Branched circle packings, discrete complex polynomials, and the approximation of analytic functions

The aim of this dissertation was to investigate circle packings more general than those which had been studied in the past. In particular, new techniques were developed to explore these broader concepts. We found necessary and sufficient conditions for the existence of finite and infinite branched circle packings; we proved the uniqueness of hexagonal circle packings of finite valence; we also obtained a finite-valence generalization of the Ring Lemma. As a result, the theory of circle packings was extended to incorporate branched circle packings. Moreover, the extended theory was shown to be, in many aspects, a discrete parallel of the theory of analytic functions. Especially, via branched circle packings, we created objects termed discrete Blaschke products and discrete complex polynomials which are discrete analogues of their classical counterparts. Then we proved that any classical complex polynomial (resp. Blaschke product) can be approximated uniformly on compacta by discrete complex polynomials (resp. Blaschke products)

Social distancing enhanced automated optimal design of physical spaces in the wake of the COVID-19 pandemic

NoAs the COVID-19 pandemic unfolds, manually enhanced ad-hoc solutions have helped the physical space designers and decision makers to cope with the dynamic nature of space planning. Due to the unpredictable nature by which the pandemic is unfolding, the standard operating procedures also change, and the protocols for physical interaction require continuous reconsideration. Consequently, the development of an appropriate technological solution to address the current challenge of reconfiguring common physical environments with prescribed physical distancing measures is much needed. To do this, we propose a design optimization methodology which takes the dimensions, as well as the constraints and other necessary requirements of a given physical space to yield optimal redesign solutions on the go. The methodology we propose here utilizes the solution to the well-known mathematical circle packing problem, which we define as a constrained mathematical optimization problem. The resulting optimization problem is solved subject to a given set of parameters and constraints – corresponding to the requirements on the social distancing criteria between people and the imposed constraints on the physical spaces such as the position of doors, windows, walkways and the variables related to the indoor airflow pattern. Thus, given the dimensions of a physical space and other essential requirements, the solution resulting from the automated optimization algorithm can suggest an optimal set of redesign solutions from which a user can pick the most feasible option. We demonstrate our automated optimal design methodology by way of a number of practical examples, and we discuss how this framework can be further taken forward as a design platform that can be implemented practically.University of Bradford's COVID-19 Response Fund, the Spanish Ministry of Science, Innovation, and Universities (Computer Science National Program) under grant #TIN2017-89275-Rof the Agencia Estatal de Investigacion and European Funds (AEI/FEDER, UE

The System of State Aid to the Entrepreneur – the Solutions in Poland and in the European Union

The EC Treaty does not give a definition of the term `State aid', it only lists in Article 87 (1) EC the conditions under which State aid is incompatible with the common market. Neither has the ECJ ruled on what exactly constitutes State aid, it has merely stated: `the concept of State aid must be understood wider than the one of a subsidy as it embraces not only positive benefits such as subsidies themselves, but also interventions which, in various forms, mitigate the charges normally included in the budget of an undertaking and which, without therefore being subsidies in the strict sense of the word, are similar in character and have the same effect'. According to Article 87 (1) EC, a measure must fulfil the following conditions to constitute State aid:–it must favour certain undertakings or the production of certain goods;–it must be granted by a Member State or through State resources;–it must distort or threaten to distort competition, and–it must affect trade between Member States.Polish State Aid Law imposes a general prohibition of State aid in order to prevent competition distortion. Simultaneously the Law specifies the conditions for admissibility of aid principles of monitoring the aid granted by State agencies, local authorities and other bodies administering public resources, including the aid granted by public enterprises. The Law is on one hand aimed at disciplining public spendings and losses in public revenues which are used to support entrepreneurs and on the other hand at the approximation of the Polish State aid law to the standards of the European Union