M. Karam Ouharou

This document explores advanced methods for solving the damped oscillator equation, a fundamental second-order differential equation derived from Newton's Second Law. The study addresses three primary cases of damping: over-damped, critically damped, and under-damped oscillators. For over-damped oscillators, solutions are found by analyzing the roots of the auxiliary equation, leading to a general solution composed of exponential functions. Critically damped oscillators, characterized by a single repeated root, require a different approach involving a linear combination of exponential and polynomial terms. Under-damped oscillators, which exhibit oscillatory behavior with gradually decreasing amplitude, are solved using complex roots, resulting in a solution that combines exponential decay with sinusoidal functions. This comprehensive analysis provides a detailed mathematical framework for understanding the dynamic behavior of damped oscillatory systems under various damping conditi...

Cette feuille propose une analyse détaillée des propriétés des matrices stochastiques et de la formule du rayon spectral. J'ai commencé par établissement des lemmes essentiels, démontrant d'abord que pour toute norme subordonnée, le rayon spectral ρ(A) d'une matrice A est toujours inférieur ou égal à ∥A∥. J'ai poursuit avec une preuve plus complexe démontrant l'existence d'une norme subordonnée pour laquelle cette inégalité se resserre à ρ(A) + ε. Le document présente ensuite le Théorème 1, qui affirme que ρ(A) = lim ∥A n ∥ 1/n pour toute norme subordonnée, consolidant cette équivalence par une série de démonstrations mathématiques rigoureuses. La preuve utilise des propriétés des valeurs propres et des vecteurs propres de la matrice, ainsi que des techniques de trigonalisation et des normes de Householder. Le document conclut que cette propriété est indépendante de la norme matricielle choisie, généralisant ainsi le résultat à toutes les normes subordonnées. Li...

We  investigate  diffusive  processes  characterized  by  drift  within  the  n dimensional  Euclidean  space  R  n   ,  particularly  focusing  on  stochastic  differential  equations.  We  summarize  the  interdependencies  between  the  drift  vector,  the  diffusion  matrix,  and  invariant  distributions,  utilizing  the  appropriate  variables .  This  approach  reveals  a  decomposition  into  'potential'  and  'geometric'  components,  with  the  geometric  part  encapsulating  non reversible  dynamics.  Physical  Langevin type  processes,  defined  on  the  phase  space  of  position  and  velocity,  where  stochastic  perturbations...

Avant-propos (Résumé) Nous présentons ici la cohomologie de Čech pour les faisceaux. L'objectif est d'exposer la construction fondamentale de Čech dans divers contextes : la définition d'objets cohomologiques globaux à partir du recollement d'objets définis sur les intersections des ouverts d'un recouvrement d'un espace. Nous débutons en motivant la construction de Čech avec l'exemple simple des complexes. Ensuite, nous utilisons la cohomologie de Čech des faisceaux pour fournir une démonstration particulièrement élégante du théorème de De Rham, qui établit l'isomorphisme entre la cohomologie de De Rham et celle de Čech à coefficients dans R pour une variété différentiable. Enfin, nous appliquons ces méthodes à l'étude des fibrés en droites (complexes) et définissons notamment leur première classe de Chern. Sur cette dernière, nous démontrons une sorte de formule de Gauß-Bonnet. Plusieurs théorèmes d'algèbre homologique concernant la cohomologie ...

Abstract: In the realm of liquid identification technology, a groundbreaking innovation has emerged in the form of a self-powered triboelectric taste-sensing system with dual-sensory fusion capabilities. This research presents a comprehensive study of this novel system, detailing its design, functionality, and potential applications. By harnessing the principles of triboelectricity and dual-sensory fusion, this system offers an effective, low-cost solution for liquid identification, opening new avenues in fields such as food safety, beverage quality control, and environmental monitoring. Infusing human taste perception into smart sensing devices to mimic the processing ability of gustatory organs to perceive liquid substances remains challenging. Here we developed a self-powered droplet-tasting sensor system based on the dynamic morphological changes of droplets and liquid–solid contact electrification. The sensor system has achieved accuracies of liquid recognition higher than 90% in ...

Forty years ago it was proposed that gas phase organic chemistry in the interstellar medium can be initiated by the methyl cation CH 3 +  ( 1–3 ), but hitherto it has not been observed outside the Solar System ( 4, 5 ). Alternative routes involving processes on grain surfaces have been invoked ( 6, 7 ). Here we report JWST observations of CH+3CH3+ in a protoplanetary disk in the Orion star forming region. We find that gas-phase organic chemistry is activated by UV irradiation. Introduction : In the realm of cosmic evolution, protoplanetary disks serve as the cradles of new star systems and planetary bodies. These disks, composed of gas and dust, harbor an intricate web of chemical reactions that sculpt the materials essential for planet formation. Among the diverse array of chemical species within these disks, the methyl cation (CH3+) stands out as a key molecular fragment. Recent research has shed light on the intriguing formation of the methyl cation through photochem...

Abstract Element isotopes are characterized by distinct atomic masses and nuclear spins, which can significantly influence material properties. Notably, however, isotopes in natural materials are homogenously distributed in space. Here, we propose a method to configure material properties by repositioning isotopes in engineered van der Waals (vdW) isotopic heterostructures. We showcase the properties of hexagonal boron nitride (hBN) isotopic heterostructures in engineering confined photon-lattice waves—hyperbolic phonon polaritons. By varying the composition, stacking order, and thicknesses of h 10 BN and h 11 BN building blocks, hyperbolic phonon polaritons can be engineered into a variety of energy-momentum dispersions. These confined and tailored polaritons are promising for various nanophotonic and thermal functionalities. Due to the universality and importance of isotopes, our vdW isotope heterostructuring method can be applied to engineer the properties of a broad range of mater...