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

Zariski closures of reductive linear differential algebraic groups

AbstractLinear differential algebraic groups (LDAGs) appear as Galois groups of systems of linear differential and difference equations with parameters. These groups measure differential-algebraic dependencies among solutions of the equations. LDAGs are now also used in factoring partial differential operators. In this paper, we study Zariski closures of LDAGs. In particular, we give a Tannakian characterization of algebraic groups that are Zariski closures of a given LDAG. Moreover, we show that the Zariski closures that correspond to representations of minimal dimension of a reductive LDAG are all isomorphic. In addition, we give a Tannakian description of simple LDAGs. This substantially extends the classical results of P. Cassidy and, we hope, will have an impact on developing algorithms that compute differential Galois groups of the above equations and factoring partial differential operators

Reductive inear differential algebraic groups and the Galois groups of parameterized linear differential equations

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL(2) in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it.Comment: 61 page

Electrodeposition of Silicon Fibers from KI–KF–KCl–K₂SiF₆ Melt and Their Electrochemical Performance during Lithiation/Delithiation

The possibility of using Si-based anodes in lithium-ion batteries is actively investigated due to the increased lithium capacity of silicon. The paper reports the preparation of submicron silicon fibers on glassy carbon in the KI–KF–KCl–K2SiF6 melt at 720 °C. For this purpose, the parameters of silicon electrodeposition in the form of fibers were determined using cyclic voltammetry, and experimental samples of ordered silicon fibers with an average diameter from 0.1 to 0.3 μm were obtained under galvanostatic electrolysis conditions. Using the obtained silicon fibers, anode half-cells of a lithium-ion battery were fabricated, and its electrochemical performance under multiple lithiations and delithiations was studied. By means of voltametric studies, it is observed that charging and discharging the anode based on the obtained silicon fibers occurs at potentials from 0.2 to 0.05 V and from 0.2 to 0.5 V, respectively. A change in discharge capacity from 520 to 200 mAh g−1 during the first 50 charge/discharge cycles at a charge current of 0.1 C and a Coulombic efficiency of 98–100% was shown. The possibility of charging silicon-based anode samples at charging currents up to 2 C was also noted; the discharge capacity ranged from 25 to 250 mAh g−1

Developing specialized software for investigating interference in complex optical systems

Abstract The paper describes a physical model and primary calculation algorithms we developed and implemented in a software package for simulating interference in complex optical systems. A model of a highly monochromatic laser beam interacting with various optical elements is proposed. We verified and validated our model by creating an interference pattern on a screen. Model verification involved an experiment with a Mach-Zehnder interferometer. Model validation consisted of simulating our own optical experiment.</jats:p