Let G be a group scheme of finite type over a field, and consider the cohomology ring H∗(G) with coefficients in the structure sheaf. We show that H∗(G) is a free module of finite rank over its component of degree 0, and is the exterior algebra of its component of degree 1. When G is connected, we determine the Hopf algebra structure of H∗(G).

We study a one-dimensional ring chain of length L with N particles interacting via Morse potentials and influenced by dissipative forces (passive and active friction). We show that by negative friction the system can be driven far from the thermodynamic equilibrium states. For over-critical pumping with free energy several types of coherent motions including uniform rotations, optical oscillati...

The formulas for coherent instabilities are summarized with the emphasis on realistic fill patterns.

Among the attractive coherent light sources resulting from the interaction between femtosecond lasers and relativistic electron beams, simultaneous Coherent Synchrotron Radiation (CSR) in the THz region, slicing and UV-VUV Coherent Harmonic Generation (CHG) can be achieved on synchrotron radiation facilities. Recently, a Ti:Sa laser at high repetition rate (1 kHz) has been seeded in the optical...

We report the observation of multiparticle coherent emission in the very far infrared from bunched electrons in an electron storage ring. The emission occurs in quasi-periodic bursts, and only when the electron beam current exceeds a threshold value, suggesting an instability-driven modulation of the electron bunch density. For the operating conditions reported, the spectral content of the cohe...

We present the first experimental demonstration of coherent perfect absorption (CPA) in an integrated device. By leveraging the effects of CPA, phase-controlled modulation is achieved in a silicon photonic ring-resonator with an extinction of 8.12dB. OCIS codes: 130.0130, 030.1670, 060.1810, 130.3120

Harmonic Generation schemes on Free Electron Laser devices are very promising. The injection of a traditional laser source inside the first undulator of an optical klystron leads to an efficient energy modulation of the electron bunch, and therefore, its spatial modulation, resulting in a more coherent light emission along the second undulator. Experiments have been performed on the UVSOR-II St...

Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we invest...

The equations for spin evolution of a particle in a storage ring are obtained considering contributions from the tensor electric and magnetic polarizabilities of the particle along with the contributions from spin rotation and birefringence effect in polarized matter of an internal target. Study of the spin rotation and birefringence effects for a particle in a high energy storage ring provides...

The directional dynamics of the resonant Rayleigh scattering from a semiconductor microcavity is investigated. When optically exciting the lower polariton branch, the strong dispersion results in a directional emission on a ring. The coherent emission ring shows a reduction of its angular width for increasing time after excitation, giving direct evidence for the time-energy uncertainty in the d...