Neural networks and Machine Learning By Marius Yamakou Neural Networks with time delayed connections Neurons communicate with each other through electrical signals. It is well known that these signals are oscillatory and that the properties of the oscillations depend on the characteristics of the individual neurons, how the neurons are […]
Math
62 posts
Stochastic Synchronization of Chaotic Neurons By Marius Yamakou Real biological neurons can show chaotic dynamics when excited by the certain external input current. The behavior of these neurons is characterized by instability and, as a result, limited predictability in time. Mathematically, a system is chaotic if it has a […]
Nonlocal population balance equations and applications By Michele Spinola Motivational example: look ahead behavior of car drivers When analyzing traffic situations, one possible way to observe the current state is from bird’s eye view. The velocity of a car driver at time at location depends on the traffic density at […]
Inverse Design For Hamilton-Jacobi Equations By Carlos Esteve, Enrique Zuazua In many evolution models, the reconstruction of the initial state given an observation of the system at time represents a major challenge in mathematical modelling. Especially if it involves irreversible processes, where sometimes, different initial conditions can lead the system […]
Stochastic Neural Dynamics By Marius Yamakou Neural activity shows fluctuations and unpredictable transitions in its dynamics. This randomness can be an integral aspect of neuronal function; examples range from discrete fluctuations of ion channels to sudden sleep stage transitions involving the entire brain. To understand brain function as well as […]
Controllability properties of fractional PDE By Umberto Biccari Controllability of the fractional heat equation Let be an open and nonempty subset. Consider the following non-local one-dimensional heat equation defined on the domain where is a given initial datum. In (1), for all , denotes the one-dimensional fractional Laplace operator, […]
Flows on Networks By Enrique Zuazua, Nicola de Nitti PDE models on Networks In the last few decades, models based on partial differential equations have been very effective in tackling many applied problems dealing with flows on networks. The areas of application include mainly the study of vehicular traffic, […]
Stochastic optimization for simultaneous control By Umberto Biccari What is a simultaneous control problem? Consider the following parameter-dependent linear control system with The matrix is associated with the Brunovsky canonical form of the linear ODE where denotes the -th derivative of the function . In (1)-(2), , , […]
Convexity and Starshapedness of feasible sets in Stationary Flow Networks This research was funded by DFG in the SFB Transregio 154: Mathematical modelling, simulation and optimization using the example of gas networks. Uncertainty often plays an important role in application driven modeling. This often leads to optimization problems […]
Collective dynamics modelling, Control and Simulation By Dongnam Ko Collective dynamics Herds, packs, bird flocks, and fish schools are common examples of the collective behaviors arising from the interactions of individuals. Each individual has its own decision policy, as in the stock market or game theory, which interacts with […]
Classical models By Cyprien Neverov Compartmental epidemiological models [1] where introduced almost a century ago and are still considered the standard way of modeling a disease in a population. They are also called SIR models because they divide the population into different compartments like Susceptible, Exposed, Infected, Recovered and model […]
Non-local population balance equations By Michele Spinola Nichtlokale Populationsbilanzgleichungen. Der Verlauf des Weges wie zur Schule oder zur Arbeit hängt stark von der entsprechenden Verkehrslage ab. Genauso spielen chemisch synthetisierte Produkte wie Pharmaka oder Kosmetika eine wichtige Rolle im Alltag. Dementsprechend relevant ist es, mathematische Modelle zu entwickeln, die diese […]