Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools

Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial p(x,y) by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools.

SPECTRAL PROPERTIES OF ADMITTANCE MATRICES OF RESISTIVE TREE NETWORKS

In this paper spectral properties of the admittance matrix of a resistive network whose underlying graph forms a general tree are studied. The algebraic presentation of the network is provided by its real node admittance matrix with respect to one of its terminal vertices, considered to be the root of the tree. The spectral properties of this matrix are studied by application of the theory of two-element-kind (R, C) networks. A mechanical analogue of a particular case of a similar problem, corresponding to a linear tree has been studied in the classical work of Gantmacher and Krein.7 Generalization of the study to networks based on trees of arbitrary structure calls for a modification of the mathematical approach. Instead of polynomial Sturm sequences applied in Ref. 7 the paper applies sequences of rational functions obeying the two basic Sturm conditions. In the special case of a linear tree these rational functions turn out to be polynomials, and the results are equivalent to those in Ref. 7. For a general tree the paper takes into consideration any root—leaf path of the tree. It is shown that the conditions on such a path are similar to those taking place on a linear tree. Some difference occurs in the number of sign reversals in the sequence of coordinates of characteristic vectors. In the case of a linear tree this number depends only on the position of the corresponding characteristic frequency in the spectrum of the matrix. In the case of a root-leaf path of a general tree, this number has to be normally decreased. The correction (which might be zero) is equal to the number of poles of the determinant of the reduced admittance matrix corresponding to the path considered, which does not exceed the characteristic frequency.