Theorem Of Geometry Research Articles

Symplectic topology and ideal-valued measures

We adapt Gromov’s notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it allows us to discuss three “big fiber theorems”—the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology—from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgüneş.

The Radius of an Internally Tangent Circle

Summary We offer a visual proof for a theorem in Euclidean geometry regarding the radius of the incircle of an arbelos.

Conformal Image Viewpoint Invariant

In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space R3,1 leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ∐αHα of R3,1 by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ∐αHα for the Minkowski metric of the ambient space.

On Using GeoGebra and ChatGPT for Geometric Discovery

This paper explores the performance of ChatGPT and GeoGebra Discovery when dealing with automatic geometric reasoning and discovery. The emergence of Large Language Models has attracted considerable attention in mathematics, among other fields where intelligence should be present. We revisit a couple of elementary Euclidean geometry theorems discussed in the birth of Artificial Intelligence and a non-trivial inequality concerning triangles. GeoGebra succeeds in proving all these selected examples, while ChatGPT fails in one case. Our thesis is that both GeoGebra and ChatGPT could be used as complementary systems, where the natural language abilities of ChatGPT and the certified computer algebra methods in GeoGebra Discovery can cooperate in order to obtain sound and—more relevant—interesting results.

Minkowski Space from Quantum Mechanics

Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.

Adjacency preserving maps between tensor spaces

Let r and s be positive integers such that r⩾3. Let U1,…,Ur be vector spaces over a field F and V1,…,Vs be vector spaces over a field K such that dim⁡Ui,dim⁡Vj⩾2 for all i,j. In this paper, we characterize maps ψ:⨂i=1rUi→⨂i=1sVi that preserve adjacency in both directions, which extends Hua's fundamental theorem of geometry of rectangular matrices. We also characterize related results concerning locally full maps preserving adjacency in both directions between tensor spaces, maps preserving adjacency in both directions between tensor spaces over a field all whose nonzero endomorphisms are automorphisms, and injective continuous adjacency preserving maps on finite dimensional tensor spaces over the real field.

New Math: How an AI proved dozens of geometry theorems from the International Mathematical Olympiad.

New Math: How an AI proved dozens of geometry theorems from the International Mathematical Olympiad.

Adjacency preserving maps on symmetric tensors

Let r and s be positive integers such that r⩾2. Let U and V be vector spaces over fields F and K, respectively, such that dim⁡U⩾3 and F has at least r+1 elements. In this paper, we characterize surjective maps ψ:SrU→SsV preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions dim⁡U⩾3 and the cardinality |F|⩾r+1 in our result.

The Ptolemy-Euler Theorem via Reflection

Summary Although there are several generalizations of the Ptolemy-Euler theorem in plane geometry, its higher dimensional cases are still interesting enough to deserve attention. In this note, we give a new proof of the Ptolemy-Euler theorem in R n , via a modified matrix version (reflection transformation) of a classical algebraic identity.

Derivation of Differential Geometry Theorems in High Dimensional Spaces

Derivation of Differential Geometry Theorems in High Dimensional Spaces

Challenges of Engineering Applications of Descriptive Geometry

Descriptive geometry has indispensable applications in many engineering activities. A summary of these is provided in the first chapter of this paper, preceded by a brief introduction into the methods of representation and mathematical recognition related to our research area, such as projection perpendicular to a single plane, projection images created by perpendicular projection onto two mutually perpendicular image planes, but placed on one plane, including the research of curves and movements, visual representation and perception relying on a mathematical approach, and studies on toothed driving pairs and tool geometry in order to place the development presented here among them. As a result of the continuous variability of the technological environment according to various optimization aspects, the engineering activities must also be continuously adapted to the changes, for which an appropriate approach and formulation are required from the practitioners of descriptive geometry, and can even lead to improvement in the field of descriptive geometry. The imaging procedures are always based on the methods and theorems of descriptive geometry. Our aim was to examine the spatial variation in the wear of the tool edge and the machining of the components of toothed drive pairs using two cameras. Resolving contradictions in spatial geometry reconstruction research is a constant challenge, to which a possible answer in many cases is the searching for the right projection direction, and positioning cameras appropriately. A special method of enumerating the possible infinite viewpoints for the reconstruction of tool surface edge curves is presented in the second part of this paper. In the case of the monitoring the shape geometry, taking into account the interchangeability of the projection directions, i.e., the property of symmetry, all images made from two perpendicular directions were taken into account. The procedure for determining the correct directions in a mathematically exact way is also presented through examples. A new criterion was formulated for the tested tooth edge of the hob to take into account the shading of the tooth next to it. The analysis and some of the results of the Monge mapping, suitable for the solution of a mechanical engineering task to be solved in a specific technical environment, namely defining the conditions for camera placements that ensure reconstructibility are also presented. Taking physical shadowing into account, conclusions can be drawn about the degree of distortion of the machined surface from the spatial deformation of the edge curve of the tool reconstructed with correctly positioned cameras.

The fundamental theorem of affine geometry

We deal with a natural generalization of the classical Fundamental Theorem of Affine Geometry to the case of non bijective maps. This extension geometrically characterizes semiaffine morphisms. It was obtained by W. Zick in 1981, although it is almost unknown. Our aim is to present and discuss a simplified proof of this result.

Volume comparison theorems in Finsler geometry

Volume comparison theorems in Finsler geometry

On Menelaus’ and Ceva’s theorems in Nil geometry

Abstract In this paper we deal with Nil geometry, which is one of the homogeneous Thurston 3-geometries. We define the “surface of a geodesic triangle” using generalized Apollonius surfaces. Moreover, we show that the “lines” on the surface of a geodesic triangle can be defined by the famous Menelaus’ condition and prove that Ceva’s theorem for geodesic trianglesistruein Nil space. In our work we will use the projective model of Nil geometry described by E. Molnár in [6].

Hartogs-type theorems in real algebraic geometry, II

AbstractLet $$f:X\rightarrow \mathbb {R}$$ f : X → R be a function defined on a connected nonsingular real algebraic set X in $$\mathbb {R}^n$$ R n with $$\textrm{dim}X\ge 2.$$ dim X ≥ 2 . We prove that f is a regular function whenever the restriction $$f|_C$$ f | C is a regular function for every algebraic curve C in X that is an analytic submanifold homeomorphic to the unit circle and has at most one singular point. We also have a suitable version of this result for X not necessarily connected.

Convexity in (Colored) Affine Semigroups

In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.

Exploring tropical differential equations

Abstract The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p-adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.

Gaussian kernels on nonsimply connected closed Riemannian manifolds are never positive definite

AbstractWe show that the Gaussian kernel on any nonsimply connected closed Riemannian manifold , where is the geodesic distance, is not positive definite for any , combining analyses in the recent preprint by Da Costa–Mostajeran–Ortega and classical comparison theorems in Riemannian geometry.

Proving geometry theorems: Student prospective teachers’ perseverance and mathematical reasoning

Proof of geometry is a topic that involves mathematical reasoning abilities and relates to perseverance involving hard work, the spirit of achievement, and self-confidence. The current important problem that occurs at this time is that students who are future teachers of mathematics still experience difficulties in compiling proofs, especially those who are not challenged to work hard. This qualitative research explores mathematics teacher candidates' reasoning abilities and perseverance in proving geometric theorems. Therefore, the research design used a case study. There were three participants in this study, and they were student prospective mathematics teachers' s taking geometry courses. Data were collected through working documents, open questionnaires, and semi-structured interviews and were analyzed using iterative techniques consisting of data condensation, data exposure, and verification. The study's results showed that students' prospective teachers did not prioritize proof in solving geometry problems, even though they worked hard to solve the problems independently until they were finished. The students' perseverance also impacts their mathematical reasoning in proving geometric theorems. Students with more hard work values tend to have more reasoning values. The results of this study have implications that there needs to be an effort from the teacher to get used to giving proof questions to support students' perseverance and mathematical reasoning abilities.

EFFECTIVENESS OF USING GEOGEBRA IN TEACHING AND LEARNING CIRCLE THEOREMS ON STUDENT-TEACHERS’ PERFORMANCE

This paper aims to explore the effect of using GeoGebra in teaching and learning circle theorems on student-teachers’ performance. This paper is based on the results of the four-week experiment of GeoGebra software in teaching and learning the concepts of circle theorems of the geometry of student-teachers at the College of Education. The effectiveness of using GeoGebra software was explored by measuring the achievement level with the help of achievement tests and perceptions of using GeoGebra software through the questionnaire. The data were analyzed and interpreted by using a frequency distribution table, percentage, mean, standard deviation, and t-test. The findings showed that there was a higher mean score in achievement in circle theorems of the experimental group, that is, those taught with GeoGebra software than that of the control group, that is, those taught without GeoGebra software, and the difference was significant at 0.05 level. Similarly, the student-teachers of the experimental group had a positive perception to support the use of GeoGebra software in teaching and learning circle theorems in geometry and mathematics in general. Thus, the use of GeoGebra software is an effective tool to increase achievement; promote curiosity and creativity; make clear sense of concepts and encourage overall learning of students in mathematics. Finally, this study suggests that mathematics teachers need to use ICT tools including GeoGebra for effective teaching and learning.<p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/soc/0073/a.php" alt="Hit counter" /></p>