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Zoltán Kovács

Publications and source records attributed to Zoltán Kovács.

At least 37 records · Page 2Linked to original sources
arXiv2022

Parametric Root Finding for Supporting Proving and Discovering Geometric Inequalities in GeoGebra

We introduced the package/subsystem GeoGebra Discovery to GeoGebra which supports the automated proving or discovering of elementary geometry inequalities. In this case study, for inequality exploration problems related to isosceles and right angle triangle subclasses, we demonstrate how our general real quantifier elimination (RQE) approach could be replaced by a parametric root finding (PRF) algorithm. The general RQE requires the full cell decomposition of a high dimensional space, while the new method can avoid this expensive computation and can lead to practical speedups. To obtain a solution for a 1D-exploration problem, we compute a Groebner basis for the discriminant variety of the 1-dimensional parametric system and solve finitely many nonlinear real (NRA) satisfiability (SAT) problems. We illustrate the needed computations by examples. Since Groebner basis algorithms are available in Giac (the underlying free computer algebra system in GeoGebra) and freely available efficient NRA-SAT solvers (SMT-RAT, Tarski, Z3, etc.) can be linked to GeoGebra, we hope that the method could be easily added to the existing reasoning tool set for educational purposes.

math.AG↗
arXiv2021

Proceedings of the 13th International Conference on Automated Deduction in Geometry

Automated Deduction in Geometry (ADG) is a forum to exchange ideas and views, to present research results and progress, and to demonstrate software tools at the intersection between geometry and automated deduction. Relevant topics include (but are not limited to): polynomial algebra, invariant and coordinate-free methods; probabilistic, synthetic, and logic approaches, techniques for automated geometric reasoning from discrete mathematics, combinatorics, and numerics; interactive theorem proving in geometry; symbolic and numeric methods for geometric computation, geometric constraint solving, automated generation/reasoning and manipulation with diagrams; design and implementation of geometry software, automated theorem provers, special-purpose tools, experimental studies; applications of ADG in mechanics, geometric modelling, CAGD/CAD, computer vision, robotics and education. Traditionally, the ADG conference is held every two years. The previous editions of ADG were held in Nanning in 2018, Strasbourg in 2016, Coimbra in 2014, Edinburgh in 2012, Munich in 2010, Shanghai in 2008, Pontevedra in 2006, Gainesville in 2004, Hagenberg in 2002, Zurich in 2000, Beijing in 1998, and Toulouse in 1996. The 13th edition of ADG was supposed to be held in 2020 in Hagenberg, Austria, but due to the COVID-19 pandemic, it was postponed for 2021, and held online (still hosted by RISC Institute, Hagenberg, Austria), September 15-17, 2021 (https://www.risc.jku.at/conferences/adg2021).

cs.AI↗
arXiv2021

Offsets of a regular trifolium

The non-uniqueness of a rational parametrization of a rational plane curve may influence the process of computing envelopes of 1-parameter families of plane curves. We study envelopes of family of circles centred on a regular trifolium and its offsets, paying attention to different parametrizations. We use implicitization both to show that two rational parametrizations of a curve are equivalent, and to determine an implicit equation for the envelope under study. The derivation of an implicit equation of an offset follows another path, leading to new developments of the package GeoGebra Discovery. As an immediate symbolic result, we obtain that in the general case the offset curve of a regular trifolium is an algebraic curve of degree 14. We illustrate this fact by providing a GeoGebra applet that computes such curves automatically and visualizes them in a web browser.

math.HO↗
arXiv2020

Towards Automated Discovery of Geometrical Theorems in GeoGebra

We describe a prototype of a new experimental GeoGebra command and tool Discover that analyzes geometric figures for salient patterns, properties, and theorems. This tool is a basic implementation of automated discovery in elementary planar geometry. The paper focuses on the mathematical background of the implementation, as well as methods to avoid combinatorial explosion when storing the interesting properties of a geometric figure.

cs.AI↗
arXiv2020

Two almost-circles, and two real ones

Implicit locus equations in GeoGebra allow the user to do experiments with generalization of the concept of ellipses, namely with $n$-ellipses. By experimenting we obtain a geometric object that is very similar to a set of two circles.

math.HO↗
arXiv2020

Teaching fractals for gifted learners at age 12 by using novel technologies

A summary of an experimental course on fractals is given that was held for young learners at age 12. The course was a part of Epsilon camp, a program designed for very gifted students who have already demonstrated high interest in studying mathematics. Prerequisites for the course were mastery of Algebra I, experience and fluency in skills like exponentials and square roots, solving equations. Also, at least two preliminary years were required in a prior Epsilon camp. The summary gives an overview of the flow of teaching, the achieved results and some evaluation of the given feedback.

math.HO↗
arXiv2020

On Euler's inequality and automated reasoning with dynamic geometry

Euler's inequality $R\geq 2r$ can be investigated in a novel way by using implicit loci in GeoGebra. Some unavoidable side effects of the implicit locus computation introduce unexpected algebraic curves. By using a mixture of symbolic and numerical methods a possible approach is sketched up to investigate the situation. By exploiting fast GPU computations, a web application written in CindyJS helps in understanding the situation even better.

cs.SC↗
arXiv2020

Towards a Geometry Automated Provers Competition

The geometry automated theorem proving area distinguishes itself by a large number of specific methods and implementations, different approaches (synthetic, algebraic, semi-synthetic) and different goals and applications (from research in the area of artificial intelligence to applications in education). Apart from the usual measures of efficiency (e.g. CPU time), the possibility of visual and/or readable proofs is also an expected output against which the geometry automated theorem provers (GATP) should be measured. The implementation of a competition between GATP would allow to create a test bench for GATP developers to improve the existing ones and to propose new ones. It would also allow to establish a ranking for GATP that could be used by "clients" (e.g. developers of educational e-learning systems) to choose the best implementation for a given intended use.

cs.AI↗
arXiv2020

Distinguishing Brans-Dicke-Kerr type naked singularities and black holes with their thin disk electromagnetic radiation properties

The possible existence of naked singularities, hypothetical astrophysical objects, characterized by a gravitational singularity without an event horizon is still an open problem in present day astrophysics. From an observational point of view distinguishing between astrophysical black holes and naked singularities also represents a major challenge. One possible way of differentiating naked singularities from black holes is through the comparative study of thin accretion disks properties around these different types of compact objects. In the present paper we continue the comparative investigation of accretion disk properties around axially-symmetric rotating geometries in Brans-Dicke theory in the presence of a massless scalar field. The solution of the field equations contains the Kerr metric as a particular case, and, depending on the numerical values of the model parameter $γ$, has also solutions corresponding to non-trivial black holes and naked singularities, respectively. Due to the differences in the exterior geometries between black holes and Brans-Dicke-Kerr naked singularities, the thermodynamic and electromagnetic properties of the disks (energy flux, temperature distribution and equilibrium radiation spectrum) are different for these two classes of compact objects, consequently giving clear observational signatures that could discriminate between black holes and naked singularities.

gr-qc↗
arXiv2019

Teaching algebraic curves for gifted learners at age 11 by using LEGO linkages and GeoGebra

A summary of an experimental course on algebraic curves is given that was held for young learners at age 11. The course was a part of Epsilon camp, a program designed for very gifted students who have already demonstrated high interest in studying mathematics. Prerequisites for the course were mastery of Algebra I and at least one preliminary year in a prior Epsilon camp. The summary gives an overview of the flow of teaching, the achieved results and some evaluation of the given feedback.

math.HO↗
arXiv2019

The epsilon-delta game and the chess

The limit of a sequence by the definition with $\varepsilon$ is introduced by the notion of checkmate in two moves. The idea is also extended to define the limit of a function with $\varepsilon$ and $δ$.

math.HO↗
arXiv2018

Reply to "Comment on 'Can accretion disk properties observationally distinguish black holes from naked singularities?'"

In the Comment on "Can accretion disk properties observationally distinguish black holes from naked singularities?", by Bertrand Chauvineau, Phys. Rev. D {\bf 98}, 088501 (2018), the author did show that the metric used in Z. Kovács and T. Harko, Phys. Rev. D {\bf 82}, 124047 (2010), and initially introduced in K. D. Krori and D. R. Bhattacharjee, J. Math. Phys. \textbf{23}, 637 (1982) and K. K. Nandi, P. M. Alsing, J. C. Evans, and T. B. Nayak, Phys. Rev. D \textbf{63}, 084027 (2001), does not satisfy the Einstein gravitational field equations with a minimally coupled scalar field. In our reply we would like to point out that this result is actually not new, but it was already published in the literature. Moreover, a rotating solution that generalizes the Kerr metric for a nonminimally coupled scalar field does exist. We briefly discuss the nature of the singularities for the generalized metric, and point out that it can be used as a testing ground to differentiate black holes from naked singularities. We also mention the existence of some other typing or technical errors existing in the literature.

gr-qc↗
arXiv2018

Detecting truth, just on parts

We introduce and discuss, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplified through the implementation of this test on the dynamic geometry program GeoGebra.

cs.AI↗
arXiv2018

On the generalization of Erdős-Vincze's theorem about the approximation of closed convex plane curves by polyellipses

A polyellipse is a curve in the Euclidean plane all of whose points have the same sum of distances from finitely many given points (focuses). The classical version of Erdős-Vincze's theorem states that regular triangles can not be presented as the Hausdorff limit of polyellipses even if the number of the focuses can be arbitrary large. In other words the topological closure of the set of polyellipses with respect to the Hausdorff distance does not contain any regular triangle and we have a negative answer to the problem posed by E. Vázsonyi (Weissfeld) about the approximation of closed convex plane curves by polyellipses. It is the additive version of the approximation of simple closed plane curves by polynomial lemniscates all of whose points have the same product of distances from finitely many given points (focuses). Here we are going to generalize the classical version of Erdős-Vincze's theorem for regular polygons in the plane. We will conclude that the error of the approximation tends to zero as the number of the vertices of the regular polygon tends to the infinity. The decreasing tendency of the approximation error gives the idea to construct curves in the topological closure of the set of polyellipses. If we use integration to compute the average distance of a point from a given (focal) set in the plane then the curves all of whose points have the same average distance from the focal set can be given as the Hausdorff limit of polyellipses corresponding to partial sums.

math.MG↗
arXiv2017

A compilation of LEGO Technic parts to support learning experiments on linkages

We present a compilation of LEGO Technic parts to provide easy-to-build constructions of basic planar linkages. Some technical issues and their possible solutions are discussed. To solve questions on fine details---like deciding whether the motion is an exactly straight line or not---we refer to the dynamic mathematics software tool GeoGebra.

math.HO↗
arXiv2017

No, This is not a Circle

A popular curve shown in introductory maths textbooks, seems like a circle. But it is actually a different curve. This paper discusses some elementary approaches to identify the geometric object, including novel technological means by using GeoGebra. We demonstrate two ways to refute the false impression, two suggestions to find a correct conjecture, and four ways to confirm the result by proving it rigorously. All of the discussed approaches can be introduced in classrooms at various levels from middle school to high school.

math.HO↗