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.

Sublinear Circuits for Polyhedral Sets

Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube [− 1,1]n.

Methods and tools formation of general indexes for automation of devices in regenerative medicine for post-stroke patients

The current processes of recovery of post-infarction and post-stroke patients in the context of the establishment of the institution of family doctors and insurance medicine are considered. It was proposed to introduce modules for automation of recovery process (MARP) to ensure procedures, quality of life and reduce labor costs during the period of long-term recovery. The forms of presentation of the model of the integral indicator are substantiated, which, in accordance with the requirements of the Ministry of Health, assesses the generalized indicator of the patient's condition (GPC), the quality of medical services and increases the efficiency of data compression. A consistent application of two Euclidean norms is proposed, which leads indicators of dissimilar physical nature to a limited metric space. The relationship between the lower and upper bounds of the GPC, the error, the width of the sliding window, and the values of the derivatives was established on the basis of the Taylor series expansion, geometric inequality and limited space. The model for evaluating the GPS as a lower bound and the method for generating information about its properties are substantiated. A three-level comparator is applied and an indicator vector (IV) is introduced as an informational addition to the time series. Additional capabilities for intelligent analysis are demonstrated. The model of GPC through IV is presented. The examples of IV values are used to demonstrate its applicability to the intelligent analysis of the recovery process. Openness, accessibility, transparency of GPC and IV as tools of KIT is implemented by the princes of public administration (PA) by reducing it to quantitative control and comparison if there are quantitative and qualitative indicators in the list. IV, sliding windows, as PA and KIT tools in software (SW) for a diagnostic conclusion and correction of the course of procedures, are numerically investigated. It is demonstrated on examples of a numerical experiment with software how the combined application of the method for calculating the GPC and IV effectively affects the compression ratio, increasing it to 60–75 %

Normal bundle and Almgren’s geometric inequality for singular varieties of bounded mean curvature

In this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We investigate the second-order properties of these varieties, obtaining results that are new also in the varifold’s setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, a property that allows to prove a Coarea-type formula for their generalized Gauss map. Then we use this formula to extend a sharp geometric inequality of Almgren and the associated soap bubble theorem. As a consequence of the geometric inequality we obtain sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.

A proof of Garfunkel inequality and of some related results in inner product spaces

In this paper, we will present a inner product space proof of a geometric inequality proposed by J. Garfunkel in American Mathematical Monthly [Garfunkel, J., Problem 2505, American Mathematical Monthly, 81 (1974), No. 11] and consider some other similar results.

Geometry of warped product pointwise semi-slant submanifolds of Kaehler manifolds

In this paper, we study warped product pointwise semi-slant submanifolds of a Kaehler manifold. First, we prove some characterizations results in terms of the tensor fields T and F and then, we obtain a geometric inequality for the second fundamental form in terms of intrinsic invariants. Furthermore, the equality case is also discussed. Moreover, we give some applications for Riemannian and compact Remannian submanifolds as well, i.e., we construct necessary and sufficient conditions for the non-existence of compact warped product pointwise semi-slant submanifold in complex space forms.