In this work we provide several improvements in the study of phase transitions of interacting particle systems: - We determine a quantitative relation between non-extremality of the limiting Gibbs measure of a tree-based spin system, and the temporal mixing of the Glauber Dynamics over its finite projections. We define the concept of 'sensitivity' of a reconstruction scheme to establish such a relation. In particular, we focus on the independent sets model, determining a phase transition for the mixing time of the Glauber dynamics at the same location of the extremality threshold of the simple invariant Gibbs version of the model. - We develop the technical analysis of the so-called spatial mixing conditions for interacting particle systems to account for the connectivity structure of the underlying graph. This analysis leads to improvements regarding the location of the uniqueness/non-uniqueness phase transition for the independent sets model over amenable graphs; among them, the elusive hard-square model in lattice statistics, which has received attention since Baxter's solution of the analogous hard-hexagon in 1980. - We build on the work of Montanari and Gerschenfeld to determine the existence of correlations for the coloring model in sparse random graphs. In particular, we prove that correlations exist above the 'clustering' threshold of such a model; thus providing further evidence for the conjectural algorithmic 'hardness' occurring at such a point.

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In this dissertation we study free boundary problems that model the evolution of interfaces in the presence of elasticity, such as thin film profiles and material void boundaries. These problems are characterized by the competition between the elastic bulk energy and the anisotropic surface energy. First, we consider the evolution equation with curvature regularization that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate. The film is strained due to the mismatch between the crystalline lattices of the two materials and anisotropy is taken into account. We present the results contained in [62] where the author establishes short time existence, uniqueness and regularity of the solution using De Giorgi’s minimizing movements to exploit the L2 -gradient flow structure of the equation. This seems to be the first analytical result for the evaporation-condensation case in the presence of elasticity. Second, we consider the relaxed energy introduced in [20] that depends on admissible pairs (E, u) of sets E and functions u defined only outside of E. For dimension three this energy appears in the study of the material voids in solids, where the pairs (E, u) are interpreted as the admissible configurations that consist of void regions E in the space and of displacements u of the atoms of the crystal. We provide the precise mathematical framework that guarantees the existence of minimal energy pairs (E, u). Then, we establish that for every minimal configuration (E, u), the function u is C 1,γ loc -regular outside an essentially closed subset of E. No hypothesis of starshapedness is assumed on the voids and all the results that are contained in [18] hold true for every dimension d ≥ 2.

La presente tesi si propone di comprendere come le organizzazioni chiamate ad agire in mercati complessi siano in grado di produrre, sedimentare, utilizzare, condividere e rendere disponibile la propria conoscenza. La ricerca si focalizza sulle organizzazioni creative legate all'artigianato d’eccellenza in quanto realtà ancora poco esplorate e all'interno delle quali il tema proposto è percepito in misura crescente. Dallo studio emergono due risultati significativi raggiunti grazie all’elaborazione di una review della letteratura e di un’indagine qualitativa condotta sul campo. Il primo risultato è la constatazione di un recente interesse da parte degli studi accademici per il tema della conoscenza e del suo trasferimento con riferimento alle industrie creative dell’artigianato d’eccellenza. Il secondo risultato mette in risalto la relazione tra trasferimento del saper fare e creazione di manufatti d’eccellenza, e riconosce il maestro d’arte come figura cruciale di questo processo di trasferimento della conoscenza (knowledge transfer) in quanto detentore di un sapere che costituisce ed alimenta il valore unico ed esclusivo dei manufatti. Il lavoro proposto riconosce quindi nel sapere dei maestri d’arte una risorsa cardine da valorizzare e trasmettere, in quanto leva di sviluppo di sistemi produttivi basati sulla qualità, sull’eccellenza e sulla differenziazione. / The goal of this study is to understand how organizations operating in a complex market are able to produce, leave sediment, use, share and make available their knowledge. The focus is on arts & crafts creative industries as still insufficiently investigated organizations, although the proposed theme is increasingly perceived. This study presents two significant results achieved thanks to a literature review and a qualitative field research. The first result is the identification of a unprecedented interest of academic studies for the issue of knowledge and its transfer relevant to arts & crafts creative industries. The second result focuses on the relationship between knowledge transfer and creation of artifacts of excellence. In this second part of the study, the craftsman is a critical factor in the process of knowledge transfer as holder of a knowledge which constitutes and nurtures the sole and the exclusive value of the products. In the present dissertation the knowledge of these craftsmen emerges as a key resource that must be valued and transmitted as lever of development of production systems based on quality, excellence and differentiation.

Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts. In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches. Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed. Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems.

This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories. I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I examine the fictionalist views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is false. I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them irrelevant to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an internal and an external perspective on existence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are internal to mathematics and, hence, that mathematical truth cannot be used to draw the conclusion that mathematical objects exist in an external/ontological sense.

Title: Flows of incompressible fluids with pressure-dependent viscosity (and their application to modelling the flow in journal bearing) Author: Martin Lanzendörfer Department: Mathematical Institute of Charles University Supervisor: prof. RNDr. Josef Málek, DSc. Abstract: The viscosity of the fluids involved in hydrodynamic lubrication typically depends on pressure and shear rate. The thesis is concerned with steady isothermal flows of such fluds. Generalizing the recent results achieved in the case of homogeneous Dirichlet boundary conditions, the existence and uniqueness of weak solutions subject to the boundary conditions employed in practical applications will be established. The second part is concerned with numerical simulations of the lubrication flow. The experiments indicate that the presented finite element method is successful as long as certain restrictions on the constitutive model are met. Both the restrictions involved in the theoretical results and those indicated by the numerical experiments allow to accurately model real-world lubricants in certain ranges of pressures and shear rates. The last part quantifies those ranges for three representative lubricants. Keywords: existence and uniqueness of weak solutions, finite element method, pressure- thickening, shear-thinning, incompressible fluids,...

This thesis consists of three parts: The first and second parts focus on long-time asymptotics of macroscopic and kinetic models respectively, while in the third part we connect these regimes using different scaling approaches. (1) Keller–Segel-type aggregation-diffusion equations: We study a Keller–Segel-type model with non-linear power-law diffusion and non-local particle interaction: Does the system admit equilibria? If yes, are they unique? Which solutions converge to them? Can we determine an explicit rate of convergence? To answer these questions, we make use of the special gradient flow structure of the equation and its associated free energy functional for which the overall convexity properties are not known. Special cases of this family of models have been investigated in previous works, and this part of the thesis represents a contribution towards a complete characterisation of the asymptotic behaviour of solutions. (2) Hypocoercivity techniques for a fibre lay-down model: We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in non-woven textile production. Further, we prove convergence to equilibrium with an explicit rate. This part of the thesis is an extension of previous work which considered the case of a stationary conveyor belt. Adding the movement of the belt, the global equilibrium state is not known explicitly and a more general hypocoercivity estimate is needed. Although we focus here on a particular application, this approach can be used for any equation with a similar structure as long as it can be understood as a certain perturbation of a system for which the global Gibbs state is known. (3) Scaling approaches for collective animal behaviour models: We study the multi-scale aspects of self-organised biological aggregations using various scaling techniques. Not many previous studies investigate how the dynamics of the initial models are preserved via these scalings. Firstly, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local kinetic 1D and 2D models to simpler models existing in the literature. Secondly, we investigate how some of the kinetic spatio-temporal patterns are preserved via these scalings using asymptotic preserving numerical methods.

Die Johannese geskrifte (Evangelie en Briewe) word veralgemenend binne die Christendom gelees en die uniekheidsbeklemtoninge ten opsigte van Jesus as enige Verlosser word ook so verstaan. Hierdie hoofsaaklik sosiaal-wetenskaplike ondersoek was gefokus op Johannes se aansprake ten opsigte van Jesus en is gedoen om bewus te maak van die sosiaal-kulturele onderbou van die geskrifte. As sodanig is dit bedoe! om heuristies in te werk ten einde bestaande hermeneuse beter te dien sodat die boodskap van die Nuwe Testament effektiewer oorgedra kan word. Dit is ook gepoog om bevindinge uit die studie deur te trek na die gesekulariseerde samelewing en pluralistiese religieuse standpunt wat te Iande bestaan. Die teksgedeeltes wat eksegeties ondersoek is, bevat die aspekte van aanvaarding of verwerping van Jesus as die unieke Godsagent wat ewige /ewe meedeel. Dit het geblyk dat die begrip in Johannes nie net op oneindigheid in die hiemamaals dui nie, maar veral op kwaliteit in die hede. Johannes se postulaat is dat Jesus konstant hierdie lewe meedeel aan hulle wat in Hom glo. Die vraag na relevansie van 'n religie uit Judaistiese oorsprong in 'n AfroWesterse samelewing en kultuur is vanuit die aspekte van kulturele relatiwisme en relatiwiteit hanteer. Dit word aanvaar dat die ingrype van God deur Jesus Christus binne die Israelitiese volksmilieu en Mediterreense kultuur plaasgevind het, maar dat dit wat Johannes aan sy lesers herbevestig het, vir aile mense relevant is. Jesus is die unieke Godsagent wat ewige lewe meedeel, wat, as sodanig, nie menslike sterflikheid negeer nie, maar dit transendeer. / The social-scientific research for this treatise concentrated on John's assertion of Jesus' uniqueness. The selected Scripture portions for exegesis contain the aspects of receiving or rejecting Him as God's Agent who bestows eternal life. Eternal lifo in John does not so much denote never ending life, but rather excellent quality of life in the present. The relevancy of a religion from a Judaistic origin in an Afro-Western cultural society was treated on the aspects of cultural relativism and cultural relativity. The conclusion accedes to the fact that God's interaction with man in the person of Jesus Christ took place within an Israelite national milieu and Mediterranean culture, but what John reasserted is relevant for all time- the life that Jesus bestows does not negate mortality, but transcends it. / New Testament / M. Th. (Nuwe Testament)