Macdonald processes

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $q,t \in [0,1)$ . We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case $t=0$ , we find a Fredholm determinant formula for a $q$ -Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new “integrable” 2d and 1d interacting particle systems. (4) In a large time limit transition, and as $q$ goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O’Connell’s Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent $1/3$ and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas.     

Femtosecond supercontinuum characteristics control

Supercontinuum generation in the spectral range 530–1100 nm in series of photonic-crystal fibers pumped by femtosecond Ti:S laser Mira Oprima 900-F is achieved. The evolution of spectral characteristics of femtosecond supercontinuum is proofed to be dependent on pump pulse wavelength and power radiation. Polarization characteristics of supercontinuum spectral components are analyzed. We demonstrated experimentally the possibility of control of femtosecond supercontinuum generation.     

Acyclic 5‐choosability of planar graphs without adjacent short cycles

The conjecture on acyclic 5‐choosability of planar graphs [Borodin et al., 2002] as yet has been verified only for several restricted classes of graphs. None of these classes allows 4‐cycles. We prove that a planar graph is acyclically 5‐choosable if it does not contain an i‐cycle adjacent to a j‐cycle where 3?j?5 if i = 3 and 4?j?6 if i = 4. This result absorbs most of the previous work in this direction. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:169‐176, 2011     

General β‐Jacobi Corners Process and the Gaussian Free Field

We prove that the two‐dimensional Gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general β‐Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to the Heckman‐Opdam hypergeometric functions (of type A). We also discuss the β → ∞ limit. © 2015 Wiley Periodicals, Inc.     

A <Superscript>31</Superscript>P NMR study of ligand substitution in heterometallic Ru/Zn complexes

Based on the ³¹P NMR data, equilibriums in the [RuNO(NO₂)₄OHZn(TPPO)₃]-Py system are quantitatively described, and equilibrium constants of dissociation of the initial complex and organic ligand substitution are determined. The stability of heterometallic complexes is found to increase in passing from TPPO to pyridine. The mixed ligand [RuNO(NO₂)₄OHZn(TPPO)₂Py] complex is structurally characterized, and it is shown that the TPPO molecule in the trans-position to the μ²-bridging hydroxo group is substituted first.     

Cumulative protons in the 9Be(12C,p)X reaction at 0.2–3.2 GeV/nucleon

The spectra of protons were measured in fragmentation of ¹²C ions in the cumulative region at energies of 0.2–3.2 GeV/nucleon. The energy dependence of the slope parameters in the obtained spectra was traced. The probability of the existence of two- and three-nucleon clusters in the carbon nucleus was estimated within the model of quark-gluon strings.     

Multiphoton ionization of atoms by a two-color laser pulse

The probability of multiphoton ionization of atoms in a laser radiation field containing an additional second harmonic is calculated by the imaginary time method [9, 10]. The conditions are found when the second-harmonic contribution to the ionization of atoms dominates over that of the first harmonic. It is shown that the average momentum of photoelectrons ejected from atoms depends on the phase shift between the first and second harmonics and their mutual polarization. The obtained asymptotic expressions can be used for the qualitative explanation of experiments on generation of terahertz radiation from the optical breakdown region in gas in the focus of a femtosecond laser.     

Viscoelasticity and plasticity mechanisms of human dentin

Theoretical models of viscoelastic behavior and plastic deformation mechanisms of human dentin are considered. Using the linear viscoelasticity theory in which creep and relaxation kernels have the form of fraction-exponential functions, numerical values of instantaneous and long-time Young’s moduli and other characteristics of dentin viscoelasticity under uniaxial compression are found. As dentin plastic deformation mechanisms, mutual collagen fiber sliding in the region of contact of their side surfaces, separation of these fibers from each other, and irreversible tension of some collagen fibers, are proposed. It is shown that the second mechanism activation requires a smaller stress than that for activating others. The models of plastic zones at the mode I crack tip, which correspond to these mechanisms, are studied. It is shown that the plastic zone size can increase from a few hundreds of nanometers to hundreds of micrometers with increasing applied stress.