Nine-point Circle Research Articles

Hyperautomation as a vital optimization tool in organizations: cognitive approach with the use of Euler circles

Purpose Hyperautomation is a technological concept whose popularity has been growing continuously since the German manufacturing industry “initiated” the Fourth Industrial Revolution (Industry 4.0), whereas, on the basis of theory, hyperautomation is a term still new and little recognized. This applies equally to scientific studies (articles, conference reports) and empirical studies (quantitative, qualitative). Therefore, this article attempts to fill definition gap that exists in the literature on management and quality sciences on the term hyperautomation.Design/methodology/approach The authors use literature review approach to identify the gaps in the existing literature on hyperautomation. They present a nominal definition of hyperautomation, discuss related issues and provide a comparative perspective between hyperautomation and automation.Findings The article’s findings include a precise definition of hyperautomation and the problems it raises. The authors point out that the term “hyperautomation” is still relatively new and underutilized in the management and quality sciences literature. It also compares hyperautomation to automation from several angles and emphasizes how it affects businesses, industries and other economic sectors.Practical implications Authors emphasize that in order to deploy hyperautomation successfully, enterprises must take a distributed and integrated approach.Originality/value This article addresses a gap in the management and quality sciences literature about the definition of hyperautomation. Authors give a thorough explanation of hyperautomation, along with relevant problems, useful implications and a comparison between hyperautomation versus automation.

Reasonable Risk and the Exercise of Professional Functions as Circumstances Precluding the Criminality of an Act: Characteristics of Similarities and Differences, Features of Interaction

The article presents a comparative characteristic of the circumstance fixed in Article 41 of the Criminal Code of the Russian Federation, excluding the criminality of the act: “reasonable risk”, and the circumstance existing in the theory of criminal law, excluding the criminality of the act: “causing harm in the exercise of professional functions”. The specifics of their interaction in the conditions of military service are also considered. The conclusion is made about the close relationship of these circumstances, which can be characterized as intersecting Euler circles.

Brianchon and Poncelet’s joint memoir, the nine-point circle, and beyond

Brianchon and Poncelet’s joint memoir, the nine-point circle, and beyond

The fruit of the meditations of a young Lieutenant of the Engineers: A Theorem of Major Impact and Its Applications. Two Centuries from the Discovery of the Nine-Point Circle

The Nine-Point Circle Theorem was first obtained in a joint work written by C.-J. Brianchon and J.-V. Poncelet. Since the authors of this Theorem presented it in their original work as a principle in geometry, we are invited to reflect whether there are some direct applications of this Theorem in advanced Euclidean geometry. Our exploration only confirms the original authors’ foretelling intuition, since un nouveau principe à la géométrie élémentaire is expected to admit applications. We regard the Nine-Point Circle Theorem in the overall context of the history of ideas and, pursuing Philip Davis’ thoughts, we regard the geometry of triangle as a chapter in the history of mathematical cultures.

Historical origins of the nine-point conic. The contribution of Eugenio Beltrami

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the nine-point conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami – Cremona, preserved in the Istituto Mazziniano, Genoa.

Müderris Şükrü [Sayan (1884-1943)

<p class="ql-align-justify">An article recently published in Osmanlı Bilimi Araştırmaları [20, 1 (2019): 102123] by Semiha Betül Takıcak analysed Şükrü Sayan’s article on imaginary quantities that was appended to his book on analytic geometry, namely the Hendese-i Tahlîliyye (vol.1, İstanbul, 1915). The same article noted that no detailed information was available on Şükrü Sayan’s life, a professor of analytic geometry at the Faculty of Science of the Darülfünun. Based on documents kept in author’s personal archives, the present addendum aims to present biographical information on Şükrü Sayan, including his year of birth, birth place, education and professional life, Besides it discloses Sayan’s work on Feuerbach’s nine-point circle published in Hasan Fehmi Çayköy’s Föyerbah Temas Nazariyesi (Feurebach’s theorem), dated 1921.&nbsp;

New points that belong to the nine-point circle

In the present paper, we show that the point of intersection of the bimedians of a cyclic quadrilateral belongs to the nine-point circle (Euler’s circle) of the triangle with one vertex at the point of intersection of the quadrilateral’s diagonals and the other vertices at the points of intersection of the extensions of its pairs of opposite sides.

METHOD FOR DETERMINING AND ADJUSTING THE EXPECTATIONS OF STAKEHOLDERS IN THE CONTEXT OF STRATEGIC MANAGEMENT OF THE PROJECT PROGRAM

The subject matter of the article is the management of the stakeholders of projects while implementing the program strategy. The goal of the work is to develop a method for determining the expectations of stakeholders and their adjustment in the context of the strategic management of the program of projects. The following tasks are solved in the article: project stakeholders are classified depending on the nature of their relationship to the strategy of the program of the project; a matrix for assessing the level of the involvement of stakeholders in the implementation of the program strategy is developed; the power/interest matrix is modified to represent the nature of the interest more accurately; the model of the strategic cube of stakeholders of projects and programs is built; recommendations for organizing a purposeful impact on the expectations of the stakeholders of the project program are developed to adjust their expectations to facilitate the efficient implementation of the program strategy. The following methods used are – set theory, Euler circles, matrix and graph-analytic methods. The following results were obtained: a classification of project stakeholders was developed depending on the nature of their relationship to the program strategy; the matrix for assessing the level of the involvement of stakeholders in the implementation of the program strategy involves developing the current and desired state; the modified power/interest matrix with the dimension of 2 * 4 contains recommendations for managing stakeholders in each quadrant; the built model of the strategic stakeholder cube contains 20 quadrants; recommendations on the use of the five-stage method for determining the expectations of stakeholders and their adjustment in the context of the strategic management of the project program are formulated. Conclusions. The method was developed for determining the expectations of stakeholders and their adjustment based on an assessment of the level of stakeholder involvement in the implementation of the strategy of the project program, taking into account the classification according to the nature of the relationship to it, which, unlike the existing ones, involves the construction and analysis of a modified power / interest matrix and the strategic cube of stakeholders of projects and programs; this method enables expanding the horizon of analysis and speed of response to changing needs of stakeholders while planning the implementation of the program strategy.

103.10 Around the nine-point circle

103.10 Around the nine-point circle

Euler zirkuluen erabilera intentsionalak silogistika kategorikorako

Artikulu honetan proposatuko ditut intentsionalki interpreta daitezkeen Euler zirkuluen sistema bi. Ikusiko denez, sistemok zehatz-mehatz egokitzen dira Aristotelesen jatorrizko sistemarekin eta modu artezean eta intuitiboan irudikatzen dute onartze existentziala, zeina zenbait modu silogistikok baliodun izateko behar duten. Aldi berean, proposizio partikularren irudikatzeetara ere hedatzen da proposizio unibertsalen eulertar irudikatzeek duten jatorrizko intuitibotasuna, zeren eta alboratu egiten baitira zirkuluak ebakitzea eta existitzen diren izakiak ad hoc markatzea.

Token Money or Cryptocurrency: technological Content and Economic Essence

The obligatory features of the unit of account are formulated. Distinctive characteristic properties of a set of cryptocalculation units, in particular, subsets of x-type (coin) and y-type (token) calculation units, are revealed. It is proved that these sets do not intersect. It is proposed to allocate a subset of C 2 in the set of coins, which has an additional characteristic property — functioning on the basis of a smart contract. The thesis that any settlement unit carries the cost of a resource is proved. Even for units of account with the so-called “intrinsic value” (bitcoin), the presence of a resource in its functions was found. Four characteristic properties inherent in the set of “money” (M = {money}) are disclosed and a subset DM = {digital money} is allocated, containing a set of digital Fiat money secured by the state (sets F = {fat}) and a subset P = {private}, which is actually an addition to the set F). It is proved that cryptocurrencies have characteristic properties of the set M = {money} only within their own community. The paradox of the algorithmic guarantor, reflecting the absence of the need for formalization of algorithmic informal institutions, is formulated. Disclosed defnitions of such concepts as “cryptocurrency”, “token economy” and “tokenization economy”. Euler circles are used to visualize the relationship between different sets of calculation units. A coherent classifcation of calculation units based on the characteristic properties of the identifed sets is formed. The results of the study can be used to form the basis of legal regulation of the circulation of digital units of account.

ЯКЩО МАТЕМАТИКА – ТО З ГУМОРОМ: НЕТРАДИЦІЙНІ ПІДХОДИ ДО ВИКЛАДАННЯ МАТЕМАТИКИ У ВИЩІЙ ШКОЛІ

The article is devoted to the issue of changing the atmosphere of the students’ audience to make it unobtrusive, to get rid of routine and restrained formality, on the other side, psychological and intellectual freedom and creativity should prevail not only among the approaches to teaching various math materials, but also in relationships between a teacher and students, between groupmates. The means, which can implement the mentioned intends, is humour. The drollery and humour as its direct manifestation are based on paradoxical thinking; non-traditional approach to things, so mathematics requires namely them, as the successful mastering of its “secrets” necessarily provides daring thoughts, looking for non-traditional ways of solving any math problems, ability to combine externally non-interconnected elements to find their essence.Making humorous situation, while presenting lecture materials or conducting practical workshops, increases the emotional tone of classes, facilitates the perception and awareness of math knowledge due to the pun-based and humorous form of its presentation (the point about the role of Euler circles while studying geometric circuits makes its understanding more accessible if one tries to imagine them before “Euler’s eyes”).Addressing to humorous tasks (their content is based on the principle of grotesque) can be appropriate when drawing function schedules is carried out through comprehension of the phrase: “Love is a curved line that is going through the heart, costs the packet and finally goes sideways!”Humour is also useful for avoiding psychological conflicts. So the story about J. Danzig, who managed to solve two unsolvable in ancient time problems, serves as background to make shirker students understand the irony of the message to miss classes more often.The “humorous” pauses can take place during the class time; they provide a kind of rest while doing intense mental and intellectual activity. To understand the laws of “golden-section search” and scope of its application, it’s appropriate to appeal to looking at a caricature of human body and later, based on the equation x2 – x – 1 = 0, remove the deliberate caricature of the picture.Thus, mathematics teacher is equipped with the sufficient amount of humorous techniques which can be successfully used and cause the desirable didactic effect of the classes.

Formalization of the arithmetization of Euclidean plane geometry and applications

This paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of the book by Schwabhäuser, Szmielew and Tarski Metamathematische Methoden in der Geometrie. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a “back-translation” from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method. Moreover, we solve a challenge proposed by Beeson: we prove that, given two points, an equilateral triangle based on these two points can be constructed in Euclidean Hilbert planes. Finally, we derive the axioms for another automated deduction method: the area method.

Introducing Nine-Point Circle to Junior High School Students

The concept of circles is an ancient concept that has appeared since Ancient Egypt from which this concept gives many significant contributions in mathematics’ development until now. Nevertheless, the concept of circles hides many uncover mysterious features that are of applications in mathematics. One of the mysterious features is the Nine-Point Circle. This Nine-point circle is also known as Euler’s circle, six-point circle, Feuerbach’s circle, the twelve-point circle, and many others. Because of these different names, there have been misunderstand among mathematicians about the Nine-Point Circle’s history. Besides, the discussion of Nine-Point Circle can be used to be an initial material to explain elementary geometry topic in junior high school’s level curriculum of 2013. Therefore, this concept needs to be delivered to the students as a geometry introduction. A possible form of the integration historical aspect of Nine-point circle is suggested in this paper as well as its importance in the curriculum of 2013.

Guided discovery of the nine-point circle theorem and its proof

ABSTRACTThe nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms.

Optimization of the Municipal Waste Collection Route Based on the Method of the Minimum Pairing

In the present article is shown the use of Maple program for processing of data describing the position of municipal waste sources and topology of collecting area. The data are further processed through the use of graph theory algorithms, which enable creation of collection round proposal. In this case study is described method of waste pick-up solution in a certain village of approx. 1,600 inhabitants and built-up area of approx. 30 hectares. Village has approx. 11.5 kilometers of ride able routes, with approx. 1 kilometer without waste source. The first part shows topology of the village in light of location of waste sources and capacity of the routes. In the second part are topological data converted into data that can be processed by use of the Graph Theory and the correspondent graph is shown. Optimizing collection route in a certain graph means to find the Euler circle. However, this circle can be constructed only on condition that all the vertices of the graph are of an even degree. Practically this means that is necessary to introduce auxiliary edges – paths that will be passed twice. These paths will connect vertices with odd values. The optimal solution then requires that the total length of the inserted edges was minimal possible, which corresponds to the minimum pairing method. As it is a problem of exponential complexity, it is necessary to make some simplifications. These simplifications are depicted graphically and the results are displayed in the conclusion. The resulting graph with embedded auxiliary edges can be used as a basic decision making material for creation of real collection round that respects local limitations such as one way streets or streets where is the waste collection is not possible from both sides at the same time.

Simulation of Nine-point Circle Using Novel Computing Tools and Investigating the Associated Useful Properties

The point, line, triangle and circle are the first set of figures in the world of geometry. The geometries of lines, triangles and circles have been studied and analyzed to obtain some specific and useful results for a long period of time. The analysis and determining various facts in the form of theorems and findings have been carried out by innumerable learned scholars and researchers using the basic tools, logical agreement and testing the data about these figures. The combination of triangle and circle in form of circumscribing circle and in-circle has also been studied at depth. The third fabulous circle is popularly known as the nine-point circle or Feuerbach's circle or Euler circle. The nine-point circle brings in light many interesting and useful properties of the basic triangle. But the construction of the nine-point circle is little tedious and cumbersome. This is because of the facts associated with the nine point circle. Therefore; in this paper the authors have developed a dedicated program in the world famous software MATLAB®. It is attempted to create a large number of triangle data and use them for verifying the useful results in addition to the construction of nine point circle.

A comparative study of linear and region based diagrams

Therearetwo categoriesof objects spatialinformation science investigates: actual objects and their spatial properties, such as in geography, and abstract objects which are employed metaphorically, as for visual languages. A prominent example of the latter are diagrams that model knowledge of some domain. Different aspects of diagrams are of in- terest, including their formal properties or how human users work with them, for example, with diagrams representing sets. The literature about diagrammatic systems for the repre- sentation of sets shows a dominance of region-based diagrams like Euler circles and Venn diagrams. The effectiveness of these diagrams, however, is limited because region-based diagrams become quite complex for more then three sets. By contrast, linear diagrams are not equally prevalent but enable the representation of a greater number of sets with- out getting cluttered. Cluttered diagrams exhibit inherent complexity due to overlapping objects, irrelevant details, or other reasons that impinge upon their legibility. This study contrasts both types of diagrammatic systems and investigates whether the performance of users differs for both kinds of diagrams. A significant difference can be shown regard- ing the number of diagrams that can be drawn within a fixed period of time and regarding the number of errors made. The results indicate that linear diagrams are more effective by being more restrictive and because region based diagrams show much clutter due to overlapping, coincident, and tangentially touching contours, as well as an overwhelming number of empty zones. Linear diagrams are less prone to errors and do not suffer from clutter.

A new ranking method for trapezial fuzzy numbers and its application to fuzzy risk analysis

In this study, a new method for ranking trapezial fuzzy numbers, generalized trapezoidal fuzzy numbers with different left and right heights, is presented. It considers triangles of a trapezium, a quadrilateral with no parallel sides, and uses the centers' of the nine-point circles and the areas of these triangles' to calculate the ranking score of the trapezial fuzzy number. The score of the trapezial fuzzy number is also expressed in terms of vertices of the trapezium. The method is simple and efficient, making the application of this method to real life problems more effective. The advantages of the proposed method are given with examples. Certain applications of the proposed ranking method to the fuzzy risk analysis problem, based on trapezial fuzzy numbers, are also presented. The proposed method ranks not only standardized, but also arbitrary trapezial fuzzy numbers. As a result, it is unnecessary to normalize to fuzzy numbers to apply the fuzzy risk algorithm.

Visual representation of rational belief revision: another look at the Sleeping Beauty problem.

The coherence of probability judgments is influenced in predictable ways by people's internal representations of problems, which may be altered by the manner in which propositions are stated or “framed” (Mandel, 2008). Likewise, several studies find that probabilistic reasoning and judgment can be improved by externally representing statistical information visually (for a review, see Garcia-Retamero and Cokely, 2013). Visual representation is thought to facilitate performance by externalizing the set-subset relations among observational data. Although some studies have examined whether visual representations can improve Bayesian reasoning, they have tended to focus on the use of natural sampling trees (Sedlmeier and Gigerenzer, 2001), Euler circles (Sloman et al., 2003), or other means of representing set-subset relations.