Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.     

Luminescent Analysis of Blood Serum for Diagnostics of Pathological and Pre-Pathological States of Cancer Patients

Journal of Fluorescence - This study is devoted to the development of a methodological approach to mathematical analysis and data interpretation of blood serum phosphorescence intensity in cancer...     

Thermodiffusion motion of electrically charged nanoparticles

The present work deals with experimental studies to examine the theoretical model of thermodiffusion of electrically charged nanoparticles. Three different ionic magnetic colloid samples have been synthesized and profoundly analyzed. The theoretical model is a classical one, based on the calculation of the temperature and the electric potential distribution around nanoparticles. The discrepancy between experimental data and theory turns out not to exceed 20%. We focus on applying different approximations between calculated electrical double layer in the theoretical model and experimental determination of the surface charge density of colloidal particles. We assume this is the main reason for obtained discrepancy.     

Phase separation in binary systems with competing internal and external multiplicative noises

We have considered phase separation processes in binary stochastic systems with thermal diffusion and ballistic mixing representing irradiation influence. Introducing fluctuations of thermal flux and an external source of atom relocations due to ballistic diffusion into dynamics of a globally conserved field, we have shown that there are two competing mechanisms of phase transitions type of “order-disorder”: thermally assisted diffusion and irradiation induced atomic exchange. We have studied dynamics of the structure function at early stages of decomposition. In the framework of the mean field theory we have derived the effective Fokker-Planck equation to describe phase separation processes. It was shown that the ordering processes can be controlled by both regular and stochastic parts of external source influence. A reentrant behavior of a mean field order parameter versus the external noise intensity and fluctuations correlation radius is found.     

Modeling phase decomposition and patterning in binary alloy systems subjected to neutron irradiation

We study processes of phase decomposition and patterning in a model of a binary alloy system subjected to sustained irradiation. We exploit the reaction rate theory and generalize the Darken approach of vacancy diffusion to describe generation, recombination, annihilation and spatial interaction of point defects. It is shown that an increase in the defect production rate phase, decomposition processes are replaced by disordering and patterning with vacancy clusters' formation. At elevated damage rates, both phase separation and patterning are accompanied by pattern selection processes. In the framework of numerical simulations, dynamics of phase decomposition and vacancy clusters formation is studied in detail. A change in the morphology of vacancy clusters during irradiation and their statistical properties are discussed.     

Magnetic-Field-Controlled Quantum Critical Points in the Triangular Antiferromagnetic Cs2CuCl4-xBrx Mixed System

The triangular antiferromagnetic Cs₂CuCl_4-xBr_x mixed system is studied by neutron single-crystal diffraction in magnetic field. It shows a rich magnetic phase diagram consisting of four regimes depending on the Br concentration and is characterized by different exchange coupling mechanisms. For the investigated compositions from regime I (0 < x ≤ 1.5), a critical magnetic field B_c is found for a Br concentration x = 0.8 at B_c = 8.10(1) T and for x = 1.1 at B_c = 7.73(1) T and from regime IV (3.2 < x < 4) for x = 3.3 at B_c = 0.99(3) T. For magnetic fields larger than the respective B_c, magnetic superlattice reflections of these compounds are not found. The incommensurate magnetic wave vector q = (0, 0.470, 0) appears below the ordering temperature T_N = 0.51(1) K for Cs₂CuCl_3.2Br_0.8, and q = (0, 0.418, 0) below T_N = 1.00(6) K for Cs₂CuCl_0.3Br_3.7. Neutron diffraction experiments at around 60 mK for x = 3.7 in a magnetic field show the critical magnetic field at B_c = 7.94(16) T and the formation of the second magnetic phase at around 8.5 T depending on the temperature. Inelastic neutron scattering experiments for the compound from regime III (2 < x ≤ 3.2) with x = 2.2 show dynamical correlations at a temperature around 50 mK giving evidence for a spin liquid phase.     

Pattern selection processes and noise induced pattern-forming transitions in periodic systems with transient dynamics

We apply the phase field crystal method for nonequilibrium patterning to stochastic systems with an external source in which transient dynamics is essential. Considering a prototype model for a one-component periodic system subjected to external influence kind of irradiation we study properties of pattern selection processes and external noise induced pattern-forming transitions. These processes are examined by means of the structure function dynamics analysis. Nonequilibrium pattern-forming transitions are analyzed numerically.     

Modeling self-organization of nano-size vacancy clusters in stochastic systems subjected to irradiation

We study the self-organization of vacancy clusters in irradiated materials under reactor and accelerator conditions. Using a continuum stochastic model we take into account dynamics of point defects and their sinks with elastic interactions of vacancies. Dynamics of vacancy clusters formation is studied analytically and numerically. We have shown a difference in patterning dynamics at irradiation under reactor and accelerator conditions. The external noise influence related to fluctuation in a defect production rate is studied in detail. Applying our approach to pure nickel irradiated under different conditions we have shown that vacancy clusters having a linear size ～eq 6 nm can arrange in a statistical periodic structure with nano-meter range. We have found that the linear size of vacancy clusters at accelerator conditions decreases down to 20%, whereas a period of vacancy clusters reduces to 6.5%.