Social choice using moral metrics

PurposeThis paper aims to develop a geometry of moral systems. Existing social choice mechanisms predominantly employ simple structures, such as rankings. A mathematical metric among moral systems allows us to represent complex sets of views in a multidimensional geometry. Such a metric can serve to diagnose structural issues, test existing mechanisms of social choice or engender new mechanisms. It also may be used to replace active social choice mechanisms with information-based passive ones, shifting the operational burden.Design/methodology/approachUnder reasonable assumptions, moral systems correspond to computational black boxes, which can be represented by conditional probability distributions of responses to situations. In the presence of a probability distribution over situations and a metric among responses, codifying our intuition, we can derive a sensible metric among moral systems.FindingsWithin the developed framework, the author offers a set of well-behaved candidate metrics that may be employed in real applications. The author also proposes a variety of practical applications to social choice, both diagnostic and generative.Originality/valueThe proffered framework, derived metrics and proposed applications to social choice represent a new paradigm and offer potential improvements and alternatives to existing social choice mechanisms. They also can serve as the staging point for research in a number of directions.

Applying Multidimensional Geometry to Basic Data Centre Designs

Fog computing deployments are catching up by the day due to their advantages on latency and bandwidth compared to cloud implementations. Furthermore, the number of required hosts is usually far smaller, and so are the amount of switches needed to make the interconnections among them. In this paper, an approach based on multidimensional geometry is proposed for building up basic switching architectures for Data Centres, in a way that the most common convex regular N-polytopes are first introduced, where N is treated in an incremental manner in order to reach a generic high-dimensional N, and in turn, those resulting shapes are associated with their corresponding switching topologies. This way, N-simplex is related to a full mesh pattern, N-orthoplex is linked to a quasi full mesh structure and N-hypercube is referred to as a certain type of partial mesh layout. In each of those three contexts, a model is to be built up, where switches are first identified, afterwards, their downlink ports leading to the end hosts are exposed, along with those host identifiers, as well as their uplink ports leading to their neighboring switches, and eventually, a pseudocode algorithm is designed, exposing how a packet coming in from any given port of a switch is to be forwarded through the proper outgoing port on its way to the destination host by using the appropriate arithmetic expressions in each particular case. Therefore, all those algorithmic models represent how their corresponding switches may work when dealing with user data traffic within a Data Centre, guiding it towards its destination.

SOME PROPERTIES OF THE HYPERSPHERE IN N-DIMENSIONAL SPACE

The study of the properties of surfaces contributes to the expansion of their use in solving various practical problems, especially if such properties can be generalized to manifolds of n-dimensional space. The most thoroughly studied are the properties of the simplest surfaces, including the properties of a sphere. That is why the simplest surfaces are most often used in practice. Each property not covered in the existing literature expands the indicated possibilities. Therefore, the purpose of this article is to identify the properties of the hypersphere unknown from the literature. Most of the properties of a circle and a sphere have been known since ancient times [1, 4, 5]. The generalized concept of a sphere into multidimensional spaces is based on the general principles of multidimensional geometry [3]. In [4], eleven basic properties of the sphere are listed and analyzed. In works [8, 10] it is shown that a circle can be considered as an isoline, and a sphere as an isosurface when modeling energy fields. In geometric modeling of energy fields with point energy sources, an essential role is played by the distances from the points of the field to the given energy sources [6, 7]. In [9], two schemes are given for determining the parameter t, taking into account the effect of the distance from the points of the field to the point sources of energy on the potentials of the points of the field. In a particular case, if this parameter is determined according to a simplified scheme with f(l)=al2, then the formula for calculating the potential of an arbitrary point of the energy field is a mathematical model of the energy field generated by the number n of point energy sources. The geometric model of the field will be a manifold that can be foliated into a one-parameter set of isospheres [8, 10]. Abstracting from the physical nature of the field, simplifying the equation for calculating the potential of an arbitrary point of the energy field and generalizing it to n-dimensional space, we can formulate the following properties: Property 1. A hypersphere can be considered as a locus of points, the sum of the squared distances from which to n given points is a constant value. Property 2. Arbitrary coefficients ki at distances li affect the parameters of the hypersphere without changing the type of surface.

Geometric support of algorithms for solving Problems of higher mathematics

The need to improve the level of mathematical in particular geometric training of students of technical universities is due to modern technologies of computer-aided design. They are based on mathematical models of designed products, technological processes, etc., taking into account a large variety of source data. Therefore, from the first years of technical universities, when studying the cycle of mathematical disciplines, it is advisable to interpret a number of issues in terms and concepts of multidimensional geometry. At the same time, the combination of constructive (graphical) algorithms for solving problems in descriptive geometry with analytical algorithms in linear algebra and matanalysis allows us to summarize their advantages: the constructive approach provides the imagery inherent in engineering thinking, and the analytical approach provides the final result. The article shows the effectiveness of combining constructive and analytical algorithms for solving problems involving linear and nonlinear forms of many variables using specific examples.

Representation of Engineering Geometry Development in “Geometry and Graphics” Journal

In the paper "On the Increasing Role of Geometry", published in the electronic "Journal of Natural Science Research" in 2017, it was outspoken a hypothesis that now, at the time of innovative technologies, the importance of geometry is constantly increasing. The significance of geometry is also demonstrated by numerous Ph.D. and doctoral dissertations in the specialty No 05.01.01 - “Engineering Geometry and Computer Graphics”. It can be affirmed that all and everyone dissertations of technical and technological profile contain a geometric component to one degree or another. The "Geometry and Graphics" journal turned 8 (it was founded in June 2012). During this time, on its pages have been published numerous scientific papers, developing namely geometry and its branches: from simplest geometric constructions based on new properties of both lines and surfaces, to imaginary elements. Investigations were conducted in the following areas: “New Directions in Geometry”, “Fractal Geometry”, “Multidimensional Geometry”, “Geometric Constructions”, “Construction and Research of Surfaces”, “Imaginary Geometry”, “Practical Application of Geometry”, “Computer Graphics”, “Descriptive Geometry as Basis of other Branches of Geometry” ,”Geometry of Phase Spaces”. The journal publishes both recognized scientists and candidate for Ph.D. and doctor degrees. The considered array of papers clearly confirms the statement of the majority of authors, published in the journal, about geometry continuous development, which knocks out the ground for skeptics who decided that geometry is the science of the past centuries. As long as objects with shapes and surfaces surround us, geometry will be in demand. This, as they say, is unequivocal.

Meshless approximation method of one-dimensional oscillatory Fredholm integral equations

In this findings, a numerical meshless solution algorithm for 1D oscillatory Fredholm integral equation (OFIE) is put forward. The proposed algorithm is based on Levin?s quadrature theory (LQT) incorporating multi-quadric radial basis function (MQ-RBF). The procedure involves local approach of MQ-RBF differentiation matrix. The proposed method is specially designed to handle the case when the kernel function (KF) involves stationary point(s) (SP(s)). In addition to that, the model without SP(s) is also considered. The main advantage of the meshless procedure is that it can be easily extended to multidimensional geometry. These models have several physical applications in the area of engineering and sciences. The existence of the SP(s) in such models has numerous applications in the field of scattering and acoustics etc. (see [1, 2, 4, 6-8]). The proposed meshless method is accurate and cost-effective and provides a trustworthy platform to solve OFIE(s).

Five-Dimensional Two-Oktantal Epure Nomogram

Multidimensional experimental tasks with interdependent physical quantities can't be characterized by use of flat two-dimensional plots. Nomograms of new type solve such tasks. In this paper have been presented the nomograms with systematized both axes and planes. At construction of such models it is required a clear separation of all parametrial variable on arguing and functional ones. Nearby axes of interdependent parameters should lie alongside. Each axonometric cell should have a resultant indicator in the form of full size’s geometrical image. For the optimum choice of graphic execution on tabular data with four or five parameters, in the present paper is offered a method of its realization by means of two-oktantal nomogram. Justifications for this method have been presented in the paper. The method itself is based on descriptive geometry’s opportunities expansion at the solution of technical tasks by means of multidimensional geometry. The main lever for the task implementation is, certainly, communication lines. Formerly known from descriptive geometry such concepts as plane of reference, horizontally projecting surface, on the one hand, and pointed measurement of all experimental parameters on the other hand, provides to the nomogram possibility of its understanding for genesis in physical processes. Based on similarity of adjacent oktantal cells having the general axis are plotting two oktantal axonometric nomograms, creating interdependence between parameters by means of communication lines. This method opens a possibility for understanding of physical processes transformation. In this paper have been presented two graphic models of two oktantal nomograms competing for the right to be used by force of theirs optimal advantages. Absolute values of parameters are the real ones, taken from papers in "News of Higher Educational Institutions. North Caucasus Region" journal. Technical Sciences. No. 3, pp. 77–83, and No. 2, pp. 112–119. 2016.

Obtaining of Four-Dimensional Nomograms Based on Similarity Theorem

In some performed experimental works there is no total characteristic of study processes with regard to their physical understanding. It’s possible to achieve such understanding by means of epure and graphic interdependency for parameters illustrating experiment process, and by means of geometrical images which testify regularity of their outline, and are characteristic of physical process. The use of epure multidimensional octant nomogram can promote the solution for a number of application-oriented problems. By means of two octant epure nomograms constructed on experimental tabular data with four or five parameters, the optimum choice of graphic execution and its implementation in the area of physical genesis is offered. This method’s justifications have been given in the paper. A basis for extension of new opportunities is certainly descriptive geometry that facilitates a solution to technical tasks on multidimensional geometry. At the heart of similarity of adjacent octant cells with general axis are plotted two octant axonometric nomograms creating interdependencies between parameters by means of communication lines. This method opens a possibility for physical processes’ nature understanding. In this paper have been presented two graphic models for two octant nomograms in which advantages of their creation and reading have been presented. Foundations on which derivations are built when constructing the nomograms are validated by the similarity theorem and the axiom of projected surface. Absolute values of parameters are actual ones, and presented from papers of journals, as well as from peer-reviewed scientific publications recommended by Higher Attestation Commission.

Multidimensional Space and Visual Geometry in the Curriculum Geometric Preparation for Undergraduate

Reviewed and shown the ability to embed concepts of the multidimensional space in modern curricula for geometric preparation for an undergraduate degree. Multidimensional geometry allows the actual material to abandon the introduction in consideration of the evidence of mathematical calculations and formulas, and use it to achieve the final results. Given that historically multidimensional geometry was based on a compilation of material three-dimensional geometry, today you can go from the General to the particular, that is, to approach three-dimensional space as a visual component. You can raise the question about how the proposed curriculum organically combine Многомерная геометрия в наглядном изложении позволяет при изучении фактического материала отказаться от введения в рассмотрение доказательств математических выкладок и формул и использовать ее достижения в виде окончательных результатов. Учитывая, что исторически многомерная геометрия строилась на основе обобщения материалов трех- мерной геометрии, сегодня можно пойти по пути от общего к частному, т.е. подойти к рассмотрению трех- мерного пространства как наглядной составляющей, позволяющей увидеть одно-, двух-, трехмерный объект из общего ряда многомерного пространства [1; 3–6]. Таким образом, вопрос можно поставить о том, как в рамках предлагаемой учебной программы ор- ганически соединить наглядную многомерную гео- метрию с теми разделами геометрии, которые тра- диционно представляют теоретический интерес при геометрической подготовке конструкторских кадров во втузах, причем в условиях использования совре- менных компьютерных средств. Отметим, что в такой постановке вопроса предлагаемая учебная програм- ма отвечает требованиям современного инноваци- онного образования. Далее будет рассмотрено, как с позиций нового подхода могут быть изложены некоторые традици- онные для геометрии разделы на конкретных при- мерах. Любая из рассматриваемых тем может быть пред- ставлена как практическая работа при лабораторных занятиях на компьютере, что позволит наглядно закрепить изучаемый материал. Прежде всего геометрический объект будем рас- сматривать как совокупность геометрического тела и его поверхности (рис. 1) и изучать по 3D-моделям, построенным в модельном пространстве компьютера. Напомним, что одной из основных характеристик пространства является его размерность n. Пространство, в котором введены декартовы координаты (х1, …, xn), называется n-мерным декартовым пространством и обозначается Rn. Если в содержащем (вмещающем) пространстве Rn содержатся, например, два линейных подпро- visual multidimensional geometry with those parts of geometry, which traditionally are of theoretical interest in geometric training design staff in higher technical educational institutions, especially in terms of using modern computer tools. Considered the intersection of geometric objects in the plane in three-dimensional, four-dimensional and five-dimensional space. Presents the construction of intersection of the line with the surface of a three-dimensional object in three-dimensional Prospace and four-dimensional space. The concept of multidimensional space contains a large scientific potential required for geometric preparation of future designersky frames. Graduates will not only gain professional competence, but also be able to participate in solving various applied for cottages in neighboring disciplines.