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In this thesis the theorems of Poncelet and Cayley will be discussed. We will use algebraic geometry and in particular the theory of elliptic curves to present a proof following the works of Griffiths and Harris (refrence 1).
The P2-version of poncelet’s theorem that we prove in our thesis says:let C , D be smooth conic in general position. Suppose there is an n-sided olygon inscribed in C and circumscribed about D.Then for any point of C, there exists an n-sided polygen, also inscribed in C and circumscribe about D, which has this point for one of its vertics.the Cayley’s theorem provides a criterion on the existence of such an n-gon given the conics C and D as above  

As the fourth most common neurological disorder, epilepsy affects lots of people all around the world, some of whom have to live with unpredictable seizures uncontrollable by surgery or medication. Hence, Developing systems for detection and prediction of the epileptic seizures will help the patients to avoid the possible damages caused by sudden seizures. This study addresses the task of epileptic seizure prediction, using three different novel approaches. The first approach, which is based on anomaly detection, contains three steps: feature extraction from EEG signals, training a one-class SVM classifier, and a post-processing step. The second method exploits a recurrent neural network to... 

Creating talking heads from audio input is interesting from both scientific and practical viewpoints, e.g. constructing virtual computer generated characters, aiding hearing-impaired people, live dubbing of videos with translated audio, etc. Due to its wide variety of applications, audio to video has been the focus of intensive research in recent years. Mapping audio to facial images with accurate lip-sync is an extremely difficult task because it is a mapping form 2-Dimensional to 3-Dimensional space and also because humans are expert at detecting any out-of-sync lip movements with respect to an audio.Approaches to automatically generating natural looking speech animation usually involve... 

In this study, the solution of compressible flows is performed using finite volume lattice Boltzmann method (FVLBM). A model associated with 13 discrete velocity vectors and 2 energy levels is used and the Boltzmann transport equation is solved using a cell-centered finite volume on structured meshes. The values of distribution functions on each cell faceare determined by averaging from their values at the two control points located on the center of two neighboring cells. The fourth-order Runge-Kutta time-stepping scheme is applied to discretize temporal derivative term. The second- and fourth-order numerical dissipation termsareadded to the algorithm to stabilize the solution when solving... 

Alzheimer's disease is the most common type of dementia, which has been recognized as the seventh most common fatal disease in the elderly over 65 years of age. Despite all the research done to recognize and treat this disease, so far there is no cure for this disease, and even most of the chemical treatments that are prescribed for Alzheimer's patients are only effective towards reducing the symptoms of this disease and lose their effectiveness as it progresses. Therefore, in the last two decades, in order to find a way to better understand and even treat AD, scientists have reached a concept called brain frequency stimulation, which can improve people's cognitive performance without the... 

The olfactory sense in rats provides crucial information for identifying food sources, detecting threats, and facilitating social interactions. The olfactory bulb is one of the key brain regions involved in the initial processing of olfactory information and its transmission to higher brain areas for more complex processing. However, more advanced cognitive processes such as evaluation and decision-making rely on interactions between other brain regions. Among these, the striatum, as a part of the basal ganglia, plays a role in reward evaluation, reward-based learning, and shaping behaviors associated with sensory stimuli. This region contributes to learning, decision-making, and action... 

The brain is constantly placed in stochastic environments. This leads to the brain developing a probabilistic model for the environment. However, sometimes unexpected events occur that lead to surprise and a need for updating the probabilistic model. It can be concluded from this argument that reduction in brian’s surprise is a sign of learning. On the other hand, a probability distribution obtained from a probabilistic model contains an amount of uncertainty. Reducing this uncertainty can also be considered a sign of learning. Surprise and uncertainty can be obtained from a probability distribution using information theory concepts. Another important issue in learning is habituation, which...