Cergy-Pontoise University (French: Université de Cergy-Pontoise) is a French university, in the Academy of Versailles. Cergy-Pontoise University is a public university and a leading centre of teaching and research, which welcomes 18,000 students and 1,500 international students interested in studying abroad. The university is located in the west of Paris (30 km from central Paris), in the Val-d'Oise department. The university also managed the Institut d'études politiques de Saint-Germain-en-Laye (in cooperation with the Versailles Saint-Quentin-en-Yvelines University).
This is an advanced introduction to the various types of normal random variables, with most of the needed preliminaries included. We first discuss the probability basics, standard central limits, and the theory of the usual, real normal variables, with examples, illustrations and numerous formulae. Then we go on a similar discussion regarding the complex normal variables, and the Rayleigh variables too. We then move to arbitrary dimensions, with a discussion regarding the Gaussian vectors, and related probability laws, featuring some functional analysis, and geometry and physics too. Finally, we provide an introduction to the quantum versions of the normal variables, and notably to those coming from free probability and random matrices.
We have studied the emergence of slow relaxation oscillations in next generation neural mass models with spike frequency adaptation. Relaxation oscillations connect low firing state (Down state) to high firing state (Up state) via the slow adaptation. In the examined cases, the orbit relaxes towards the Up State via a sequence of collective damped oscillations (peaks of activity), thus revealing population bursting dynamics. The slower is the adaptation time scale the higher is the complexity (number of peaks) displayed by the relaxation oscillations. In particular, a chaos-induced spike-adding mechanism regulates the increase in the number of peaks. In analogy to what found in the Hidmarsh- Rose neuron model, two different types of chaotic behaviors have been identified: Population Spiking and Population Bursting Chaos. The increase of the adaptation strength leads to shorter (longer) Up (Down) state durations somehow mimicking the effect of charbachol in in vitro experiments, where spontaneous slow waves are observed. Indeed, the scenario depicted in [1], where an increase of the concentration of carbachol induces a transition from anesthesia-like to sleep-like dynamics is consistent with our results based on the variation of the adaptation strength.
The celebrated Evans-Searles, respectively Gallavotti-Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state. In this paper, we consider and compare several possible extensions of these fluctuation theorems to quantum systems. In addition to the direct two-time measurement approach whose discussion is based on (LMP 114:32 (2024)), we discuss a variant where measurements are performed indirectly on an auxiliary system called ancilla, and which allows to retrieve non-trivial statistical information using ancilla state tomography. We also show that modular theory provides a way to extend the classical notion of phase space contraction rate to the quantum domain, which leads to a third extension of the fluctuation theorems. We further discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in (Nonlinearity 24, 699 (2011)). Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators. The obtained results shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.
A result is obtained showing how global stability of equilibrium, under the standard tâtonnement dynamic, arises in large economies having a continuous, diffuse distribution of consumers, as well as many goods.
Missing observations and unevenly spaced data are problems common to different disciplines in the context of time series analysis. This paper introduces a new approach to deal with both issues, by considering an irregularly spaced autoregressive moving average process of order (1,1) that is stationary (and therefore homoscedastic) and invertible allowing temporal variations in its coefficients. We test our model in the analysis of greenhouse time series by comparing it with a standard benchmark in the literature. As a result, our methodology leads to a huge advantage in the computational time with respect to the competitor.
