Extensions of differential representations of SL(2) and tori

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. The differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with SL(2) and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of SL(2). In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde

Physical aspects of dust–plasma interactions

Low-pressure gas discharge plasmas are known to be strongly affected by the presence of small dust particles. This issue plays a role in the investigations of dust particle-forming plasmas, where the dust-induced instabilities may affect the properties of synthesized dust particles. Also, gas discharges with large amounts of microparticles are used in microgravity experiments, where strongly coupled subsystems of charged microparticles represent particle-resolved models of liquids and solids. In this field, deep understanding of dust–plasma interactions is required to construct the discharge configurations which would be able to model the desired generic condensed matter physics as well as, in the interpretation of experiments, to distinguish the plasma phenomena from the generic condensed matter physics phenomena. In this review, we address only physical aspects of dust–plasma interactions, that is, we always imply constant chemical composition of the plasma as well as constant size of the dust particles. We also restrict the review to two discharge types: dc discharge and capacitively coupled rf discharge. We describe the experimental methods used in the investigations of dust–plasma interactions and show the approaches to numerical modelling of the gas discharge plasmas with large amounts of dust. Starting from the basic physical principles governing the dust–plasma interactions, we discuss the state-of-the-art understanding of such complicated, discharge-type-specific phenomena as dust-induced stratification and transverse instability in a dc discharge or void formation and heartbeat instability in an rf discharge

Three-dimensional structure of a string-fluid complex plasma

Three-dimensional structure of complex (dusty) plasmas was investigated under long-term microgravity conditions in the International-Space-Station-based Plasmakristall-4 facility. The microparticle suspensions were confined in a polarity-switched dc discharge. The experimental results were compared to the results of the molecular dynamics simulations with the interparticle interaction potential represented as a superposition of isotropic Yukawa and anisotropic quadrupole terms. Both simulated and experimental data exhibited qualitatively similar structural features indicating the bulk liquid-like order with the inclusion of solid-like strings aligned with the axial electric field. Individual strings were identified and their size spectrum was calculated. The decay rate of the size spectrum was found to decrease with the enhancement of string-like structural features

Slowing of acoustic waves in electrorheological and string-fluid complex plasmas

The PK-4 laboratory consists of a direct current plasma tube into which microparticles are injected, forming a complex plasma. The microparticles acquire many electrons from the ambient plasma and are thus highly charged and interact with each other. If ion streams are present, wakes form downstream of the microparticles, which lead to an attractive term in the potential between the microparticles, triggering the appearance of microparticle strings and modifying the complex plasma into an electrorheological form. Here we report on a set of experiments on compressional waves in such a string fluid in the PK-4 laboratory during a parabolic flight and on board the International Space Station. We find a slowing of acoustic waves and hypothesize that the additional attractive interaction term leads to slower wave speeds than in complex plasmas with purely repulsive potentials. We test this hypothesis with simulations, and compare with theory

Effect of Negative Ion Generation on Complex Plasma Structure Properties

We propose a low-density discharge plasma model that takes into account the impact of oxygen admixture in typical conditions of complex (dusty) plasmas. Numerical simulations based on this model show that the concentration of negative ions turns out to be very high, and they play an important role in the overall kinetics in this particular range of plasma conditions. The ambipolar diffusion electric field drags these negative ions into the center of the plasma. The density of negative ions is high enough to push the negatively charged dust component out of the center, both by weakening the radial electric field and by increasing the thermophoretic force. This phenomenon was observed in the published experiment and qualitatively supports the proposed model. Additionally, the proposed model allows an alternative explanation of the experiment

Excitation of progressing dust ionization waves on PK-4 facility

We report observation of the dust ionization waves (DIWs) excited by an external oscillating electric field on the Plasma Kristall-4 facilityunder microgravity conditions. It is shown that at the smallest excitation amplitude, the waves are linear, and the dispersion relation can bededuced from the experimental data. The microparticle oscillations are represented as a superposition of two longitudinal waves propagatingin the opposite directions. In the investigated range of excitation frequency, the wavenumber is not directly proportional to the frequency,and the phase velocity is almost proportional to the frequency. We propose an interpretation of DIW assuming that the microparticle effecton the recombination rate rather than the microparticle subsystem compressibility is responsible for the wave propagation. The calculatedphase velocity of DIW is compatible with the experimental one