On moduli spaces of semistable sheaves on K3 surfaces

Ich untersuche die nicht bereits durch die Arbeit "Singular symplectic moduli spaces" von Kaledin, Lehn und Sorger (Invent. Math. 164 (2006), no. 3) abgedeckten Fälle von Modulräumen halbstabiler Garben auf projektiven K3-Flächen - die Fälle mit Mukai-Vektor (0,c,0) sowie die Modulräume zu nichtgenerischen amplen Divisoren - hinsichtlich der möglichen Konstruktion neuer Beispiele von kompakten irreduziblen symplektischen Mannigfaltigkeiten. Ich stelle einen Zusammenhang zu den bereits untersuchten Modulräumen und Verallgemeinerungen derselben her und erweitere bekannte Ergebnisse auf alle offenen Fälle von Garben vom Rang 0 und viele Fälle von Garben von positivem Rang. Insbesondere kann in diesen Fällen die Existenz neuer Beispiele von kompakten irreduziblen symplektischen Mannigfaltigkeiten, die birational über Komponenten des Modulraums liegen, ausgeschlossen werden.

Knots and links in lens spaces

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.

Moduli spaces of (G,h)-constellations

Given a reductive group G acting on an affine scheme X over C and a Hilbert function h: Irr G → N_0, we construct the moduli space M_Ө(X) of Ө-stable (G,h)-constellations on X, which is a common generalisation of the invariant Hilbert scheme after Alexeev and Brion and the moduli space of Ө-stable G-constellations for finite groups G introduced by Craw and Ishii. Our construction of a morphism M_Ө(X) → X//G makes this moduli space a candidate for a resolution of singularities of the quotient X//G. Furthermore, we determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of Sl_2 on (C²)⁶ as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for the realisation of such calculations. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.

Higher order Sobolev Spaces and polyharmonic boundary value problems in Carnot Groups

The main task of this work is to present a concise survey on the theory of certain function spaces in the contexts of Hörmander vector fields and Carnot Groups, and to discuss briefly an application to some polyharmonic boundary value problems on Carnot Groups of step 2.

Fibered knots and links in lens spaces

La tesi propone alcuni esempi di link fibrati in spazi lenticolari. Sfruttando la compatibilità fra le mosse di chirurgia intera e la nozione di open book decomposition, si ricava un esempio di link fibrato prima in L(p,1), per poi generalizzarlo a L(p,q). Si conclude determinando una struttura di contatto equivalente alla open book relativa agli spazi del tipo L(p,1).

Cooperative Cognitive Wireless Networks over TV White and Gray Spaces

Wireless networks rapidly became a fundamental pillar of everyday activities. Whether at work or elsewhere, people often benefits from always-on connections. This trend is likely to increase, and hence actual technologies struggle to cope with the increase in traffic demand. To this end, Cognitive Wireless Networks have been studied. These networks aim at a better utilization of the spectrum, by understanding the environment in which they operate, and adapt accordingly. In particular recently national regulators opened up consultations on the opportunistic use of the TV bands, which became partially free due to the digital TV switch over. In this work, we focus on the indoor use of of TVWS. Interesting use cases like smart metering and WiFI like connectivity arise, and are studied and compared against state of the art technology. New measurements for TVWS networks will be presented and evaluated, and fundamental characteristics of the signal derived. Then, building on that, a new model of spectrum sharing, which takes into account also the height from the terrain, is presented and evaluated in a real scenario. The principal limits and performance of TVWS operated networks will be studied for two main use cases, namely Machine to Machine communication and for wireless sensor networks, particularly for the smart grid scenario. The outcome is that TVWS are certainly interesting to be studied and deployed, in particular when used as an additional offload for other wireless technologies. Seeing TVWS as the only wireless technology on a device is harder to be seen: the uncertainity in channel availability is the major drawback of opportunistic networks, since depending on the primary network channel allocation might lead in having no channels available for communication. TVWS can be effectively exploited as offloading solutions, and most of the contributions presented in this work proceed in this direction.

Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Recycling the city: a sustainable planning framework to reduce, reuse and recycle urban residual spaces

Cities are key locations where Sustainability needs to be addressed at all levels, as land is a finite resource. However, not all urban spaces are exploited at best, and land developers often evaluate unused, misused, or poorly-designed urban portions as impracticable constraints. Further, public authorities lose the challenge to enable and turn these urban spaces into valuable opportunities where Sustainable Urban Development may flourish. Arguing that these spatial elements are at the centre of SUD, the paper elaborates a prototype in the form of a conceptual strategic planning framework, committed to an effective recycling of the city spaces using a flexible and multidisciplinary approach. Firstly, the research focuses upon a broad review of Sustainability literature, highlighting established principles and guidelines, building a sound theoretical base for the new concept. Hence, it investigates origins, identifies and congruently suggests a definition, characterisation and classification for urban “R-Spaces”. Secondly, formal, informal and temporary fitting functions are analysed and inserted into a portfolio meant to enhance adaptability and enlarge the choices for the on-site interventions. Thirdly, the study outlines ideal quality requirements for a sustainable planning process. Then, findings are condensed in the proposal, which is articulated in the individuation of tools, actors, plans, processes and strategies. Afterwards, the prototype is tested upon case studies: Solar Community (Casalecchio di Reno, Bologna) and Hyllie Sustainable City Project, the latter developed via an international workshop (ACSI-Camp, Malmö, Sweden). Besides, the qualitative results suggest, inter alia, the need to right-size spatial interventions, separate structural and operative actors, involve synergies’ multipliers and intermediaries (e.g. entrepreneurial HUBs, innovation agencies, cluster organisations…), maintain stakeholders’ diversity and create a circular process open for new participants. Finally, the paper speculates upon a transfer of the Swedish case study to Italy, and then indicates desirable future researches to favour the prototype implementation.

Un modello di percezione del colore, espresso da un problema di Cauchy in spazi di Banach

Scopo della tesi è studiare un modello di percezione cromatica, che descrive la propagazione dell'attività mediante un problema di Cauchy in spazi di Banach. Presentiamo dapprima il problema della stabilità delle soluzioni al problema di Cauchy tramite il metodo Lyapunov; prima in dimensione finita, e poi in spazi di Banach. Poi verifichiamo che l'equazione fondamentale di percezione cromatica ricade nel setting considerato e che il funzionale di Lyapunov associato verifica le ipotesi che assicurano la stabilità.

Compattezza debole in spazi di Banach riflessivi

Lo scopo di questa tesi è dare la nozione di topologia debole in spazi di Banach e fornire una caratterizzazione della compattezza debole in spazi riflessivi e separabili. Infatti in spazi di Banach riflessivi e separabili la compattezza è equivalente alla compattezza sequenziale, mentre in mancanza delle ipotesi di riflessività e di separabilità dello spazio ciò non è vero.

Comparison of physiological triggering schemes for diffusion-weighted magnetic resonance imaging in kidneys

To determine the potential benefit of combined respiratory-cardiac triggering for diffusion-weighted imaging (DWI) of kidneys compared to respiratory triggering alone (RT).

Evaluation of renal allograft function early after transplantation with diffusion-weighted MR imaging

To determine the inter-patient variability of apparent diffusion coefficients (ADC) and concurrent micro-circulation contributions from diffusion-weighted MR imaging (DW-MRI) in renal allografts early after transplantation, and to obtain initial information on whether these measures are altered in histologically proven acute allograft rejection (AR).

Predicting and monitoring cancer treatment response with diffusion-weighted MRI

An imaging biomarker that would provide for an early quantitative metric of clinical treatment response in cancer patients would provide for a paradigm shift in cancer care. Currently, nonimage based clinical outcome metrics include morphology, clinical, and laboratory parameters, however, these are obtained relatively late following treatment. Diffusion-weighted MRI (DW-MRI) holds promise for use as a cancer treatment response biomarker as it is sensitive to macromolecular and microstructural changes which can occur at the cellular level earlier than anatomical changes during therapy. Studies have shown that successful treatment of many tumor types can be detected using DW-MRI as an early increase in the apparent diffusion coefficient (ADC) values. Additionally, low pretreatment ADC values of various tumors are often predictive of better outcome. These capabilities, once validated, could provide for an important opportunity to individualize therapy thereby minimizing unnecessary systemic toxicity associated with ineffective therapies with the additional advantage of improving overall patient health care and associated costs. In this report, we provide a brief technical overview of DW-MRI acquisition protocols, quantitative image analysis approaches and review studies which have implemented DW-MRI for the purpose of early prediction of cancer treatment response.

Synthetic-echo time postprocessing technique for generating images with variable T2-weighted contrast: diagnosis of meniscal and cartilage abnormalities of the knee

To assess the feasibility and accuracy of a synthetic-echo time (TE) magnetic resonance (MR) postprocessing technique for the diagnostic evaluation of abnormalities of menisci and articular cartilage in the knee.

Diffusion-weighted imaging for the follow-up of patients after matrix-associated autologous chondrocyte transplantation

To evaluate the use of diffusion-weighted imaging (DWI) for the assessment of cartilage maturation in patients after matrix-associated autologous chondrocyte transplantation (MACT).