are described by invariants of the Riemann tensor, in particular of its irreducible part: the Weyl tensor. The convergence towards the affirmation by Lusanna and Pauri that the existence of a holistic, non-local and relational structure from which the properties quantitatively identified of point-events depend (in addition to their own intrinsic detection), even if it is obtained from different considerations, is realized, in our opinion, in the assignment of a crucial role to the degree of curvature of spacetime that is defined by the Weyl tensor even in the case of empty spacetimes (as in the analysis conducted by Lusanna and Pauri). In the end, matter, regarded as the physical counterpart of spacetime curvature, whose expression is the Weyl tensor, changes the value of this tensor even in spacetimes without matter. In this way, going back to the approach of Lusanna and Pauri, it affects the DOs evolution and, consequently, the physical identification of point-events (as our authors claim). In conclusion, we think that it is possible to see the holistic, relational, and non-local structure of spacetime also through the "behavior" of the Weyl tensor in terms of the Riemann tensor. This "behavior" that leads to geometrical effects of curvature is characterized from the beginning by the fact that it concerns extensive domains of the manifold (although it should be pointed out that the values of the Weyl tensor change from point to point) by virtue of the fact that the action of matter elsewhere indefinitely acts. Finally, we think that the characteristic relationality of spacetime structure should be identified in this "primitive level of organization" of spacetime.

Algoritmi esatti per il Job Shop Scheduling: approcci Mathematical Programming

In questa tesi ci occuperemo di fornire un modello MIP di base e di alcune sue varianti, realizzate allo scopo di comprenderne il comportamento ed eventualmente migliorarne l’efficienza. Le diverse varianti sono state costruite agendo in particolar modo sulla definizione di alcuni vincoli, oppure sui bound delle variabili, oppure ancora nell’obbligare il risolutore a focalizzarsi su determinate decisioni o specifiche variabili. Sono stati testati alcuni dei problemi tipici presenti in letteratura e i diversi risultati sono stati opportunamente valutati e confrontati. Tra i riferimenti per tale confronto sono stati considerati anche i risultati ottenibili tramite un modello Constraint Programming, che notoriamente produce risultati apprezzabili in ambito di schedulazione. Un ulteriore scopo della tesi è, infatti, comparare i due approcci Mathematical Programming e Constraint Programming, identificandone quindi i pregi e gli svantaggi e provandone la trasferibilità al modello raffrontato.

Ein inverses Problem für die degeneriert parabolische Richardsgleichung

In der vorliegenden Arbeit werden zwei physikalischeFließexperimente an Vliesstoffen untersucht, die dazu dienensollen, unbekannte hydraulische Parameter des Materials, wiez. B. die Diffusivitäts- oder Leitfähigkeitsfunktion, ausMeßdaten zu identifizieren. Die physikalische undmathematische Modellierung dieser Experimente führt auf einCauchy-Dirichlet-Problem mit freiem Rand für die degeneriertparabolische Richardsgleichung in derSättigungsformulierung, das sogenannte direkte Problem. Ausder Kenntnis des freien Randes dieses Problems soll dernichtlineare Diffusivitätskoeffizient derDifferentialgleichung rekonstruiert werden. Für diesesinverse Problem stellen wir einOutput-Least-Squares-Funktional auf und verwenden zu dessenMinimierung iterative Regularisierungsverfahren wie dasLevenberg-Marquardt-Verfahren und die IRGN-Methode basierendauf einer Parametrisierung des Koeffizientenraumes durchquadratische B-Splines. Für das direkte Problem beweisen wirunter anderem Existenz und Eindeutigkeit der Lösung desCauchy-Dirichlet-Problems sowie die Existenz des freienRandes. Anschließend führen wir formal die Ableitung desfreien Randes nach dem Koeffizienten, die wir für dasnumerische Rekonstruktionsverfahren benötigen, auf einlinear degeneriert parabolisches Randwertproblem zurück.Wir erläutern die numerische Umsetzung und Implementierungunseres Rekonstruktionsverfahrens und stellen abschließendRekonstruktionsergebnisse bezüglich synthetischer Daten vor.

Mathematical Modeling to Investigate Antiarrhythmic Drug Side Effects: Rate-Dependence Role in Ionic Currents and Action Potentials Shape in the O’Hara Model

Sudden cardiac death due to ventricular arrhythmia is one of the leading causes of mortality in the world. In the last decades, it has proven that anti-arrhythmic drugs, which prolong the refractory period by means of prolongation of the cardiac action potential duration (APD), play a good role in preventing of relevant human arrhythmias. However, it has long been observed that the “class III antiarrhythmic effect” diminish at faster heart rates and that this phenomenon represent a big weakness, since it is the precise situation when arrhythmias are most prone to occur. It is well known that mathematical modeling is a useful tool for investigating cardiac cell behavior. In the last 60 years, a multitude of cardiac models has been created; from the pioneering work of Hodgkin and Huxley (1952), who first described the ionic currents of the squid giant axon quantitatively, mathematical modeling has made great strides. The O’Hara model, that I employed in this research work, is one of the modern computational models of ventricular myocyte, a new generation began in 1991 with ventricular cell model by Noble et al. Successful of these models is that you can generate novel predictions, suggest experiments and provide a quantitative understanding of underlying mechanism. Obviously, the drawback is that they remain simple models, they don’t represent the real system. The overall goal of this research is to give an additional tool, through mathematical modeling, to understand the behavior of the main ionic currents involved during the action potential (AP), especially underlining the differences between slower and faster heart rates. In particular to evaluate the rate-dependence role on the action potential duration, to implement a new method for interpreting ionic currents behavior after a perturbation effect and to verify the validity of the work proposed by Antonio Zaza using an injected current as a perturbing effect.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.