AROUND THE ERDÖS–GALLAI CRITERION

By an (integer) partition we mean a non-increasing sequence $\lambda=(\lambda_1, \lambda_2, \dots)$ of non-negative integers that contains a finite number of non-zero components. A partition $\lambda$ is said to be graphic if there exists a graph $G$ such that $\lambda = \mathrm{dpt}\,G$, where we denote by  $\mathrm{dpt}\,G$ the degree partition of $G$ composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper

ON SEQUENCES OF ELEMENTARY TRANSFORMATIONS IN THE INTEGER PARTITIONS LATTICE

An integer partition, or simply, a  partition is a nonincreasing sequence $\lambda = (\lambda_1, \lambda_2, \dots)$ of nonnegative integers that contains only a finite number of nonzero components. The  length $\ell(\lambda)$ of a partition $\lambda$ is the number of its nonzero components. For convenience, a partition $\lambda$ will often be written in the form $\lambda=(\lambda_1, \dots, \lambda_t)$, where $t\geq\ell(\lambda)$; i.e., we will omit the zeros, starting from some zero component, not forgetting that the sequence is infinite. Let there be natural numbers $i,j\in\{1,\dots,\ell(\lambda)+1\}$ such that (1) $\lambda_i-1\geq \lambda_{i+1}$; (2) $\lambda_{j-1}\geq \lambda_j+1$; (3) $\lambda_i=\lambda_j+\delta$, where $\delta\geq2$. We will say that the partition $\eta={(\lambda_1, \dots, \lambda_i-1, \dots, \lambda_j+1, \dots, \lambda_n)}$ is obtained from a partition $\lambda=(\lambda_1, \dots, \lambda_i, \dots, \lambda_j, \dots, \lambda_n)$ by an elementary transformation of the first type. Let $\lambda_i-1\geq \lambda_{i+1}$, where $i\leq \ell(\lambda)$. A transformation that replaces $\lambda$ by $\eta=(\lambda_1, \dots, \lambda_{i-1}, \lambda_i-1, \lambda_{i+1}, \dots)$ will be called an elementary transformation of the second type. The authors showed earlier that a partition $\mu$ dominates a partition  $\lambda$ if and only if $\lambda$ can be obtained from $\mu$ by a finite number (possibly a zero one) of elementary transformations of the pointed types. Let $\lambda$ and $\mu$ be two arbitrary partitions such that $\mu$ dominates $\lambda$. This work aims to study the shortest sequences of elementary transformations from $\mu$ to $\lambda$. As a result, we have built an algorithm that finds all the shortest sequences of this type

Senchonok, T.A., On maximal graphical partitions that are the nearest to a given graphical partition

A graphical partition is called maximal if it is maximal under domination among graphical partitions of a given weight. Let λ and μ be partitions such that μ ≤ λ. The height of λ over μ is the number of transformations in some shortest sequence of elementary transformations which transforms λ to μ, denoted by height(λ; μ). For a given graphical partition μ, a maximal graphical partition λ such that μ ≤ λ and sum(μ) = sum(λ) is called the h-nearest to μ if it has the minimal height over μ among all maximal graphical partitions λ' such that μ ≤ λ' and sum(μ) = sum(λ'). The aim is to prove the following result: Let μ be a graphical partition and λ be an h-nearest maximal graphical partition to μ. Then (1) either r(λ) = r(μ)-1, l(tl(μ) < r(μ) or r(λ) = r(μ), (2) height(λ; μ) = height(tl(μ); hd(μ)-[sum(tl(μ)-sum(hd(μ)] = tl(μ)i-hd(μ)i; where r = r(μ) is the rank, hd(μ) is the head and tl(μ) is the tail of the partition μ, l(tl(μ) is the length of tl(μ). We provide an algorithm that generates some h-nearest to μ maximal graphical partition λ such that r(λ) = r(μ). For the case l(tl(μ) < r(μ), we also provide an algorithm that generates some h-nearest to μ maximal graphical partition λ such that r(λ) = r(μ)-1. In addition we present a new proof of the Kohnert's criterion for a partition to be graphical not using other criteria. © 2020 Baransky V.A., Senchonok T.A

Magnetohydrodynamic equilibria of a cylindrical plasma with poloidal mass flow and arbitrary cross section shape

The equilibrium of a cylindrical plasma with purely poloidal mass flow and cross section of arbitrary shape is investigated within the framework of the ideal MHD theory. For the system under consideration it is shown that only incompressible flows are possible and, conscequently, the general two dimensional flow equilibrium equations reduce to a single second-order quasilinear partial differential equation for the poloidal magnetic flux function $\psi$, in which four profile functionals of $\psi$ appear. Apart from a singularity occuring when the modulus of Mach number associated with the Alfv\'en velocity for the poloidal magnetic field is unity, this equation is always elliptic and permits the construction of several classes of analytic solutions. Specific exact equlibria for a plasma confined within a perfectly conducting circular cylindrical boundary and having i) a flat current density and ii) a peaked current density are obtained and studied.Comment: Accepted to Plasma Physics & Controlled Fusion, 14 pages, revte

About possibility to locate an EQ epicenter using parameters of ELF/ULF preseismic emission

A relation between parameters of preseismic ULF/ELF emissions and EQ is studied. The magnetic data measured at Karymshino station (Kamchatka, Russia) along with data on local seismic activity during eight years of observations (2001–2008) are taken for the analysis. Source azimuth is detected in different techniques, based on the analysis of the total field and its polarized pulsed component. The latter technique shows a better accuracy in the source azimuth detection. The errors of the method are associated with existence of non-seismic sources and with use of one-point observation. The second error can be eliminated by development of multi-point observations

Analysis of the rotation period of asteroids (1865) Cerberus, (2100) Ra-Shalom, and (3103) Eger - search for the YORP effect

The spin state of small asteroids can change on a long timescale by the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect, the net torque that arises from anisotropically scattered sunlight and proper thermal radiation from an irregularly-shaped asteroid. The secular change in the rotation period caused by the YORP effect can be detected by analysis of asteroid photometric lightcurves. We analyzed photometric lightcurves of near-Earth asteroids (1865) Cerberus, (2100) Ra-Shalom, and (3103) Eger with the aim to detect possible deviations from the constant rotation caused by the YORP effect. We carried out new photometric observations of the three asteroids, combined the new lightcurves with archived data, and used the lightcurve inversion method to model the asteroid shape, pole direction, and rotation rate. The YORP effect was modeled as a linear change in the rotation rate in time d\omega /dt. Values of d\omega/ dt derived from observations were compared with the values predicted by theory. We derived physical models for all three asteroids. We had to model Eger as a nonconvex body because the convex model failed to fit the lightcurves observed at high phase angles. We probably detected the acceleration of the rotation rate of Eger d\omega / dt = (1.4 +/- 0.6) x 10^{-8} rad/d (3\sigma error), which corresponds to a decrease in the rotation period by 4.2 ms/yr. The photometry of Cerberus and Ra-Shalom was consistent with a constant-period model, and no secular change in the spin rate was detected. We could only constrain maximum values of |d\omega / dt| < 8 x 10^{-9} rad/d for Cerberus, and |d\omega / dt| < 3 x 10^{-8} rad/d for Ra-Shalom

The astrometric Gaia-FUN-SSO observation campaign of 99 942 Apophis

Astrometric observations performed by the Gaia Follow-Up Network for Solar System Objects (Gaia-FUN-SSO) play a key role in ensuring that moving objects first detected by ESA's Gaia mission remain recoverable after their discovery. An observation campaign on the potentially hazardous asteroid (99 942) Apophis was conducted during the asteroid's latest period of visibility, from 12/21/2012 to 5/2/2013, to test the coordination and evaluate the overall performance of the Gaia-FUN-SSO . The 2732 high quality astrometric observations acquired during the Gaia-FUN-SSO campaign were reduced with the Platform for Reduction of Astronomical Images Automatically (PRAIA), using the USNO CCD Astrograph Catalogue 4 (UCAC4) as a reference. The astrometric reduction process and the precision of the newly obtained measurements are discussed. We compare the residuals of astrometric observations that we obtained using this reduction process to data sets that were individually reduced by observers and accepted by the Minor Planet Center. We obtained 2103 previously unpublished astrometric positions and provide these to the scientific community. Using these data we show that our reduction of this astrometric campaign with a reliable stellar catalog substantially improves the quality of the astrometric results. We present evidence that the new data will help to reduce the orbit uncertainty of Apophis during its close approach in 2029. We show that uncertainties due to geolocations of observing stations, as well as rounding of astrometric data can introduce an unnecessary degradation in the quality of the resulting astrometric positions. Finally, we discuss the impact of our campaign reduction on the recovery process of newly discovered asteroids.Comment: Accepted for publication in A&