WHITEHEAD GROUPS OF MODULE EXTENSIONS

When R satisfies (strong unit) 1-stable range and M is a R-R-bimodule, the authors calculate that K_1(R(?) M)≌U(R(?) M)/L(R(?) M). As applications, the authors also calculate some Whitehead groups for exchange rings with artinian primitive factors.

Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian geometry. We also discovered in the meantime that, in our first GB book of 1978, we had already used a new way for studying the compatibility conditions (CC) of an operator that may not be necessarily formally integrable (FI) in order to construct canonical formally exact differential sequences on the jet level. The purpose of this paper is to prove that the combination of these two facts clearly shows the specific importance of the Spencer operator and the Spencer δ-cohomology, totally absent from mathematical physics today. The results obtained are unavoidable because they only depend on elementary combinatorics and diagram chasing. They also provide for the first time the purely intrinsic interpretation of the respective numbers of successive first, second, third and higher order generating CC. However, if they of course agree with the linearized Killing operator over the Minkowski metric, they largely disagree with recent publications on the respective numbers of generating CC for the linearized Killing operator over the Schwarzschild and Kerr metrics. Many similar examples are illustrating these new techniques, providing in particular a few resolutions in which the orders of the successive operators may go “up and down” surprisingly, like in the conformal situation for various dimensions.

Killing Operator for the Kerr Metric

When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) →F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 →F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 οD = 0 implies ad (D) οad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

The decadally modulating eddy field in the upstream Kuroshio Extension and its related mechanisms

Both the level of the high-frequency eddy kinetic energy（HF-EKE） and the energy-containing scale in the upstream Kuroshio Extension（KE） undergo a well-defined decadal modulation, which correlates well with the decadal KE path variability. The HF-EKE level and the energy-containing scales will increase with unstable KE path and decrease with stable KE path. Also the mesoscale eddies are a little meridionally elongated in the stable state, while they are much zonally elongated in the unstable state. The local baroclinic instability and the barotropic instability associated with the decadal modulation of HF-EKE have been investigated. The results show that the baroclinic instability is stronger in the stable state than that in the unstable state, with a shorter characteristic temporal scale and a larger characteristic spatial scale. Meanwhile, the regional-averaged barotropic conversion rate is larger in the unstable state than that in the stable state. The results also demonstrate that the baroclinic instability is not the dominant mechanism influencing the decadal modulation of the mesoscale eddy field, while the barotropic instability makes a positive contribution to the decadal modulation.

Design and Implementation of Hardware Architecture in Ionospheric Oblique Backscattering Sounding System

The ionospheric oblique backscattering sounding system can not only be used to detect the state of the ionosphere and the condition of high frequency channel in real time, but also be used for over-the-horizon sounding. Therefore, it has a very high military and civil value. For the characteristics of ionospheric oblique backscattering sounding, such as long sounding distance, wake echo, strong background noise, slow moving target, etc., a hardware platform of ionospheric oblique backscattering sounding system is designed. This platform adopts the technology of software radio and is designed as a new kind of general purpose, modularized, software-based ionosonde that is based on the VXI （Versa module eurocard eXtensions for Instrumentation） bus. This hardware platform has been successfully used in actual ionospheric oblique backscattering sounding, and the experimental results demonstrate that this system can satisfy the requirements.

Hopf Ore Extension over Dihedral Group

In this paper, the Hopf Ore extension and corresponding module extension of the group algebra over dihedral group are studied. It turns out that the 1-dimensional and 2- dimensional simple representations can both be extended to the simple representations over a class of Hopf Ore extension.