Search

Let $ \mu_1 $ and $ \mu_2 $ be two, in general complex-valued, Borel measures on the real line such that $ \mathrm{supp} \,\mu_1 =[\alpha_1,\beta_1] < \mathrm{supp}\,\mu_2 =[\alpha_2,\beta_2] $ and $ d\mu_i(x) = -\rho_i(x)dx/2\pi\mathrm{i} $, where $ \rho_i(x) $ is the restriction to $ [\alpha_i,\beta_i] $ of a function non-vanishing and holomorphic in some neighborhood of $ [\alpha_i,\beta_i] $. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices $ (n_1,n_2) $ tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to $ \min\{n_1,n_2\} $.

Motivated by the simplest case of tt*-Toda equations, we study the large and small $x$ asymptotics for $x>0$ of real solutions of the sinh-Godron Painlev\'e III($D_6$) equation. These solutions are parametrized through the monodromy data of the corresponding Riemann-Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions.

We describe the pole-free regions of the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.

Our studies of the cyclotron resonance (CR) photoresistance in GaAs-based two-dimensional electron systems (2DES) reveal an anomalously low sensitivity to the helicity of the incoming circularly polarized terahertz radiation. We find that this anomaly is strongly intensity dependent, and the ratio of the low-temperature photoresistance signals for the CR-active (CRA) and CR-inactive (CRI) polarities of magnetic field increases with lowering power, but, nevertheless, remains substantially lower than expected from conventional theory assuming interaction of the plane electromagnetic wave with the uniform 2DES. Our analysis shows that all data can be well described by the nonlinear CR-enhanced electron gas heating in both CRA and CRI regimes. This description, however, requires a source of anomalous absorption of radiation in the CRI regime. It can stem from evanescent electromagnetic fields originating from the near-field diffraction within or in the vicinity of the quantum well hosting the 2DES., Comment: 15 pages, 8 figures

We show that the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions, as well as associated solutions of P$_\mathrm{XXXIV}$ and S$_\mathrm{II}$, can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P$_\mathrm{II}$ that depend only on the first Airy function $ \mathrm{Ai} $ (but not on $ \mathrm{Bi} $) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.

We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_\Delta s^kQ_n(s)\rho(s)ds =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ \Delta $ is a Chebotar\"ev (minimal capacity) contour connecting three non-collinear points and $ \rho(s) $ is a Jacobi-type weight including a possible power-type singularity at the Chebotar\"ev center of $ \Delta $.

$\bar\partial$-extension of the matrix Riemann-Hilbert method is used to study asymptotics of the polynomials $P_n(z)$ satisfying orthogonality relations \[ \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \] where $\rho(x)$ is a positive $m$ times continuously differentiable function on $[-1,1]$, $m\geq3$.

Let $ f(z)=\int(z-x)^{-1}d\mu(x) $, where $ \mu $ is a Borel measure supported on several subintervals of $ (-1,1) $ with smooth Radon-Nikodym derivative. We study strong asymptotic behavior of the error of approximation $ (f-r_n)(z) $, where $ r_n(z) $ is the $ L_{\mathbb R}^2$-best rational approximant to $ f(z) $ on the unit circle with $ n $ poles inside the unit disk.

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential $V(M)=-\frac{1}{3}M^3+tM$ where $t$ is a complex parameter. As proven in our previous paper, the whole phase space of the model, $t\in\mathbb C$, is partitioned into two phase regions, $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$, such that in $O_{\mathsf{one-cut}}$ the equilibrium measure is supported by one Jordan arc (cut) and in $O_{\mathsf{two-cut}}$ by two cuts. The regions $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$ are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In our previous work the one-cut phase region was investigated in detail. In the present paper we investigate the two-cut region. We prove that in the two-cut region the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter $t$, but not of the parameter $t$ itself. We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann--Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of $S$-curves and quadratic differentials.

Let $ \{\varphi_i(z;\alpha)\}_{i=0}^\infty $, corresponding to $ \alpha\in(-1,1) $, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(\alpha) $, of random polynomials \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z;\alpha), \] where $ \eta_0,\dots,\eta_n $ are i.i.d. standard Gaussian random variables. When $ \alpha=0 $, $ \varphi_i(z;0)=z^i $ and $ P_n(z)$ are called Kac polynomials. In this case it was shown by Wilkins that $ \mathbb E_n(0)$ admits an asymptotic expansion of the form \[ \mathbb E_n(0) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of $ \mathbb E(\alpha) $ for $ \alpha\neq 0 $. As it turns out, the leading term of the asymptotics in this case is $ (1/\pi)\log(n+1) $.

We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.

A new class of one-dimensional solar wind models is developed within the general polytropic, single-fluid hydrodynamic framework. The particular case of quasi-adiabatic radial expansion with a localized heating source is considered. We consider analytical solutions with continuous Mach number over the entire radial domain while allowing for jumps in the flow velocity, density, and temperature, provided that there exists an external source of energy in the vicinity of the critical point which supports such jumps in physical quantities. This is substantially distinct from both the standard Parker solar wind model and the original nozzle solutions, where such discontinuous solutions are not permissible. We obtain novel sample analytic solutions of the governing equations corresponding to both slow and fast wind., Comment: 13 pages, 3 figures, MNRAS, Accepted for publication

The observed anomalous excess of high-energy cosmic ray (CR) positrons is widely discussed as possible indirect evidence for dark matter (DM). However, any source of cosmic positrons is inevitably the source of gamma radiation. The least model dependent test of CR anomalies interpretation via DM particles decays (or annihilation) is connected with gamma-ray background due to gamma overproduction in such processes. In this work, we impose an observational constraint on gamma ray production from DM. Then, we study the possible suppression of gamma yield in the DM decays into identical final fermions. Such DM particles arise in the multi-component dark atom model. The influence of the interaction vertices on the gamma suppression was also considered. No essential gamma suppression effects are found. However, some minor ones are revealed., Comment: The article was published in MDPI "Particle". The authors used to the article structure suggested by the journal (MDPI "Particles")

We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.

We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_\Delta s^kQ_n(s)\rho(s)\dd s =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ \Delta := [-a,a]\cup [-\ic b,\ic b] $, $ a,b>0 $, and $ \rho(s) $ is a Jacobi-type weight.

We suggest to use the Hubble Space Telescople (HST) follow-up observations of the TESS targets for detecting possible plasma tori along the orbits of exoplanets orbiting M dwarfs. The source of the torus could be planetary volcanic activity due to tidal or electromagnetic induction heating. Fast losses to space for planets orbiting these active stars can lead to the lost material forming a torus along the planetary orbit, similar to the Io plasma torus. We show that such torus would be potentially detectable by the HST in the UV., Comment: White paper submitted on the use of HST for TESS follow-up observations, 5 pages, 2 figures

Let $ \{\varphi_i\}_{i=0}^\infty $ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure $ \mu $ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(\mu) $, of random polynomials \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z), \] where $ \eta_0,\dots,\eta_n $ are i.i.d. standard Gaussian random variables. When $ \mu $ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that $ \mathbb E_n(|\mathrm d\xi|) $ admits an asymptotic expansion of the form \[ \mathbb E_n(|\mathrm d\xi|) \sim \frac2\pi\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where $ \mu $ is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case $ \mathbb E_n(\mu) $ admits an analogous expansion with coefficients the $ A_p $ depending on the measure $ \mu $ for $ p\geq 1 $ (the leading order term and $ A_0 $ remain the same).

We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $\mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval $[-2,2]$ on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakly to the arcsine distribution supported on $[-2,2]$. Outside this region the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from $[-2,2]$. These kernels, as well as the scaling limits for the kernels in the bulk $(-2,2)$ and at the endpoints $\{-2,2\}$ are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.

A set of 23 observations of coronal jet events that occurred in coronal bright points has been analyzed. The focus was on the temporal evolution of the mean brightness before and during coronal jet events. In the absolute majority of the cases either single or recurrent coronal jets were preceded by slight precursor disturbances observed in the mean intensity curves. The key conclusion is that we were able to detect quasi-periodical oscillations with characteristic periods from sub-minute up to 3-4 min values in the bright point brightness which precede the jets. Our basic claim is that along with the conventionally accepted scenario of bright point evolution through new magnetic flux emergence and its reconnection with the initial structure of the bright point and the coronal hole, certain MHD oscillatory and wave-like motions can be excited and these can take an important place in the observed dynamics. These quasi-oscillatory phenomena might play the role of links between different epochs of the coronal jet ignition and evolution. They can be an indication of the MHD wave excitation processes due to the system entropy variations, density variations or shear flows. It is very likely a sharp outflow velocity transverse gradients at the edges between the open and closed field line regions. We suppose that magnetic reconnections can be the source of MHD waves due to impulsive generation or rapid temperature variations, and shear flow driven nonmodel MHD wave evolution (self-heating and/or overreflection mechanisms)., Comment: 9 pages, 3 figures, The Astrophysical Journal Letters, Accepted for publication

Let \( \{\varphi_i\}_{i=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure \( \mu \). We study zero distribution of random linear combinations of the form \[ P_n(z)=\sum_{i=0}^{n-1}\eta_i\varphi_i(z), \] where \( \eta_0,\dots,\eta_{n-1} \) are i.i.d. standard Gaussian variables. We use the Christoffel-Darboux formula to simplify the density functions provided by Vanderbei for the expected number real and complex of zeros of \( P_n \). From these expressions, under the assumption that \( \mu \) is in the Nevai class, we deduce the limiting value of these density functions away from the unit circle. Under the mere assumption that \( \mu \) is doubling on subarcs of \( \T \) centered at \( 1 \) and \( -1 \), we show that the expected number of real zeros of \( P_n \) is at most \[ (2/\pi) \log n +O(1), \] and that the asymptotic equality holds when the corresponding recurrence coefficients decay no slower than \( n^{-(3+\epsilon)/2} \), \( \epsilon>0 \). We conclude with providing results that estimate the expected number of complex zeros of \( P_n \) in shrinking neighborhoods of compact subsets of \( \T \).

The present paper discusses results of a statistical study of the characteristics of coronal hole (CH) rotation in order to find connections to the internal rotation of the Sun. The goal is to measure CH rotation rates and study their distribution over latitude and their area sizes. In addition, the CH rotation rates are compared with the solar photospheric and inner layer rotational profiles. We study coronal holes observed within $\pm 60$ latitude and longitude degrees from the solar disc centre during the time span from the 1 January 2013 to 20 April 2015, which includes the extended peak of solar cycle 24.We used data created by the Spatial Possibilistic Clustering Algorithm (SPoCA), which provides the exact location and characterisation of solar coronal holes using SDO=AIA 193 {\AA} channel images. The CH rotation rates are measured with four-hour cadence data to track variable positions of the CH geometric centre. North-south asymmetry was found in the distribution of coronal holes: about 60 percent were observed in the northern hemisphere and 40 percent were observed in the southern hemisphere. The smallest and largest CHs were present only at high latitudes. The average sidereal rotation rate for 540 examined CHs is $13:86 (\pm 0:05)$ degrees/d. Conclusions. The latitudinal characteristics of CH rotation do not match any known photospheric rotation profile. The CH angular velocities exceed the photospheric angular velocities at latitudes higher than 35-40 degrees. According to our results, the CH rotation profile perfectly coincides with tachocline and the lower layers of convection zone at around 0.71 $R_{\odot}$; this indicates that CHs may be linked to the solar global magnetic field, which originates in the tachocline region., Comment: 8 pages, 8 figures, Accepted for publication in A&A

The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Pad\'e approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is "symmetric" with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single-valued continuation with non-identically zero jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author. We consider rational interpolants with free poles of Cauchy transforms of non-vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize \( \bar\partial \)-extension of the Riemann-Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation.

Growth kinetics of AlGaN in NH3 MBE under significant Ga desorption was studied. It was found that the addition of gallium stimulates 2D growth and provides better morphology of films compared to pure AlN. The effect was experimentally observed at up to 98% desorption of the impinging gallium. We found that, under the conditions of significant thermal desorption, larger amounts of gallium were retained at lateral boundaries of 3D surface features than at flat terraces because of the higher binding energy of Ga atoms at specific surface defects. The selective accumulation of gallium resulted in an increase in lateral growth component through the formation of the Ga-enriched AlGaN phase at boundaries of 3D surface features. We studied temperature dependence of AlGaN growth rate and developed a kinetic model analytically describing this dependence. As the model was in good agreement with the experimental data, we used it to estimate the increase in the binding energy of Ga atoms at surface defects compared to terrace surface cites using data on the Ga content in different AlGaN phases. We also applied first-principles calculations to the thermodynamic analysis of stable configurations on the AlN surface and then used these surface configurations to compare the binding energy of Ga atom at terraces and steps. Both first-principles calculations and analytical estimations of the experimental results gave similar values of difference in binding energies; this value is 0.3 eV., Comment: 19 pages, 10 figures

Let $\widehat\sigma$ be a Cauchy transform of a possibly complex-valued Borel measure $\sigma$ and $\{p_n\}$ be a system of orthonormal polynomials with respect to a measure $\mu$, $\mathrm{supp}(\mu)\cap\mathrm{supp}(\sigma)=\varnothing$. An $(m,n)$-th Frobenius-Pad\'e approximant to $\widehat\sigma$ is a rational function $P/Q$, $\mathrm{deg}(P)\leq m$, $\mathrm{deg}(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the linear form $Q\widehat\sigma-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Pad\'e approximants to $\widehat\sigma$ along ray sequences $\frac n{n+m+1}\to c>0$, $n-1\leq m$, when $\mu$ and $\sigma$ are supported on intervals on the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the respective interval are holomorphic functions.