Squarefree Powers of Edge Ideals of Forests

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

The Carbon-to-H2, CO-to-H2 conversion factors, and carbon abundance on kiloparsec scales in nearby galaxies

ABSTRACT Using the atomic carbon [C i] ($^{3} \rm P_{1} \rightarrow {\rm ^3 P}_{0}$) and [C i] ($^{3} \rm P_{2} \rightarrow {\rm ^3 P}_{1}$) emission {hereafter [C i] (1 − 0) and [C i] (2 − 1), respectively} maps observed with the Herschel Space Observatory, and CO (1 − 0), H i, infrared and submm maps from literatures, we estimate the [C i]-to-H2 and CO-to-H2 conversion factors of α[C i](1 − 0), α[C i](2 − 1), and αCO at a linear resolution $\sim 1\,$kpc scale for six nearby galaxies of M 51, M 83, NGC 3627, NGC 4736, NGC 5055, and NGC 6946. This is perhaps the first effort, to our knowledge, in calibrating both [C i]-to-H2 conversion factors across the spiral disks at spatially resolved $\sim 1\,$kpc scale though such studies have been discussed globally in galaxies near and far. In order to derive the conversion factors and achieve these calibrations, we adopt three different dust-to-gas ratio (DGR) assumptions that scale approximately with metallicity taken from precursory results. We find that for all DGR assumptions, the α[C i](1 − 0), α[C i](2 − 1), and αCO are mostly flat with galactocentric radii, whereas both α[C i](2 − 1) and αCO show decrease in the inner regions of galaxies. And the central αCO and α[C i](2 − 1) values are on average ∼2.2 and 1.8 times lower than its galaxy averages. The obtained carbon abundances from different DGR assumptions show flat profiles with galactocentric radii, and the average carbon abundance of the galaxies is comparable to the usually adopted value of 3 × 10−5. We find that both metallicity and infrared luminosity correlate moderately with the αCO, whereas only weakly with either the α[C i](1 − 0) or carbon abundance, and not at all with the α[C i](2 − 1).

Vector bundles 𝐸 on ℙ3 with homological dimension 2 and 𝜒(End 𝐸) = 1

We find the complete integer solutions of the equation X 2 + Y 2 + Z 2 - 4 ⁢ X ⁢ Y - 4 ⁢ Y ⁢ Z + 10 ⁢ X ⁢ Z = 1 X^{2}+Y^{2}+Z^{2}-4XY-4YZ+10XZ=1 . As an application, we prove that, for each solution ( a , b , c ) (a,b,c) such that a > 0 a>0 , b - 2 ⁢ a > 0 b-2a>0 and ( b - 2 ⁢ a ) 2 ≥ 4 ⁢ a (b-2a)^{2}\geq 4a , there is a vector bundle 𝐸 on P 3 \mathbb{P}^{3} defined by a minimal linear resolution 0 → O P 3 ⁢ ( - 2 ) a → O P 3 ⁢ ( - 1 ) b → O P 3 c → E → 0 0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)^{a}\to\mathcal{O}_{\mathbb{P}^{3}}(-1)^{b}\to\mathcal{O}_{\mathbb{P}^{3}}^{c}\to E\to 0 . In particular, 𝐸 satisfies χ ⁢ ( End ⁡ E ) = 1 \chi(\operatorname{End}E)=1 .

Edge rings of bipartite graphs with linear resolutions

On the one hand, Ohsugi and Hibi characterized the edge ring of a finite connected simple graph with a 2-linear resolution. On the other hand, Hibi, Matsuda and the author conjectured that the edge ring of a finite connected simple graph with a [Formula: see text]-linear resolution, where [Formula: see text], is a hypersurface and proved the case [Formula: see text]. In this paper, we solve this conjecture for the case of finite connected simple bipartite graphs.

A transition from parabolic to conical shape as a common effect in nearby AGN jets

ABSTRACT Observational studies of collimation in jets in active galactic nuclei (AGN) are a key to understanding their formation and acceleration processes. We have performed an automated search for jet shape transitions in a sample of 367 AGN using VLBA data at 15 and 1.4 GHz. This search has found 10 out of 29 nearby jets at redshifts z < 0.07 with a transition from a parabolic to conical shape, while the full analysed sample is dominated by distant AGN with a typical z ≈ 1. The ten AGN are UGC 00773, NGC 1052, 3C 111, 3C 120, TXS 0815−094, Mrk 180, PKS 1514+00, NGC 6251, 3C 371, and BL Lac. We conclude that the geometry transition may be a common effect in AGN jets. It can be observed only when sufficient linear resolution is obtained. Supplementing these results with previously reported shape breaks in the nearby AGN 1H 0323+342 and M87, we estimate that the break occurs at 105–106 gravitational radii from the nucleus. We suggest that the jet shape transition happens when the bulk plasma kinetic energy flux becomes equal to the Poynting energy flux, while the ambient medium pressure is assumed to be governed by Bondi accretion. In general, the break point may not coincide with the Bondi radius. The observational data support our model predictions on the jet acceleration and properties of the break point.

Mathematical means and software for digital image processing when evaluating the linear resolution of aerial photographic systems

The authors consider the issues of evaluating digital aerial photographic systems based on the modulation transfer function. Special attention is paid to the issues of synthesis of mathematical and software for processing digital images when evaluating the linear resolution of aerial photographic systems. The basis of the mathematical support is the original method for evaluating the linear resolution on the ground using the modulation transfer function, taking into account the results of the flight experiment. The software is implemented in an automated complex for image processing of digital aerial photographic systems, which automatically calculates the modulation transfer function based on the results of the flight experiment. The research methodology combines methods of systems analysis, digital image processing, probability theory, mathematical modeling and software engineering. The main conclusion of the study, confirmed by the results of determining the experimental function of modulation transfer using an automated image processing complex based on images obtained during flight tests of a digital camera of the complex from the Orlan-10 unmanned aerial vehicle, is to confirm the operability of the developed automated complex for digital image processing, which makes it possible to use it to estimate the linear resolution on the ground during flight tests of aerial photographic systems of aerial reconnaissance and surveillance.

Disc kinematics and stability in high-mass star formation

Context. In the disc-mediated accretion scenario for the formation of the most massive stars, high densities and high accretion rates could induce gravitational instabilities in the disc, forcing it to fragment and produce companion objects. Aims. We investigate the effects of inclination and spatial resolution on the observable kinematics and stability of discs in high-mass star formation. Methods. We studied a high-resolution 3D radiation-hydrodynamic simulation that leads to the fragmentation of a massive disc. Using RADMC-3D we produced 1.3 mm continuum and CH3CN line cubes at different inclinations. The model was set to different distances, and synthetic observations were created for ALMA at ~80 mas resolution and NOEMA at ~0.4′′. Results. The synthetic ALMA observations resolve all fragments and their kinematics well. The synthetic NOEMA observations at 800 pc with linear resolution of ~300 au are able to resolve the fragments, while at 2000 pc with linear resolution of ~800 au only a single structure slightly elongated towards the brightest fragment is observed. The position–velocity (PV) plots show the differential rotation of material best in the edge-on views. A discontinuity is seen at a radius of ~250 au, corresponding to the position of the centrifugal barrier. As the observations become less resolved, the inner high-velocity components of the disc become blended with the envelope and the PV plots resemble rigid-body-like rotation. Protostellar mass estimates from PV plots of poorly resolved observations are therefore overestimated. We fit the emission of CH3CN (12K−11K) lines and produce maps of gas temperature with values in the range of 100–300 K. Studying the Toomre stability of the discs, we find low Q values below the critical value for stability against gravitational collapse at the positions of the fragments and in the arms connecting the fragments for the resolved observations. For the poorly resolved observations we find low Q values in the outskirts of the disc. Therefore, although we could not resolve any of the fragments, we are able to predict that the disc is unstable and fragmenting. This conclusion is valid regardless of our knowledge about the inclination of the disc. Conclusions. These synthetic observations reveal the potential and limitations of studying discs in high-mass star formation with current (millimetre) interferometers. While the extremely high spatial resolution of ALMA reveals objects in extraordinary detail, rotational structures and instabilities within accretion discs can also be identified in poorly resolved observations.

PERANCANGAN SIMULASI MEDIA PEMBELAJARAN DENGAN METODE DIVISION REMAINDER UNTUK PENCARIAN ALAMAT RELATIF PADA PROSES PENEMPATAN DATA

Learning media is one medium that can help students understand the subject matter. No file system in the course material on how to put the data in a table. The material is the use of hashing technique in the search for the relative address of the data. In applying the hash function to find the relative address is probable some data have the same hash function results. This causes a collision (collision) at the time of placement data. Collision (collision) can be addressed by using multiple strategies, among others, Resolution Linear (Linear Resolution), and Double Hashing Overflow. The design of this simulation was organized which begins with the concept, design, material collecting, and assembly. At the design stage is divided into four stages, namely stages of the application process flowchart design, design app menu hierarchy, the design of the main menu display and design process flow aplikasi.Aplikasi calculation aims to assist lecturers in teaching so that students can easily understand and comprehend the material on the hash function particularly the division remainder method to find the relative address.

Containment Problem for Quasi Star Configurations of Points in ℙ2

In this paper, the containment problem for the defining ideal of a special type of zero-dimensional subscheme of ℙ2, the so-called quasi star configuration, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. As an application of this result, for every real number [Formula: see text], we construct an infinite family of homogeneous radical ideals of points in 𝕂[ℙ2] such that their resurgences lie in the interval [2−ε, 2). Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that all of these ordinary powers have a linear resolution.