Some remarks on the Jacobian conjecture and polynomial endomorphisms

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization, which is that under some conditions, a polynomial endomorphism with $r$ homogeneous parts of positive degree does not have $r$ times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree $r$ does not take the same values on $r > 1$ collinear points, provided $r$ is a unit in the base field. Next, we show that for invertible maps $x + H$ of degree $d$, such that \ker \jac H has $n-r$ independent vectors over the base field, in particular for invertible power linear maps $x + (Ax)^{*d}$ with \rk A = r, the degree of the inverse of $x + H$ is at most $d^r$.Comment: 11 page

Irreducibility properties of Keller maps

Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducible properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski, that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.Comment: 22 page

What are the Effects of Mothers’ and Fathers’ Depression and Thoughts of Death on Their Children’s Level of Parental Connectedness?

Mental health outcomes such as depression are often passed down in families. While links between the mental health conditions of parents and their children have been established, there is a limited understanding of these outcomes over time and the impact that mothers and fathers have on their children independently. Analyzing data from the Fragile Families and Child Wellbeing study, PRC faculty research associates Susan De Luca and Yolanda Padilla and co-author Yan Yueqi show that children felt less connected to both mothers and fathers with mental health symptoms, but the effects varied somewhat based on the sex of the parent.Population Research Cente

A Novel Antenna Selection Scheme for Spatially Correlated Massive MIMO Uplinks with Imperfect Channel Estimation

We propose a new antenna selection scheme for a massive MIMO system with a single user terminal and a base station with a large number of antennas. We consider a practical scenario where there is a realistic correlation among the antennas and imperfect channel estimation at the receiver side. The proposed scheme exploits the sparsity of the channel matrix for the effective selection of a limited number of antennas. To this end, we compute a sparse channel matrix by minimising the mean squared error. This optimisation problem is then solved by the well-known orthogonal matching pursuit algorithm. Widely used models for spatial correlation among the antennas and channel estimation errors are considered in this work. Simulation results demonstrate that when the impacts of spatial correlation and imperfect channel estimation introduced, the proposed scheme in the paper can significantly reduce complexity of the receiver, without degrading the system performance compared to the maximum ratio combining.Comment: in Proc. IEEE 81st Vehicular Technology Conference (VTC), May 2015, 6 pages, 5 figure

Age Problem in Lemaitre-Tolman-Bondi Void Models

As is well known, one can explain the current cosmic acceleration by considering an inhomogeneous and/or anisotropic universe (which violates the cosmological principle), without invoking dark energy or modified gravity. The well-known one of this kind of models is the so-called Lema\^{\i}tre-Tolman-Bondi (LTB) void model, in which the universe is spherically symmetric and radially inhomogeneous, and we are living in a locally underdense void centered nearby our location. In the present work, we test various LTB void models with some old high redshift objects (OHROs). Obviously, the universe cannot be younger than its constituents. We find that an unusually large $r_0$ (characterizing the size of the void) is required to accommodate these OHROs in LTB void models. There is a serious tension between this unusually large $r_0$ and the much smaller $r_0$ inferred from other observations (e.g. SNIa, CMB and so on). However, if we instead consider the lowest limit 1.7\,Gyr for the quasar APM 08279+5255 at redshift $z=3.91$, this tension could be greatly alleviated.Comment: 17 pages, 9 figures, revtex4; v2: discussions added, Phys. Lett. B in press; v3: published versio

Density of States Monte Carlo Method for Simulation of Fluids

A Monte Carlo method based on a density-of-states sampling is proposed for study of arbitrary statistical mechanical ensembles in a continuum. A random walk in the two-dimensional space of particle number and energy is used to estimate the density of states of the system; this density of states is continuously updated as the random walk visits individual states. The validity and usefulness of the method are demonstrated by applying it to the simulation of a Lennard-Jones fluid. Results for its thermodynamic properties, including the vapor-liquid phase coexistence curve, are shown to be in good agreement with high-accuracy literature data.Comment: 5 pages, 3 figures, accepted by J Chem Phy