Limit T-subspaces and the central polynomials in n variables of the Grassmann algebra

Let F be the free unitary associative algebra over a field F on the set X = {x_1, x_2, ...}. A vector subspace V of F is called a T-subspace (or a T-space) if V is closed under all endomorphisms of F. A T-subspace V in F is limit if every larger T-subspace W \gneqq V is finitely generated (as a T-subspace) but V itself is not. Recently Brand\~ao Jr., Koshlukov, Krasilnikov and Silva have proved that over an infinite field F of characteristic p>2 the T-subspace C(G) of the central polynomials of the infinite dimensional Grassmann algebra G is a limit T-subspace. They conjectured that this limit T-subspace in F is unique, that is, there are no limit T-subspaces in F other than C(G). In the present article we prove that this is not the case. We construct infinitely many limit T-subspaces R_k (k \ge 1) in the algebra F over an infinite field F of characteristic p>2. For each k \ge 1, the limit T-subspace R_k arises from the central polynomials in 2k variables of the Grassmann algebra G.Comment: 22 page

Fast multi-dimensional scattered data approximation with Neumann boundary conditions

An important problem in applications is the approximation of a function $f$ from a finite set of randomly scattered data $f(x_j)$. A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials $\{e^{2\pi i kx}\}$. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials $\cos (\pi kx)$ as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem

Low temperature decoherence and relaxation in charge Josephson-junction qubits

In this lectures, we have described some essential features of loss of coherence by a qubit coupled to the environment. We have first presented well known semiclassical arguments that relate both decoherence and relaxation to the environmental noise. Then we have shown that models with pure decoherence (but no relaxation in qubit states) can be exactly solvable. As an example, we have treated in detail the model of fluctuating background charges which is believed to describe one of the most important channels for decoherence for the charge Josephson junction qubit. We have shown that the decoherence rate saturates at `high' temperatures while becoming linear in T at low temperatures and showing in all regimes a non-monotonic behaviour as a function of the coupling of the qubit to the fluctuating background charges. We have also considered, albeit only perturbatively, the qubit relaxation by the background charges and demonstrated that a quasi-linear behaviour of the spectral density of noise deduced from the measurements of the relaxation rate can be qualitatively explained.Comment: Lecture notes for International workshop on mesoscopic and nanoscopic systems, Kolkata, India, February 2006, to be published by Springe

Future space transport

Prospects for the mastery of space and the basic problems which must be solved in developing systems for both manned and cargo spacecraft are examined. The achievements and flaws of rocket boosters are discussed as well as the use of reusable spacecraft. The need for orbiting satellite solar power plants and related astrionics for active control of large space structures for space stations and colonies in an age of space industrialization is demonstrated. Various forms of spacecraft propulsion are described including liquid propellant rocket engines, nuclear reactors, thermonuclear rocket engines, electrorocket engines, electromagnetic engines, magnetic gas dynamic generators, electromagnetic mass accelerators (rail guns), laser rocket engines, pulse nuclear rocket engines, ramjet thermonuclear rocket engines, and photon rockets. The possibilities of interstellar flight are assessed

Mathematical modelling of the spread of contamination during fires in forests exposed to radioactive contamination

The paper suggested in the context of the general mathematical model of forest fires [1] gives a new mathematical setting and method of numerical solution of a problem of a radioactive spread above the forest region. Numerical solution of problems of radioactive smoke spread during crown fire in exemplified heat energy release in the forest fire front was found. Heat energy release in the forest fire front was found to cause further radioactive particles spread by the action of wind. In the absence of wind, radioactive smoke particles deposit again on the underlying surface after a time. As a wind velocity increases, these particles are transferred in the ground layer over distances proportional to a wind velocity

Potential for Development and Limitations of Regional Business Environment

The article considers some questions of institutional support for the development of the regional business environment. In connection with constantly growing social liabilities of regional and municipal authorities, which results in the upsurge of risks of budget imbalance at respective levels, the research provides a foundation to the necessity of forming favorable business environment promoting enterprise activity in regions. The process of creating institutes of a region’s business environment is analyzed with applying a matrix institutional model of entrepreneurship. The article reveals the mechanism stipulating the emergence of specific peculiarities of institutional configurations of the regional business environment. A conclusion is drawn about the decisive role of the institutional configuration of the business environment in the existing differences in the levels of social and economic development of regions. The researchers put forward a hypothesis that supporting a favorable institutional configuration of a region’s business environment that stimulates enterprise activity in the area cannot last indefinitely as its development is of a cyclical nature. Simulation parameters for the formation and maintenance of a favorable institutional configuration of the business environment are given. With the aim of practical application of the suggested approach, the authors suggest a methodology allowing for regional authorities to estimate the expediency of favorable institutional configuration of business environment, and on this basis to formulate rules of play for business so that these rules could boost business activity and increase the contribution of entrepreneurship to the region’s social and economic development. The approach under consideration provides an opportunity to substantiate a theoretical and methodological framework for developing practical mechanisms of projecting and further long-term maintenance of favorable institutional configuration of business environment which will contribute to expanding the region’s business activity, and which is resistant to negative effects of the external environment.Work produced with the financial support of the grant FSBI “Russian Humanitarian Science Foundation” in the framework of scientific — research project “Interaction of authority and business organizations in the municipalities: through compromise to mutual loyalty» № 15-12-02008