Mathematical Methods for 4d N=2 QFTs

In this work we study different aspects of 4d N = 2 superconformal field theories. Not only we accurately define what we mean by a 4d N = 2 superconformal field theory, but we also invent and apply new mathematical methods to classify these theories and to study their physical content. Therefore, although the origin of the subject is physical, our methods and approach are rigorous mathematical theorems: the physical picture is useful to guide the intuition, but the full mathematical rigor is needed to get deep and precise results. No familiarity with the physical concept of Supersymmetry (SUSY) is need to understand the content of this thesis: everything will be explained in due time. The reader shall keep in mind that the driving force of this whole work are the consequences of SUSY at a mathematical level. Indeed, as it will be detailed in part II, a mathematician can understand a 4d N = 2 superconformal field theory as a complexified algebraic integrable system. The geometric properties are very constrained: we deal with special K\ua8ahler geometries with a few other additional structures (see part II for details). Thanks to the rigidity of these structures, we can compute explicitly many interesing quantities: in the end, we are able to give a coarse classification of the space of "action" variables of the integrable system, as well as a fine classification -- only in the case of rank k = 1 -- of the spaces of "angle" variables. We were able to classify conical special K\ua8ahler geometries via a number of deep facts of algebraic number theory, diophantine geometry and class field theory: the perfect overlap between mathematical theorems and physical intuition was astonishing. And we believe we have only scratched the surface of a much deeper theory: we can probably hope to get much more information than what we already discovered; of course, a deeper study of the subject -- as well as its generalizations -- is required. A 4d N = 2 superconformal field theory can thus be defined by its geometric structure: its scaling dimensions, its singular fibers, the monodromy around them and so on. But giving a proper and detailed definition is only the beginning: one may be interested in exploring its physical content. In particular, we are interested in supersymmetric quantities such as BPS states, framed BPS states and UV line operators. These quantities, thanks to SUSY, can be computed independently of many parameters of the theory: this peculiarity makes it possible to use the language of category theory to analyze the aforementioned aspects. As it will be proven in part V, to each 4d N = 2 superconformal field theory we can associate a web of categories, all connected by functors, that describe the BPS states, the framed BPS states (IR) and the UV line operators. Hence, following the old ideas of \u2018t Hooft, it is possible to describe the phase space of gauge theories via categories, since the vacuum expectation values of such line operators are the order parameters of the confinement/deconfinement phase transitions. Mathematically, the (quantum) cluster algebra of Fomin and Zelevinski is the structure needed. Moreover, the analysis of BPS objects led us to a deep understanding of generalized S-dualities. Not only were we able to precisely define -- abstractly and generally -- what the S-duality group of a 4d N = 2 superconformal field theory should be, but we were also able to write a computer algorithm to obtain these groups in many examples (with very high accuracy)

Detection of Buried Inhomogeneous Elliptic Cylinders by a Memetic Algorithm

The application of a global optimization procedure to the detection of buried inhomogeneities is studied in the present paper. The object inhomogeneities are schematized as multilayer infinite dielectric cylinders with elliptic cross sections. An efficient recursive analytical procedure is used for the forward scattering computation. A functional is constructed in which the field is expressed in series solution of Mathieu functions. Starting by the input scattered data, the iterative minimization of the functional is performed by a new optimization method called memetic algorithm. (c) 2003 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works

A Reconstruction Procedure for Microwave Nondestructive Evaluation based on a Numerically Computed Green's Function

This paper describes a new microwave diagnostic tool for nondestructive evaluation. The approach, developed in the spatial domain, is based on the numerical computation of the inhomogeneous Green’s function in order to fully exploit all the available a-priori information of the domain under test. The heavy reduction of the computational complexity of the proposed procedure (with respect to standard procedures based on the free-space Green’s function) is also achieved by means of a customized hybrid-coded genetic algorithm. In order to assess the effectiveness of the method, the results of several simulations are presented and discussed

A Numerical Technique for Determining the Internal Field in Biological Bodies Exposed to Electromagnetic Fields

In this paper, the field prediction inside biological bodies exposed to electromagnetic incident waves is addressed by considering inverse scattering techniques. In particular, the aim is to evaluate the possibility of limiting the test area in order to strongly reduce the computational time, ensuring, at the same time, a quite accurate solution. The approach is based on separating the scattering contributions of the region under test and the other part of the biological body. The starting point is represented by the inverse-scattering equations, which are recast as a functional to be minimized. A Green's function approach is then developed in order to include an approximate knowledge (a model) of the biological body. The possible application of the approach for diagnostic purposes is also discussed

Synthesis of sum and difference patterns for monopulse antennas by an hybrid real/integer-coded differential evolution method

The synthesis of sum and difference patterns of monopulse antennas is considered in this paper. The synthesis problem is recast as an optimization problem by defining a suitable cost function based on the constraints on the side lobe levels. A subarray configuration is considered and the excitations of the difference pattern are approximately determined. The optimization problem is efficently solved by a differential evolution algorithm, wich is able to contemporarly handle real and integer unknowns. Numerical results are reported considering classic array configurations previusly assumed in the literature

Improved Microwave Imaging Procedure for Non-Destructive Evaluations of Two-Dimensional Structures

An improved microwave procedure for detecting defects in dielectric structures is proposed. The procedure is based on the integral equations of the inverse scattering problem. A hybrid Genetic Algorithm is applied in order to minimize the obtained nonlinear functional. Since in nondestructive evaluations the unperturbed object is completely known, it is possible off-line to numerically compute the Green's function for the configuration without defects. Consequently, a very dignificant computation saving is obtained, since the 'chromosome' of the Genetic Algorithm codes only the parameters describing the unknown defect

giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration

We introduce giotto-tda, a Python library that integrates high-performance topological data analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C++ implementations. The library's ability to handle various types of data is rooted in a wide range of preprocessing techniques, and its strong focus on data exploration and interpretability is aided by an intuitive plotting API. Source code, binaries, examples, and documentation can be found at https://github.com/giotto-ai/giotto-tda.Comment: 7 pages, 2 figure

Validation of a Simple, Rapid, and Cost-Effective Method for Acute Rejection Monitoring in Lung Transplant Recipients

Despite advances in immunosuppression therapy, acute rejection remains the leading cause of graft dysfunction in lung transplant recipients. Donor-derived cell-free DNA is increasingly being considered as a valuable biomarker of acute rejection in several solid organ transplants. We present a technically improved molecular method based on digital PCR that targets the mismatch between the recipient and donor at the HLA-DRB1 locus. Blood samples collected sequentially post-transplantation from a cohort of lung recipients were used to obtain proof-of-principle for the validity of the assay, correlating results with transbronchial biopsies and lung capacity tests. The results revealed an increase in dd-cfDNA during the first 2 weeks after transplantation related to ischemia-reperfusion injury (6.36 ± 5.36%, p < 0.0001). In the absence of complications, donor DNA levels stabilized, while increasing again during acute rejection episodes (7.81 ± 12.7%, p < 0.0001). Respiratory tract infections were also involved in the release of dd-cfDNA (9.14 ± 15.59%, p = 0.0004), with a positive correlation with C-reactive protein levels. Overall, the dd-cfDNA percentages were inversely correlated with the lung function values measured by spirometry. These results confirm the value of dd-cfDNA determination during post-transplant follow-up to monitor acute rejection in lung recipients, achieved using a rapid and inexpensive approach based on the HLA mismatch between donor and recipient

Paediatric recurrent pericarditis: Appropriateness of the standard of care and response to IL1-blockade

Objective: To analyse, in a cohort of paediatric patients with recurrent pericarditis (RP) undergoing anti-IL-1 treatment: the agent and dosing used as first line treatment, the long-term efficacy of IL1-blockers, the percentage of patients achieving a drug-free remission, the presence of variables associated with drug-free remission. Study design: Data were collected from patients' charts. Annualized relapse rate (ARR) was used for evaluation of treatment efficacy, bivariate logistic regression analysis for variables associated with drug-free remisison. Results: 58 patients, treated between 2008 and 2018, were included in the study (mean follow-up 2.6 years). 14/56 patients non-responsive to first line drugs were under-dosed. 57 patients were treated with anakinra: the ARR before and during daily treatment was 3.05 and 0.28, respectively (p<0.0001); an increase to 0.83 was observed after the reduction/withdrawal of treatment (p<.0001). The switch from anakinra to canakinumab (5 patients) was associated to an increase of the ARR (0.49 vs 1.46), but without statistical significance (p=0.215). At last follow-up only 9/58 patients had withdrawn all treatments. With the limits of a retrospective study and the heterogeneity between the patients enrolled in the study, a shorter duration of treatment with anakinra was the only variable associated with drug-free remission. Conclusion: This study shows that most of the pediatric patients with RP needing IL-1 blockade received an inadequate treatment with first line agents. The effectiveness of anakinra is supported by this study, but few patients achieved drug free-remission. The different rate of response to anakinra and canakinumab may suggest a possible role of IL1α in the pathogenesis of RP