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Register a new userQuadratic homogeneous polynomial maps $H$ and Keller maps $x+H$ with $3 le {rm rk} J H le 4$
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Authors
Michiel de Bondt
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We compute by hand all quadratic homogeneous polynomial maps $H$ and all Keller maps of the form $x + H$, for which ${rm rk} J H = 3$, over a field of arbitrary characteristic. Furthermore, we use computer support to compute Keller maps of the form $x + H$ with ${rm rk} J H = 4$, namely: $bullet$ all such maps in dimension $5$ over fields with $frac12$; $bullet$ all such maps in dimension $6$ over fields without $frac12$. We use these results to prove the following over fields of arbitrary characteristic: for Keller maps $x + H$ for which ${rm rk} J H le 4$, the rows of $J H$ are dependent over the base field.
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We classify all quadratic homogeneous polynomial maps $H$ and Keller maps of the form $x + H$, for which $rk J H = 3$, over a field $K$ of arbitrary characteristic. In particular, we show that such a Keller map (up to a square part if $char K=2$) is a tame automorphism.
Let $K$ be any field and $x = (x_1,x_2,ldots,x_n)$. We classify all matrices $M in {rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${rm rk} M le 2$. As a special case, we describe all such matrices $M$, which are the Jacobian matrix $J H$ (the matrix of partial derivatives) of a polynomial map $H$ from $K^n$ to $K^m$. Among other things, we show that up to composition with linear maps over $K$, $M = J H$ has only two nonzero columns or only three nonzero rows in this case. In addition, we show that ${rm trdeg}_K K(H) = {rm rk} J H$ for quadratic polynomial maps $H$ over $K$ such that $frac12 in K$ and ${rm rk} J H le 2$. Furthermore, we prove that up to conjugation with linear maps over $K$, nilpotent Jacobian matrices $N$ of quadratic polynomial maps, for which ${rm rk} N le 2$, are triangular (with zeroes on the diagonal), regardless of the characteristic of $K$. This generalizes several results by others. In addition, we prove the same result for Jacobian matrices $N$ of quadratic polynomial maps, for which $N^2 = 0$. This generalizes a result by others, namely the case where $frac12 in K$ and $N(0) = 0$.
As a part of the Milky Way Imaging Scroll Painting (MWISP) survey, we performed a simultaneous 12CO(1-0), 13CO(1-0), C18O(1-0) mapping toward molecular clouds in a region encompassing 3.75 square degrees. We reveal three molecular clouds, the 15 km/s cloud, the 27 km/s cloud, and the 50 km/s cloud, in the surveyed region. The 50 km/s cloud is resolved with an angular resolution of around 1 for the first time. Investigating their morphology and velocity structures, we find that the 27 km/s cloud is likely affected by feedback from the stellar association Mon OB3 and the 50 km/s cloud is characterised by three large expanding molecular shells. The region is mapped in C18O (1-0) for the first time. We discover seven C18O clumps, which are likely to form massive stars, and 15 dust clumps based on the BGPS archival data. Using infrared color-color diagrams, we find 56 Class I and 107 Class II young stellar object (YSO) candidates. According to the distribution of YSO candidates, an overdensity is found around the HII region S287 and the intersection of two shells, indicative of triggering. The star formation efficiency and rate of the 27~km/s cloud are discussed. Comparing the observed values of the filament S287-main with the models of fragmentation, we suggest that turbulence controls the large scale of fragmentation in the filament while gravitational fragmentation plays an important role in the formation of YSOs on the small scale. We find that star-forming gas tends to have higher excitation temperature, higher 13CO opacity, and higher column density than non-star-forming gas, which is consistent with the point that star formation happens in denser gas and star-forming gas is heated by YSOs. Using the 1.1 mm dust emission to trace dense gas, we obtain a dense gas fraction of 2.7%-10.4% for the 27 km/s cloud.
Four light-mass nuclei are considered by an effective two-body clusterisation method; $^6$Li as $^2$H$+^4$He, $^7$Li as $^3$H$+^4$He, $^7$Be as $^3$He$+^4$He, and $^8$Be as $^4$He$+^4$He. The low-energy spectrum of each is determined from single-channel Lippmann-Schwinger equations, as are low-energy elastic scattering cross sections for the $^2$H$+^4$He system. These are presented at many angles and energies for which there are data. While some of these systems may be more fully described by many-body theories, this work establishes that a large amount of data may be explained by these two-body clusterisations.
We have measured high-field magnetization and magnetoresistance of polycrystalline samples of the A-site ordered perovskite CaCu3Ti4-xRuxO12 (x=0 - 4) utilizing a non-destructive pulsed magnet. We find that the magnetization for x=0.5, 1.0 and 1.5 is nonlinear, and tends to saturate in high fields. This is highly nontrivial because the magnetization for x=0 and 4 is linear in external field up to the highest one. We have analyzed this field dependence based on the thermodynamics of magnetic materials, and propose that the external fields delocalize the holes on the Cu2+ ions in order to maximize the entropy. This scenario is qualitatively consistent with a large magnetoresistance of -70% observed at 4.2 K at 52 T for x=1.5.
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