Regularity of Wiener-Hopf plus Hankel operators

In this thesis we consider Wiener-Hopf-Hankel operators with Fourier symbols in the class of almost periodic, semi-almost periodic and piecewise almost periodic functions. In the first place, we consider Wiener-Hopf-Hankel operators acting between L2 Lebesgue spaces with possibly different Fourier matrix symbols in the Wiener-Hopf and in the Hankel operators. In the second place, we consider these operators with equal Fourier symbols and acting between weighted Lebesgue spaces Lp(R;w), where 1 < p < 1 and w belongs to a subclass of Muckenhoupt weights. In addition, singular integral operators with Carleman shift and almost periodic coefficients are also object of study. The main purpose of this thesis is to obtain regularity properties characterizations of those classes of operators. By regularity properties we mean those that depend on the kernel and cokernel of the operator. The main techniques used are the equivalence relations between operators and the factorization theory. An invertibility characterization for the Wiener-Hopf-Hankel operators with symbols belonging to the Wiener subclass of almost periodic functions APW is obtained, assuming that a particular matrix function admits a numerical range bounded away from zero and based on the values of a certain mean motion. For Wiener-Hopf-Hankel operators acting between L2-spaces and with possibly different AP symbols, criteria for the semi-Fredholm property and for one-sided and both-sided invertibility are obtained and the inverses for all possible cases are exhibited. For such results, a new type of AP factorization is introduced. Singular integral operators with Carleman shift and scalar almost periodic coefficients are also studied. Considering an auxiliar and simpler operator, and using appropriate factorizations, the dimensions of the kernels and cokernels of those operators are obtained. For Wiener-Hopf-Hankel operators with (possibly different) SAP and PAP matrix symbols and acting between L2-spaces, criteria for the Fredholm property are presented as well as the sum of the Fredholm indices of the Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators. By studying dependencies between different matrix Fourier symbols of Wiener-Hopf plus Hankel operators acting between L2-spaces, results about the kernel and cokernel of those operators are derived. For Wiener-Hopf-Hankel operators acting between weighted Lebesgue spaces, Lp(R;w), a study is made considering equal scalar Fourier symbols in the Wiener-Hopf and in the Hankel operators and belonging to the classes of APp;w, SAPp;w and PAPp;w. It is obtained an invertibility characterization for Wiener-Hopf plus Hankel operators with APp;w symbols. In the cases for which the Fourier symbols of the operators belong to SAPp;w and PAPp;w, it is obtained semi-Fredholm criteria for Wiener-Hopf-Hankel operators as well as formulas for the Fredholm indices of those operators.

Invertibility characterization of Wiener-Hopf plus Hankel operators on variable exponent Lebesgue spaces via even asymmetric factorization

We obtain invertibility and Fredholm criteria for the Wiener-Hopf plus Hankel operators acting between variable exponent Lebesgue spaces on the real line. Such characterizations are obtained via the so-called even asymmetric factorization which is applied to the Fourier symbols of the operators under study.

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.