Многообразия флагов, торические многообразия и надстройки: три примера бесконечной транзитивности
Let K be a ﬁeld and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerA of all K-derivations of A is an A-module in a natural way and if R is the quotient ﬁeld of A then RDerA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerA of rank k over R (i.e. such that dimR RL = k), then the derived length of L is at most k and L is ﬁnite dimensional over its ﬁeld of constants. In case of solvable Lie algebras over a ﬁeld of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector ﬁelds with polynomial, rational, or formal coeﬃcients.
The problem of receiving points with high curvature (singular points) of contours for identification of the shape of objects on images is solved. Analysis of existing methods of numerical differentiation in the given aspect is held. The new method of differentiation of the flat discretely defined curves, which are dots (pixels) of circuits, based on variations of Arch Height method is considered. Features of such method of differentiation are shown using various formulas of calcu- lation of a derivative. Dependency aspects of the accuracy of the derivative on the chord length are analyzed. It is shown, that with an increase in its length differentiation accuracy degrades, and the result tends to the module of curvature of a curve at the given point. Comparison of the developed method with other known methods is made. The analysis of area of applicability and variability of parameters of differentiation is made. The accuracy aspects of calculation of derivatives for various parameters of differentiation are investigated. Examples of differentiation of various curves, both set analytically, and the functions-contours received from real images are considered. It is shown, that the offered method allows to get rid of the ambiguity in position of points of a contour with high curvature and consequently to raise quality of recognition of the shape of objects. Possible scopes of the given method in various areas of science and technics are stated.
A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.