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This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations.
The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves.
More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem.
A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.
The paper describes the union of the physical processes in constructing the virtual model, which allows to conduct research jointly occurring electrical, thermal and mechanical processes in onboard electronic means considering their mutual influence.
At ultralow temperatures (T< 100 mK) rather small external disturbance might lead to a noticeable overheating. While electron transport measurements the inevitable external EM noise, picked-up and transmitted through electric wires, results in a mismatch between the electron Te and the phonon Tphtemperatures P~W(Te^5-Tph^5), where P is the power dissipated at the sample with volume W. Hence, for sufficiently small nanoelectronic systems the effect might be clearly pronounced. Multiple methods have been suggested to reduce the undesired electron heating. Typically various RF filters are used to cut the impact of noisy EM environment. Often the supporting amplitude vs. frequency data are obtained only at room temperatures analyzing the impact of 'isolated' elements without taking into consideration the wires. Here we describe the custom made multistage RLC filtering system for ultralow temperature nanoelectronic experiments. The amplitude vs. frequency characteristics were measured down to very low temperatures. Distributed elements theory analysis supports experimental data.
We introduce a notion of semiclassical bi-states. They arise from pairs of eigenstates corresponding to tunnel-splitted eigenlevels and generate 2-level subsystems in a given quantum system. As an example, we consider the planar Penning trap with rectangular electrodes assuming the 3:(-1) resonance regime of charge dynamics. We demonstrate that under small deviation of the rectangular shape of electrodes from the square shape (symmetry breaking), there appear instanton pseudoparticles, semiclassical bi-states and 2-level subsystems in such a quantum trap.