Special Polygons Research Articles

Improving regulation of the use of soils and subsoils dumps in the Russian Federation

The analysis of the practice for regulating the use of dumps of soils and subsoils in the Russian Federation is carried out. It is shown that the existing regulations do not meet modern requirements for the protection and rational use of dumps of soils and subsoils. The absence of a unified recycling approach has been revealed. In some regulatory documents, dumps of soils and subsoils are considered as material for the reclamation of disturbed lands, in others – as industrial waste. Disagreement with the practice of classifying all dumps of soils and subsoils as waste of hazard classes 3–5 and subject to export to special polygons is justified. Disposal of such waste materials is administratively or criminally punishable. This approach to the material of soil excavation causes great harm to both the economy and the environment as a whole. It is proposed to change the practice of irrational use of soil and subsoil dump material and make appropriate changes to regulatory documents. One of such changes may be the introduction of the gradation “environmentally safe waste” into the documents on the definition of the hazard class of waste, with the prescription for the use of soil and subsoil dumps belonging to this category without restrictions on use in the national economy. It is proposed to clarify the magnitude of the soil thickness (soil layer) of 5 m, including the maximum depth of occurrence of fertile soil layer and potentially fertile layer. This value corresponds to the accepted stratigraphic rule on the general regularity of the structure of the sedimentary shell of the Earth. According to this point of view, soils are underlain by subsurface geological rocks, which does not imply the presence of intermediate geological layers between soils and subsurface.

A characterisation of regular n-gons via (in)commensurability

In Euclidean geometry, a regular polygon is equiangular (all angles are equal in size) and equilateral (all sides have the same length) polygon. So regular polygons should be thought of as special polygons.

Углеводороды и общество: проблемы взаимодействия и экологической оптимизации в условиях глобального потепления (на начало 2022 г.)

The article discusses the history and prospects for the development of the oil and gas industry in the face of increasing environmental protection requirements, associated with global warming, on the one hand, and growing political and economic pressure, on the other. The main issues and trends of the modern climate agenda, existing and proposed ways to reduce emissions of carbon dioxide and hydrocarbon gases are analyzed. Based on comprehensive and in-depth analysis, a number of conclusions were made about the possible consequences of the signing of the so-called Pars agreement for the fuel and energy complex of the Russian Federation. To conduct the necessary research, special polygons should be built. It was especially noted that it is required to develop a new paradigm for the development of the oil and gas complex of Russia.

Hexagonal grid approximation of the solution of the heat equation on special polygons

We consider the first type boundary value problem of the heat equation in two space dimensions on special polygons with interior angles alpha _{j}pi , j=1,2,ldots,M, where alpha _{j}in { frac{1}{2},frac{1}{3},frac{2}{3} } . To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points. It is proved that the given implicit schemes in both difference problems are unconditionally stable. It is also shown that the solutions of the constructed Difference Problem 1 and Difference Problem 2 converge to the exact solution on the grids of order O ( h^{2}+tau ^{2} ) and O ( h^{4}+tau ) respectively, where h and frac{sqrt{3}}{2}h are the step sizes in space variables x_{1} and x_{2} respectively and τ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.

Institutional aspects of solid waste treatment in Russia

Institutional aspects of solid waste treatment in Russia

Area minimization of special polygons

It is quite rare that a simple area optimization result bears somebody’s name. One of these statements, called Hajos’ Lemma, became particularly known, mainly because of its esthetic appearance and due to its application at solving the densest circle packing problem. Hajos considered a pair of concentric circles and wanted to find the minimum area polygon among those polygons which contain the smaller circle and whose vertices are outside of the larger circle. In this paper we state and prove two generalizations of Hajos’ Lemma. In the first version we allow the circles to be non concentric, in the second version we consider disc polygons instead of usual polygons.

Efficient and Privacy-Preserving Polygons Spatial Query Framework for Location-Based Services

With the pervasiveness of mobile devices and the development of wireless communication technique, location-based services (LBSs) have made our life more convenient, and the polygons spatial query, which can provide more flexible LBS, has attracted considerable interest recently. However, the flourish of polygons spatial query still faces many challenges including the query information privacy. In this paper, we present an efficient and privacy-preserving polygons spatial query framework for location-based services, called Polaris. With Polaris, the LBS provider (LP) outsources the encrypted LBS data to cloud server (CS), and the registered user can query any polygon range to get accurate LBS results without divulging his/her query information to the LP and CS. Specifically, an efficient special polygons spatial query (SPSQ) algorithm over ciphertext is constructed, based on an improved homomorphic encryption technology over composite order group. With SPSQ, Polaris can search outsourced encrypted LBS data in CS by the encrypted request, and respond the encrypted polygons spatial query results accurately. Detailed security analysis shows that the proposed Polaris can resist various known security threats. In addition, performance evaluations via implementing Polaris on smartphone and workstation with real LBS dataset demonstrate Polaris' effectiveness in term of real environment.

Patio Stones and Some Special Polygons

Photographs of patio blocks are represented by geometric figures and prompt questions about area and symmetry.

On Explanation of Polygons in Galilean Geometry to High School Students

In this paper, we have tried to indicate the own properties of polygons in Galilean geometry using the Affine concepts as well. The relationships between an angle and a side as well as the relationships between altitudes and medians concepts, and comparison of some special polygons have been examined carefully. In addition, the area concept has been mentioned. Finally, the paper was completed with a new idea, Theorem 6.

Adapted hyperbolic polygons and symplectic representations for group actions on Riemann surfaces

We prove that given a finite group G together with a set of fixed geometric generators, there is a family of special hyperbolic polygons that uniformize the Riemann surfaces admitting the action of G with the given geometric generators. From these special polygons, we obtain geometric information for the action: a basis for the homology group of surfaces, its intersection matrix, and the action of the given generators of G on this basis. We then use the Frobenius algorithm to obtain a symplectic representation G of G corresponding to this action. The fixed point set of G in the Siegel upper half-space corresponds to a component of the singular locus of the moduli space of principally polarized abelian varieties.We also describe an implementation of the algorithm using the open source computer algebra system SAGE.

Target Geometry and Probability Using a Dartboard

As middle-grades students explore area problems in geometry class, they typically calculate areas of circles and squares using formulas. Sometimes they explore problems using special polygons to compare a shaded area to an unshaded area. Students are often asked to simulate probability through the use of coins and dice. However, these situations contain little motivation for finding a solution. The following dartboard activity investigates how students can use simulations to determine experimental probability and build on intuition to make connections between calculating area and finding the theoretical probability of random events.

Voronoi-based range and continuous range query processing in mobile databases

With the wide availability of mobile devices (smart phones, iPhones, etc.), mobile location-based queries are increasingly in demand. One of the most frequent queries is range search which returns objects of interest within a pre-defined area. Most of the existing methods are based on the road network expansion method – expanding all nodes (intersections and objects) and computing the distance of each node to the query point. Since road networks are extremely complex, node expansion approaches are inefficient. In this paper, we propose a method, Voronoi Range Search ( VRS) based on the Voronoi diagram, to process range search queries efficiently and accurately by partitioning the road networks to some special polygons. Then we further propose Voronoi Continuous Range ( VCR) to satisfy the requirement for continuous range search queries (moving queries) based on VRS. Our empirical experiments show that VRS and VCR surpass all their rivals for both static and moving queries.

Methods for computing the least deviation from the sums of functions of one variable

We propose a method effective (in the sense of practical application) for finding the least deviation of a function of two variables from the sums of functions of one variable in the special polygons with sides parallel to the coordinate axes.

More on cutting a polygon into triangles of equal areas

In 2000 a general conjecture was proposed:a special polygon cannot be cut into an odd number of triangles of equal areas. It has been proved that the conjecture holds for polygons with at most six sides. In this paper we prove the existence of specialn-polygon for any integern>6 and discuss the conjecture for special polygons with seven sides.

A Distributed Formation of Smallest Faulty Orthogonal Convex Polygons in 2-D Meshes

The rectangular faulty block model is the most commonly used fault model for designing fault-tolerant and deadlock-free routing algorithms in mesh-connected multicomputers. The convexity of a rectangle facilitates simple and efficient ways to route messages around fault regions using relatively few virtual channels to avoid deadlock. However, such a faulty block may include many nonfaulty nodes which are disabled, i.e., they are not involved in the routing process. Therefore, it is important to define a fault region that is convex, and at the same time, to include a minimum number of nonfaulty nodes. In this paper, we propose a simple and efficient distributed algorithm that can quickly construct a set of special convex polygons, called orthogonal convex polygons, from a given set of rectangular faulty blocks in a 2-D mesh (or 2-D torus). The formation of orthogonal convex polygons is done through a labeling scheme based on iterative message exchanges among neighboring nodes. For a given faulty block, after some nonfaulty nodes have been removed, the block is split into a set of orthogonal convex polygons, each of which is the smallest orthogonal convex polygon that contains all the faults it covers. Moreover, we show that the number of nonfaulty nodes covered in these polygons is no more than the one in the smallest orthogonal convex polygon that includes all the faulty nodes in the original faulty block. Finally, we discuss one open problem and present extensions to multidimensional meshes.

ϒ-hyperelliptic Riemann surfaces

We give some characterizations of ϒ -hyperelliptic Riemann surfaces of genus ɡ ≥ 2, that is, pairs (S, j) where S is a closed Riemann surface of genus ɡ and j : S ⟶ S is a conformal involution with exactly 2ɡ + 2 - 4ϒ fixed points. These characterizations are given by Schottky groups, special hyperbolic polygons and algebraic curves.These can be seen as generalizations of the works [5] and [11].

Evaluation of the Schwarz–Christoffel Mapping Function for Special Polygons

Evaluation of the Schwarz–Christoffel Mapping Function for Special Polygons