Two-valenced association schemes and the Desargues theorem

The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition, the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique, we prove a new result: given a prime p, any $$\{1,p\}$$ { 1 , p } -scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

FORMATION OF CRITICAL THINKING OF FUTURE MATHEMATICS TEACHERS BY MEANS OF GEOMETRY

The essence of the concept of critical thinking as a modern pedagogical phenomenon is disclosed in the article. It is revealed that among the most important competencies, which are relevant to the work of a modern teacher, teachers often refer to critical thinking as the ability to critically process various information, the ability to produce ideas, identify strategic issues and solve them, find convincing arguments, take responsibility, defend your position and correct it under the reasonable influence of the opponents. The theoretical foundations of the process of formation of critical thinking of future teachers in the process of professional training at the level of principles, properties, methods, stages are identified. The analysis of the project "The Higher Education Standard of Ukraine" in the field of knowledge 01 "Education" for specialty 014 "Secondary Education", subject specialization 014.04 "Secondary Education (Mathematics)", which provides for the formation of a number of competencies among students is carried out. It is found that the basis of the above listed competencies and learning outcomes is critical thinking. It is proved that geometry (in particular, constructive and projective), its methods and tasks have inexhaustible possibilities and means that can be used as an effective tool for critical thinking formation in the process of professional training of future mathematics teachers. For an example, we solve two problems of Euclidean geometry (a construction task and a proof) with the use of the Desargues theorem- one of the key statements of projective geometry.

Sphericity of a real hypersurface via projective geometry

In this work, we obtain an unexpected geometric characterization of sphericity of a real-analytic Levi-nondegenerate hypersurface [Formula: see text]. We prove that [Formula: see text] is spherical if and only if its Segre(-Webster) varieties satisfy an elementary combinatorial property, identical to a property of straight lines on the plane and known in Projective Geometry as the Desargues Theorem.

ON THE RELATIONSHIP BETWEEN PLANE AND SOLID GEOMETRY

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate the problem area we are after.Our strategy will be as follows. In the first part of the paper we will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. The sketch is given in broad strokes and only with the intent of acquainting the reader with some of the mathematical context against which the problem emerges. In the second part, we will look at a debate (on “fusionism”) in which for the first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. We conclude this part of the paper by remarking that only through a foundational and philosophical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the paper focuses on a specific case study which has been the subject of such an effort, namely the foundational analysis of the plane version of Desargues’ theorem on homological triangles and its implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, we begin in the fourth section the analytic work necessary for exploring various important claims about “purity,” “content,” and other relevant notions.

Desargues maps and the Hirota–Miwa equation

We study the Desargues maps , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multi-dimensional compatibility of the map is equivalent to the Desargues theorem and its higher dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the non-local -dressing method to construct Desargues maps and the corresponding solutions of the system. In particular, we identify the Fredholm determinant of the integral equation inverting the non-local -dressing problem with the τ -function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.

GENERALIZATION OF THE DESARGUES THEOREM FOR SPARSE 3D RECONSTRUCTION

Visual perception for walking machines needs to handle more degrees of freedom than for wheeled robots. For humanoid, four- or six-legged robots, camera motion is 6D instead of 3D or planar motion. Classical 3D reconstruction methods cannot be applied directly, because explicit sensor motion is needed. In this paper, we propose an algorithm for 3D reconstruction of an unstructured environment using motion-free uncalibrated single camera. Computer vision techniques are employed to obtain an incremental geometrical reconstruction of the environment, therefore using vision as a sensor for robot control tasks like navigation, obstacle avoidance, manipulation, tracking, etc. and 3D model acquisition. The main contribution is that the offline 3D reconstruction problem is considered as a point trajectory search through the video stream. The algorithm takes into account the temporal aspect of the sequence of images in order to have an analytical expression of the geometrical locus of the point trajectories through the sequence of images. The approach is a generalization of the Desargues theorem applied to multiple views taken from nearby viewpoints. Experiments on both synthetic and real image sequences show the simplicity and efficiency of the proposed method. This method provides an alternative technical solution easy to use, flexible in the context of robotic applications and can significantly improve the 3D estimation accuracy.