# Gallery — Math

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V-Formation as Optimal Control
We present a new formulation of the V-formation problem for migrating birds in terms of model predictive control (MPC).
Junxing
SOLAR SALES ON YOUR TRIP TO MARS
We study Logarithmically Spiral Trajectories and, in particular, we look for a solution to minimize the transit time of a Spacecraft propelled by a Solar Sail, while simultaneously minimizing the area of the Solar Sail, which would allow us to carry more payload on board. We start by analyzing the forces that act on the Spacecraft taking into account that its propellant is a Solar Sail; we use the studied forces to deduce the motion equations. We then solve this motion equation with a Runge-Kutta 4 method and transform the problem of minimizing time and area to a Non-linear Optimization problem. When solving the NLP we also try to minimize the relative final speed of th spacecraft with the destination planet in order to guarantee the possibility of a safe landing on its surface. The model improves when an angle parameter α (describing the angle formed by the Solar Sail with the colliding photons) is defined as a piecewise constant function and optimized whose values are optimized in every interval to minimize transit time and Area. Using the developed model to optimize the trajectory to be followed for sending from Earth to Mars a 2000kg-spacecraft propelled by a Solar Sail, subject to the condition that at trajectory start Mars and Earth were at their closest approach, and the Arrival Relative Velocity is less than 9km/s, give us a minimal transit time of 500days and a minimal area for the Solar Sail of 183158m2, meaning that the maximal payload would be 718kg. Compared with different number of partitions of α, the optimum stays stable. This gives a solid optimal trajectory and a great result for the numerical method used. Actually, waiting until the best moment to throw the Spacecraft, id est, Mars is at 1.14 radians respectively to Earth initial position, the minimal sail area 145950 m2 and, therefore, ables to transport until 978 kg of payload with the same transit time. In addition and to conclude we tried the model to optimize the inverse trajectory.
Marco Praderio Bova, Eneko Martin Martinez, & Maria dels Àngels Guinovart Llort
Solucion parcial 3
Solución al tercer examen parcial del curso de Algebra
Memo Garro
ProofOfEuler'sFormula
Proof of Euler's Formula
Adrian D'Costa
Teorema de eliminación de corte
Comparto este trabajo para quien le pueda servir la plantilla que utilizamos, únicamente con fines educativos.
Diego Londoño
FSU-MATH2400-Project6
In this calculus project, students use infinite series to investigate Euler's Equation: $e^{i\pi} + 1 = 0$.
Sarah Wright
Trabajo practico-Fenomenos de transporte 3
Trabajo realizado en la catedra fenomenos 3
Oscar Daniel Rivas Villar
Integration of Some Elementary Integrals
Integration of some elementary integrals
Adrian D'Costa
MOOC Dropout Prediction with Model Stacking
We are asked to predict the probability of the event that a student will drop out a course. We firstly extracted many features from the huge dataset. Then we used ensemble learning machine and model stacking technique to get the final result, which ranked the 1st in 68 teams.
Qi Zhao