Post-Lie Algebras, Factorization Theorems and Isospectral-Flows

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl

Effects of coarse-graining on the scaling behavior of long-range correlated and anti-correlated signals

We investigate how various coarse-graining methods affect the scaling properties of long-range power-law correlated and anti-correlated signals, quantified by the detrended fluctuation analysis. Specifically, for coarse-graining in the magnitude of a signal, we consider (i) the Floor, (ii) the Symmetry and (iii) the Centro-Symmetry coarse-graining methods. We find, that for anti-correlated signals coarse-graining in the magnitude leads to a crossover to random behavior at large scales, and that with increasing the width of the coarse-graining partition interval $\Delta$ this crossover moves to intermediate and small scales. In contrast, the scaling of positively correlated signals is less affected by the coarse-graining, with no observable changes when $\Delta1$ a crossover appears at small scales and moves to intermediate and large scales with increasing $\Delta$. For very rough coarse-graining ($\Delta>3$) based on the Floor and Symmetry methods, the position of the crossover stabilizes, in contrast to the Centro-Symmetry method where the crossover continuously moves across scales and leads to a random behavior at all scales, thus indicating a much stronger effect of the Centro-Symmetry compared to the Floor and the Symmetry methods. For coarse-graining in time, where data points are averaged in non-overlapping time windows, we find that the scaling for both anti-correlated and positively correlated signals is practically preserved. The results of our simulations are useful for the correct interpretation of the correlation and scaling properties of symbolic sequences.Comment: 19 pages, 13 figure

Phonon-assisted radiofrequency absorption by gold nanoparticles resulting in hyperthermia

It is suggested that in gold nanoparticles (GNPs) of about 5 nm sizes used in the radiofrequency (RF) hyperthermia, an absorption of the RF photon by the Fermi electron occurs with involvement of the longitudinal acoustic vibrational mode (LAVM), the dominating one in the distribution of vibrational density of states (VDOS). This physical mechanism helps to explain two observed phenomena: the size dependence of the heating rate (HR) in GNPs and reduced heat production in aggregated GNPs. The argumentation proceeds within the one-electron approximation, taking into account the discretenesses of energies and momenta of both electrons and LAVMs. The heating of GNPs is thought to consist of two consecutive processes: first, the Fermi electron absorbs simultaneously the RF photon and the LAVM available in the GNP; hereafter the excited electron gets relaxed within the GNP's boundary, exciting a LAVM with the energy higher than that of the previously absorbed LAVM. GNPs containing the Ta and/or Fe impurities are proposed for the RF hyperthermia as promising heaters with enhanced HRs, and GNPs with rare-earth impurity atoms are also brought into consideration. It is shown why the maximum HR values should be expected in GNPs with about 5-7 nm size.Comment: proceedings at the NATO Advanced Research workshop FANEM-2015 (Minsk, May 25-27, 2015). To be published in the final form in: "Fundamental and Applied NanoElectroMagnetics" (Springer Science + Business Media B.V.

Electronic structure of Co_xTiSe_2 and Cr_xTiSe_2

The results of investigations of intercalated compounds Cr_xTiSe_2 and Co_xTiSe_2 by X-ray photoelectron spectroscopy (XPS) and X-ray emission spectroscopy (XES) are presented. The data obtained are compared with theoretical results of spin-polarized band structure calculations. A good agreement between theoretical and experimental data for the electronic structure of the investigated materials has been observed. The interplay between the M3d--Ti3d hybridization (M=Cr, Co) and the magnetic moment at the M site is discussed. A 0.9 eV large splitting of the core Cr2p{3/2} level was observed, which reveals a strong exchange magnetic interaction of 3d-2p electrons of Cr. In the case of a strong localization of the Cr3d electrons (for x<0.25), the broadening of the CrL spectra into the region of the states above the nominal Fermi level was observed and attributed to X-ray re-emission. The measured kinetic properties are in good accordance with spectral investigations and band calculation results.Comment: 14 pages, 11 figures, submitted to Phys.Rev.

Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings

We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in C^m constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of N: every N embeds as a submanifold in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a quotient of R by a finite group with fibre a torus. These properties are used to produce new examples of Hamiltonian-minimal Lagrangian submanifolds with quite complicated topology.Comment: 14 pages, published version (minor changes

DENTITION IN PATIENTS BEFORE AND AFTER ORTHODONTIC TREATMENT OF MALOCCLUSION CLASS III

Many scientists were engaged in the study of dentition anthropometry in patients before and after orthodontic treatment. The most common orthodontic device for the treatment of malocclusion class III is a regulator of Frenkel type III. We have proposed a new orthodontic device for the treatment of patients with malocclusion class III and premature removal of temporary molars. Aim - to compare changes of dentitions in children aged 6-12 before and after orthodontic treatment of malocclusion class III using the device of own design and regulator of Frenkel type III. Materials and methods. 110 patients at the age of 6-12 years with malocclusion class III were treated with the use of removable orthodontic curative devices. Anthropometry of dentitions was conducted before and after the treatment. Results. After applying the new diagnostic and treatment algorithm, the duration of orthodontic treatment of patients with malocclusion class III reduced by 1.3 times. The children's quality of life have improved after orthodontic treatment. Conclusion. In case of children aged 6-12 years with malocclusion class III associated with premature removal of temporary molars, it is necessary to use the device of our design. This will give the opportunity not only to fix the dento-facial abnormality, but also to restore the functions of chewing, swallowing and speech