Combinatorial presentation of multidimensional persistent homology

A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals.Comment: 21 pages, 3 figure

Topology of Social and Managerial Networks

With the explosion of innovative technologies in recent years, organizational and man- agerial networks have reached high levels of intricacy. These are one of the many complex systems consisting of a large number of highly interconnected heterogeneous agents. The dominant paradigm in the representation of intricate relations between agents and their evolution is a network (graph). The study of network properties, and their implications on dynamical processes, up to now mostly focused on locally defined quantities of nodes and edges. These methods grounded in statistical mechanics gave deep insight and explanations on real world phenomena; however there is a strong need for a more versatile approach which would rely on new topological methods either separately or in combination with the classical techniques. In this thesis we approach this problem introducing new topological methods for network analysis relying on persistent homology. The results gained by the new methods apply both to weighted and unweighted networks; showing that classi- cal connectivity measures on managerial and societal networks can be very imprecise and extending them to weighted networks with the aim of uncovering regions of weak connectivity. In the first two chapters of the thesis we introduce the main instruments that will be used in the subsequent chapters, namely basic techniques from network theory and persistent homology from the field of computational algebraic topology. The third chapter of the thesis approaches social and organizational networks studying their con- nectivity in relation to the concept of social capital. Many sociological theories such as the theory of structural holes and of weak ties relate social capital, in terms of profitable managerial strategies and the chance of rewarding opportunities, to the topology of the underlying social structure. We review the known connectivity measures for social networks, stressing the fact that they are all local measures, calculated on a node’s Ego network, i.e considering a nodes direct contacts. By analyzing real cases it, nevertheless, turns out that the above measures can be very imprecise for strategical individuals in social networks, revealing fake brokerage opportunities. We, therefore, propose a new set of measures, complementary to the existing ones and focused on detecting the position of links, rather than their density, therefore extending the standard approach to a mesoscopic one. Widening the view from considering direct neighbors to considering also non-direct ones, using the “neighbor filtration”, we give a measure of height and weight for structural holes, obtaining a more accurate description of a node’s strategical position within its contacts. We also provide a refined version of the network efficiency measure, which collects in a compact form the height of all structural holes. The methods are implemented and have been tested on real world organizational and managerial networks. In pursuing the objective of improving the existing methods we faced some technical difficulties which obliged us to develop new mathematical tools. The fourth chapter of the thesis deals with the general problem of detecting structural holes in weighted networks. We introduce thereby the weight clique rank filtration, to detect particular non-local structures, akin to weighted structural holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchi- cally nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasi local network properties, because of the intrinsic non-locality of homology, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. In the fifth chapter of the thesis, we develop new insights in the mathematical setting underlying multipersistent homology. More specifically we calculate combinatorial resolutions and efficient Gro ̈bner bases for multipersistence homology modules. In this new frontier of persistent homology, filtrations are parametrized by multiple elements. Using multipersistent homology temporal networks can be studied and the weight filtration and neighbor filtration can be combined

COMPARISON OF MULTI-SOURCE DATA, INTEGRATED SURVEY FOR COMPLEX ARCHITECTURE DOCUMENTATION

The metric documentation of architectural complexes requires today the use of several integrated survey methodologies. This need is an answer to the morphology of the object such as dimension, geometry, inaccessible areas and urban context. These properties inhibit the use of single surveying techniques and force the integration of Geomatics tools. In addition, the metric documentation of Cultural heritage objects not always requires uniform accuracy and resolution, therefore the integration of different surveying methodologies and techniques become the only effective solution both from a technical and economic point of view. The integration, that is today adopted as normal strategy, allows also the better understanding of the benefits which can arise to speed up the metric documentation of Cultural Heritage objects and the benefits that each of the possible surveying techniques can have thanks to the integration of the different potentialities. This study starting from an integrated survey, performed whit a combined use of Mobile Mapping System (MMS), Unmanned Aerial Vehicles (UAV) and Terrestrial Laser Scanner (TLS) and show the results of the comparisons between the possible achievable accuracies by using a correct integration between the different used technologies and the ones achievable by using the same techniques as independent tools. The case study is the architectural complex of the Ducal Palace in Gubbio (Italy), located upstream of the most important town square facing the cathedral in a very complex but realistic urban context

Homotopical decompositions of simplicial and Vietoris Rips complexes

Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components

Cancer rates and mortality in people with severe mental illness: Further evidence of lack of parity

Background: Severe mental illness (SMI) is associated with poorer physical health, however the relationship between SMI and cancer is complex and previous study findings are inconsistent. Low incidence of cancer in those with SMI has been attributed to premature mortality, though evidence for this is lacking. We aimed to investigate the relationship between SMI and cancer incidence and mortality, and to assess the effect of premature mortality on cancer incidence rates. / Methods: In this UK-wide matched cohort study using primary care records we calculated incidence and mortality rates of all-cancer, and bowel, lung, breast or prostate cancer, in patients with SMI, compared to matched patients without SMI. We used competing risks regression to account for mortality from other causes. / Findings: 69,632 patients had an SMI diagnosis. The rate of all-cancer diagnoses was reduced in those with SMI (Hazard ratio (HR):0·95; 95%CI 0·93–0·98) compared to those without SMI, and particularly in those with schizophrenia (HR:0·82; 95%CI 0·77–0·88) compared to those without SMI. When accounting for the competing risk of premature mortality, incidence remained lower only in patients with schizophrenia. All-cause mortality after cancer was increased in the SMI group, and cancer-specific mortality was increased in those with schizophrenia (hazard ratio: 1.96; 95%CI 1.57–2.44). / Interpretation: Patients with schizophrenia have lower rates of cancer diagnosis but higher all-cause and cancer-specific mortality rates following diagnosis compared to those without SMI. Premature mortality does not explain these differences, suggesting the findings reflect barriers to cancer diagnosis and treatment, which need to be identified and addressed

FROM DATA TO TANGIBLE MODELS: CASE STUDY OF A VAULT IN THE ROYAL RESIDENCE OF VENARIA REALE

Framed on a wider research project that investigates Geometry as a cultural substrate and shared language for the comprehension of Architecture and its shapes, the presented research focuses on the geometric analysis and dissemination actions of a vault of the Royal Residence of Venaria Reale, designed by Benedetto Alfieri in the XVIII century. The vault is the only one offering to visitors’ sight both its intrados and extrados surfaces. We propose an interdisciplinary approach that uses Geometry both as qualifying intangible heritage of the built shapes and as a language transversal to observation and survey, return of data and their interpretation from a dissemination point of view. To achieve this, we propose an innovative use of physical models, both in their meaning of object to be explored and in that of their design, between prototyping and seriality. Interaction between public and physical models becomes a way to promote critical shape-reading activities and to enhance spatial visualization abilities by their haptic/visual exploration, to recognize 3D built geometry and to explore architecture from different a point of view, getting closer to its shapes

Mass Spectrometry-Based Metabolomic and Proteomic Strategies in Organic Acidemias

Organic acidemias (OAs) are inherited metabolic disorders caused by deficiency of enzymatic activities in the catabolism of amino acids, carbohydrates, or lipids. These disorders result in the accumulation of mono-, di-, or tricarboxylic acids, generally referred to as organic acids. The OA outcomes can involve different organs and/or systems. Some OA disorders are easily managed if promptly diagnosed and treated, whereas, in others cases, such as propionate metabolism-related OAs (propionic acidemia, PA; methylmalonic acidemia, MMA), neither diet, vitamin therapy, nor liver transplantation appears to prevent multiorgan impairment. Here, we review the recent developments in dissecting molecular bases of OAs by using integration of mass spectrometry-(MS-) based metabolomic and proteomic strategies. MS-based techniques have facilitated the rapid and economical evaluation of a broad spectrum of metabolites in various body fluids, also collected in small samples, like dried blood spots. This approach has enabled the timely diagnosis of OAs, thereby facilitating early therapeutic intervention. Besides providing an overview of MS-based approaches most frequently used to study the molecular mechanisms underlying OA pathophysiology, we discuss the principal challenges of metabolomic and proteomic applications to OAs