Complex functions with Geogebra

Complex functions, generally feature some interesting peculiarities, seen as extensions real functions, complementing the study of real analysis. However, the visualization of some complex functions properties requires the simultaneous visualization of two-dimensional spaces. The multiple Windows of GeoGebra, combined with its ability of algebraic computation with complex numbers, allow the study of the functions defined from ℂ to ℂ through traditional techniques and by the use of Domain Colouring. Here, we will show how we can use GeoGebra for the study of complex functions, using several representations and creating tools which complement the tools already provided by the software. Our proposals designed for students of the first year of engineering and science courses can and should be used as an educational tool in collaborative learning environments. The main advantage in its use in individual terms is the promotion of the deductive reasoning (conjecture / proof). In performed the literature review few references were found involving this educational topic and by the use of a single software.

Using GeoGebra to study complex functions

The aim of this workshop to present some of the strategies studied to use GeoGebra in the analysis of complex functions. The proposed tasks focus on complex analysis topics target for students of the 1st year of higher education, which can be easily adapted to pre-university students. In the first part of this workshop we will illustrate how to use the two graphical windows of GeoGebra to represent complex functions of complex variable. The second part will present the use of the dynamic color Geogebra in order to obtain Coloring domains that correspond to the graphic representation of complex functions. Finally, we will use the threedimensional graphics window in GeoGebra to study the component functions of a complex function. During the workshop will be provided scripts orientation of the different tasks proposed to be held on computers with Geogebra version 5.0 or high.

Singlet extensions of the standard model at LHC Run 2: benchmarks and comparison with the NMSSM

The Complex singlet extension of the Standard Model (CxSM) is the simplest extension that provides scenarios for Higgs pair production with different masses. The model has two interesting phases: the dark matter phase, with a Standard Model-like Higgs boson, a new scalar and a dark matter candidate; and the broken phase, with all three neutral scalars mixing. In the latter phase Higgs decays into a pair of two different Higgs bosons are possible. In this study we analyse Higgs-to-Higgs decays in the framework of singlet extensions of the Standard Model (SM), with focus on the CxSM. After demonstrating that scenarios with large rates for such chain decays are possible we perform a comparison between the NMSSM and the CxSM. We find that, based on Higgs-to-Higgs decays, the only possibility to distinguish the two models at the LHC run 2 is through final states with two different scalars. This conclusion builds a strong case for searches for final states with two different scalars at the LHC run 2. Finally, we propose a set of benchmark points for the real and complex singlet extensions to be tested at the LHC run 2. They have been chosen such that the discovery prospects of the involved scalars are maximised and they fulfil the dark matter constraints. Furthermore, for some of the points the theory is stable up to high energy scales. For the computation of the decay widths and branching ratios we developed the Fortran code sHDECAY, which is based on the implementation of the real and complex singlet extensions of the SM in HDECAY.

Novel hardware and software combination for automotive motorsport vehicles based on direct processing of graphical functions

The main motivation for the work presented here began with previously conducted experiments with a programming concept at the time named "Macro". These experiments led to the conviction that it would be possible to build a system of engine control from scratch, which could eliminate many of the current problems of engine management systems in a direct and intrinsic way. It was also hoped that it would minimize the full range of software and hardware needed to make a final and fully functional system. Initially, this paper proposes to make a comprehensive survey of the state of the art in the specific area of software and corresponding hardware of automotive tools and automotive ECUs. Problems arising from such software will be identified, and it will be clear that practically all of these problems stem directly or indirectly from the fact that we continue to make comprehensive use of extremely long and complex "tool chains". Similarly, in the hardware, it will be argued that the problems stem from the extreme complexity and inter-dependency inside processor architectures. The conclusions are presented through an extensive list of "pitfalls" which will be thoroughly enumerated, identified and characterized. Solutions will also be proposed for the various current issues and for the implementation of these same solutions. All this final work will be part of a "proof-of-concept" system called "ECU2010". The central element of this system is the before mentioned "Macro" concept, which is an graphical block representing one of many operations required in a automotive system having arithmetic, logic, filtering, integration, multiplexing functions among others. The end result of the proposed work is a single tool, fully integrated, enabling the development and management of the entire system in one simple visual interface. Part of the presented result relies on a hardware platform fully adapted to the software, as well as enabling high flexibility and scalability in addition to using exactly the same technology for ECU, data logger and peripherals alike. Current systems rely on a mostly evolutionary path, only allowing online calibration of parameters, but never the online alteration of their own automotive functionality algorithms. By contrast, the system developed and described in this thesis had the advantage of following a "clean-slate" approach, whereby everything could be rethought globally. In the end, out of all the system characteristics, "LIVE-Prototyping" is the most relevant feature, allowing the adjustment of automotive algorithms (eg. Injection, ignition, lambda control, etc.) 100% online, keeping the engine constantly working, without ever having to stop or reboot to make such changes. This consequently eliminates any "turnaround delay" typically present in current automotive systems, thereby enhancing the efficiency and handling of such systems.

Phase transitions and nonlinear phenomena in neuronal network models

Communication and cooperation between billions of neurons underlie the power of the brain. How do complex functions of the brain arise from its cellular constituents? How do groups of neurons self-organize into patterns of activity? These are crucial questions in neuroscience. In order to answer them, it is necessary to have solid theoretical understanding of how single neurons communicate at the microscopic level, and how cooperative activity emerges. In this thesis we aim to understand how complex collective phenomena can arise in a simple model of neuronal networks. We use a model with balanced excitation and inhibition and complex network architecture, and we develop analytical and numerical methods for describing its neuronal dynamics. We study how interaction between neurons generates various collective phenomena, such as spontaneous appearance of network oscillations and seizures, and early warnings of these transitions in neuronal networks. Within our model, we show that phase transitions separate various dynamical regimes, and we investigate the corresponding bifurcations and critical phenomena. It permits us to suggest a qualitative explanation of the Berger effect, and to investigate phenomena such as avalanches, band-pass filter, and stochastic resonance. The role of modular structure in the detection of weak signals is also discussed. Moreover, we find nonlinear excitations that can describe paroxysmal spikes observed in electroencephalograms from epileptic brains. It allows us to propose a method to predict epileptic seizures. Memory and learning are key functions of the brain. There are evidences that these processes result from dynamical changes in the structure of the brain. At the microscopic level, synaptic connections are plastic and are modified according to the dynamics of neurons. Thus, we generalize our cortical model to take into account synaptic plasticity and we show that the repertoire of dynamical regimes becomes richer. In particular, we find mixed-mode oscillations and a chaotic regime in neuronal network dynamics.