scholarly journals Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes

2021 ◽  
Vol 7 (6) ◽  
Author(s):  
Emil M. Prodanov
Keyword(s):  
Stationary Points ◽  
Model Parameters ◽  
The Real ◽  
Quartic Equation ◽  
Real Roots ◽  
Quadratic Equations ◽  
Parallel Lines ◽  
Intersection Points

AbstractPresented is a very detailed two-tier analysis of the location of the real roots of the general quartic equation $$x^4 + a x^3 + b x^2 + c x + d = 0$$ x 4 + a x 3 + b x 2 + c x + d = 0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. The presented classification of the roots of the quartic equation is particularly useful in situations in which the equation stems from a model the coefficients of which are (functions of) the model parameters and solving cubic equations, let alone using the explicit quartic formulæ , is a daunting task. The only benefit in such cases would be to gain insight into the location of the roots and the proposed method provides this. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the “sub-quartic” $$x^4 + ax^3 + bx^2$$ x 4 + a x 3 + b x 2 with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.

The basic principles of the cadastral methodology

2021 ◽  
Vol 974 (8) ◽  
pp. 45-54
Author(s):  
S.A. Atamanov
Keyword(s):  
Real Estate ◽  
Directed Graph ◽  
Basic Characteristic ◽  
Basic Principles ◽  
The Real ◽  
The Real Estate ◽  
Property Relations ◽  
Characteristic Features

Cadastral activities, in addition to describing a property for the purpose of cadastral- and rights registration, include an algorithmically complex determination of the sequence of actions necessary to solve the matter. Their result is not only some prepared documentation, but the fact of bringing the description of the real estate and rights to it in accordance with reality, provided that the task of the interested person is completed. The subtype of real estate includes objects that possess certain basic characteristic features and can have definite situational ones. The classification of real estate on the basis of these features provides an opportunity to compare and formalize the requirements for implementation of cadastral works of disparate regulatory legal acts. The field regulating land-and-property relations is modeled as a directed graph, the arcs of which are assigned a direction depending on the type of the initial and final vertices

Cyclotomy and the heptadecagon

2016 ◽  
Vol 100 (548) ◽  
pp. 288-297 ◽  
Author(s):  
Peter Shiu
Keyword(s):  
Significant Contribution ◽  
Regular Polygon ◽  
Necessary And Sufficient Condition ◽  
Real Roots ◽  
Quadratic Equations ◽  
Parallel Lines ◽  
To Come ◽  
Necessary And Sufficient ◽  
Similar Triangles ◽  
Fermat Primes

Cyclotomy is concerned with the division of a circle into a given number of equal segments, amounting to the construction of a regular polygon, say a q-gon, so that we need to deliver the angle α = 2π / q, or the length cos α. The construction is by Euclidean means, which make use of only ruler and compasses. Now, from given lengths, sums and differences of lengths are easy to obtain and, with the compasses, products and quotients of lengths can be obtained from similar triangles using parallel lines; indeed even the length can be obtained by applying the intersecting chord theorem to a circle with diameter 1 + a. However, there is not much else one can do with the compasses, so that the length cos α has to come from the real roots of a sequence of quadratic equations with ‘suitable’ coefficients — the meaning of being suitable will be made clear later.Gauss made the first significant contribution to the classical theory of cyclotomy in Article 365 of his famous Disquisitiones Arithmeticae [1] in 1801. He showed that the construction is possible if q = p is a Fermat prime, that is a prime of the form 22n + 1; see §7 for a necessary and sufficient condition for q. The only known Fermat primes are p = 3, 5, 17, 257, 65537; the cases p = 3, 5 and 17 correspond to the construction of the equilateral triangle, the regular pentagon, and the regular heptadecagon, the details for which Gauss gave.

scholarly journals ‘Malenkey Robot’ in the Carpathian Basin, in Hungary – Data, Facts, Interpretations, Connections

2015 ◽  
Vol 7 (1) ◽  
pp. 5-16
Author(s):  
Zalán Bognár
Keyword(s):  
Data Interpretation ◽  
Carpathian Basin ◽  
Effective Number ◽  
The Real ◽  
Real Value ◽  
Similarities And Differences ◽  
Archive Data

Abstract Interpretation problems related to the notion of ‘malenkaya rabota,’ POW, internee, GULAG and GUPVI. Ways of classification of the victims of ‘malenkaya rabota' in the Carpathian Basin, various groups and types. Determination of the effective number of the groups, and of the total number of those deported as civilians from the 14.7 million inhabitants of the Hungary of the time, based on different data, and the difficulties of definitions. The interpretation and implementation of the central Soviet commands. The connections between the deportations. Similarities and differences between the deportations as internee and as POW. Manageability of the data, interpretation of Soviet and Hungarian archive data and the reasons why they are different. The real value of Soviet archival sources. The determination of the losses attributed to ‘malenkaya rabota’.

THE ESSENCE, THE NEED FOR EVALUATION AND TYPES OF ECONOMIC POTENTIAL OF ENTERPRISES

2017 ◽  
Author(s):  
Olexandr Yemelyanov ◽  
Keyword(s):  
Economic Activity ◽  
Business Strategies ◽  
Enterprise Management ◽  
Value Assessment ◽  
Economic Potential ◽  
Enterprise Value ◽  
Development Assessment ◽  
State Of The Environment

The formation of business strategies of enterprises should be based on a preliminary assessment of their current and future economic opportunities. Such an assessment is to establish the value of the total economic potential of enterprises and its individual varieties. The purpose of this study was to clarify the essence of the economic potential of enterprises, justify the need for its evaluation and selection of its types. The main approaches to interpreting the terms "potential" and "enterprise potential" are identified. These include resource, result, resource-result, resource-target, and result-target concepts. It is established that the potential of any object, including the enterprise, can be interpreted as a set of its external functional properties, which this object shows or can show in a certain state of the environment in which it is located. Accordingly, the assessment of the potential of an object should be based on the identification and determination of its external properties, taking into account the environment in which the object is located. The main situations in which there is a need for information about certain components of the economic potential of the enterprise are identified, and the types of this potential and the consumers of the information about their level, corresponding to these situations, are determined. In particular, such situations include management of production and sales, management of financial and economic results of the enterprise, management of enterprise development, assessment of the company's need for various types of resources, assessment of enterprise value, assessment of current and future impact of the enterprise on the economy of the country (region), etc. The features of classification of types of enterprise potential existing in the scientific literature are supplemented by the following ones: by the main types of economic activity, by the dynamics of changes in the economic and production system of the enterprise, by consequences for the subject of enterprise potential assessment, by the stages of economic activity, enterprise potential, etc. The obtained results make it possible to improve the understanding of the complex patterns that underlie the formation of the economic potential of enterprises.

State Failure and the Political Violence Phenomenon: A Comparative Analysis of Iraq and Syria Cases

2016 ◽  
Vol 4 (2) ◽  
pp. 170 ◽  
Author(s):  
K. Eylem Özkaya Lassalle
Keyword(s):  
Cold War ◽  
Political Violence ◽  
State Failure ◽  
The Political ◽  
Collective Violence ◽  
Short Run ◽  
Definition Of ◽  
The Cold War

The concept of failed state came to the fore with the end of the Cold War, the collapse of the USSR and the disintegration of Yugoslavia. Political violence is central in these discussions on the definition of the concept or the determination of its dimensions (indicators). Specifically, the level of political violence, the type of political violence and intensity of political violence has been broached in the literature. An effective classification of political violence can lead us to a better understanding of state failure phenomenon. By using Tilly’s classification of collective violence which is based on extent of coordination among violent actors and salience of short-run damage, the role played by political violence in state failure can be understood clearly. In order to do this, two recent cases, Iraq and Syria will be examined.

Method of determination of the text direction on the image with the use of convolutional neural network

2020 ◽  
pp. 96-99
Author(s):  
P.L. Nikolaev
Keyword(s):  
Neural Network ◽  
Convolutional Neural Network ◽  
Deep Neural Network ◽  
Binary Classification ◽  
Synthetic Data ◽  
Real Data ◽  
Method Of Determination ◽  
Classification Of Images

This article deals with method of binary classification of images with small text on them Classification is based on the fact that the text can have 2 directions – it can be positioned horizontally and read from left to right or it can be turned 180 degrees so the image must be rotated to read the sign. This type of text can be found on the covers of a variety of books, so in case of recognizing the covers, it is necessary first to determine the direction of the text before we will directly recognize it. The article suggests the development of a deep neural network for determination of the text position in the context of book covers recognizing. The results of training and testing of a convolutional neural network on synthetic data as well as the examples of the network functioning on the real data are presented.

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