scholarly journals Construction of a curve by using the state equation of Frenet formula

2021 ◽  
Vol 37 ◽  
pp. 454-465
Author(s):  
J T Chen ◽  
J W Lee ◽  
S K Kao ◽  
Y T Chou
Keyword(s):  
Plane Curve ◽  
State Equation ◽  
Initial Position ◽  
The State ◽  
Radius Of Curvature ◽  
Engineering Mathematics ◽  
The Matrix ◽  
Curvature And Torsion ◽  
Frenet Equation ◽  
The Given

Abstract In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.

scholarly journals Research of the Robust Stability of Control Systems Using a New Approach to the Lyapunov Functions Construction

2015 ◽  
Vol 9 (11) ◽  
pp. 176 ◽  
Author(s):  
Gulzhan Uskenbayeva
Keyword(s):  
Robust Stability ◽  
Lyapunov Functions ◽  
Closed Loop ◽  
Time Derivative ◽  
State Equation ◽  
The State ◽  
New Approach ◽  
Vector Lyapunov Functions ◽  
The Matrix ◽  
The Right

<p class="22">We investigate a new approach to the construction of vector Lyapunov functions. An approach to the construction of Lyapunov functions as vector functions is developed based on a geometrical interpretation of the second method of Lyapunov. The negative of the gradient is determined from the components of the time derivative of the state vector (i.e., the right-hand side of the state equation). The region of stability of a closed-loop linear, stationary system with uncertain parameters is governed by inequalities in the matrix elements of the closed-loop system. This study developed a method for analysing the robust stability of SISO and MIMO linear systems in canonical forms.</p>

scholarly journals Method for finding a solution to a linear nonstationary interval ОDЕ system

2021 ◽  
Vol 17 (2) ◽  
pp. 148-165
Author(s):  
Аlexander V. Fominyh ◽  
◽  
Keyword(s):  
Original Problem ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Initial Position ◽  
Time Interval ◽  
Support Functions ◽  
Final State ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
The Given

The article analyses a linear nonstationary interval system of ordinary differential equations so that the elements of the matrix of the system are the intervals with the known lower and upper bounds. The system is defined on the known finite time interval. It is required to construct a trajectory, which brings this system from the given initial position to the given final state. The original problem is reduced to finding a solution of the differential inclusion of a special form with the fixed right endpoint. With the help of support functions, this problem is reduced to minimizing a functional in the space of piecewise continuous functions. Under a natural additional assumption, this functional is Gateaux differentiable. For the functional, Gateaux gradient is found, necessary and sufficient conditions for the minimum are obtained. Оn the basis of these conditions, the method of the steepest descent is applied to the original problem. Some examples illustrate the constructed algorithm realization.

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

scholarly journals Industrial clusters in St. Petersburg as a basis for development of Arctic territories

2018 ◽  
Vol 170 ◽  
pp. 02006
Author(s):  
Elena Khutieva ◽  
Alexander Maizel ◽  
Marina Vlasova
Keyword(s):  
Industrial Clusters ◽  
The State ◽  
State Support ◽  
Strategic Development ◽  
Cluster Development ◽  
Effective Implementation ◽  
Coordination Center ◽  
Cluster Environment ◽  
The Given ◽  
Essential Prerequisite

The exploitation of Arctic resources is becoming now one of the most important directions of Russia’s strategic development. The coordination center for this project is St. Petersburg. The article assesses the potential of this region which forms an essential prerequisite for the effective implementation of the given work from the standpoint of the state and prospects of industrial clusters formed in its territory. The subjects of the cluster environment of St. Petersburg relevant programs of state support are divided into three categories: 2 innovative territorial clusters, 3 territorial clusters, 9 territorial clusters administered by the Center for cluster development. Specific recommendations for them are proposed on the basis of analysis of their strengths and weaknesses, as well as assessment of opportunities and threats to their development.

scholarly journals Integrated Recycling Process of Matrices of Organic Moulding Sands

2013 ◽  
Vol 58 (3) ◽  
pp. 809-812 ◽  
Author(s):  
R. Dańko
Keyword(s):  
Recycling Process ◽  
Conceptual Approach ◽  
Assessment Index ◽  
Experimental Stand ◽  
Synthetic Resins ◽  
The Matrix ◽  
Ignition Loss ◽  
On Line ◽  
The Given ◽  
Reclamation Process

Abstract The idea and experimental verification of assumptions of the integrated recycling process of matrices of uniform self-hardening moulding sands with synthetic resins, leading to obtaining moulding sands matrix of expected quality - is presented in the hereby paper. The basis of the presented process constitutes a combination of the method of forecasting averaged ignition losses of moulding sands after casting and defining the range of necessary matrix reclamation treatments in order to obtain its full recycling. Simultaneously, the empirically determined dependence of dusts amounts emitted during the reclamation process of the matrix from the given spent sand on the ignition loss values (which is the most proper assessment index of the obtained reclaimed material quality) was taken into account. The special experimental stand for investigations of the matrix recycling process was one of the elements of the conceptual approach and verification of its assumptions. The stand was equipped with the system of current on-line control of the purification degree of matrix grains from organic binder remains. The results of own investigations, allowing to combine ignition loss values of spent moulding sands after casting knocking out with amounts of dusts generated during the mechanical reclamation treatment of such sands, were utilized in the system.

scholarly journals Control Systems and Number Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-28
Author(s):  
Fuhuo Li
Keyword(s):  
Zeta Functions ◽  
Feedback System ◽  
State Equation ◽  
The State ◽  
Point Of View ◽  
Pid Controllers ◽  
Primary Concern ◽  
Feedback Connection ◽  
Proper Understanding ◽  
Natural Way

We try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines, and partially show how to number-theorize some practical problems. Our primary concern is linear systems from the point of view of our principle of visualization of the state, an interface between the past and the present. We view all the systems as embedded in the state equation, thus visualizing the state. Then we go on to treat the chain-scattering representation of the plant of Kimura 1997, which includes the feedback connection in a natural way, and we consider theH∞-control problem in this framework. We may view in particular the unit feedback system as accommodated in the chain-scattering representation, giving a better insight into the structure of the system. Its homographic transformation works as the action of the symplectic group on the Siegel upper half-space in the case of constant matrices. Both ofH∞- and PID-controllers are applied successfully in the EV control by J.-Y. Cao and B.-G. Cao 2006 and Cao et al. 2007, which we may unify in our framework. Finally, we mention some similarities between control theory and zeta-functions.

scholarly journals How Important is Gambling in National GDP: Case Study from Austria, Croatia, Italy and Slovenia

ECONOMICS ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 31-49
Author(s):  
Andrej Raspor ◽  
Iva Bulatović ◽  
Ana Stranjančević ◽  
Darko Lacmanović
Keyword(s):  
Research Question ◽  
The State ◽  
State Budget ◽  
Significant Information ◽  
Main Research ◽  
Benchmark Analysis ◽  
The World ◽  
State Budgets ◽  
The Given ◽  
First Time

Abstract Purpose – The situation in the field of gambling is changing due to the rise of Internet and Mobile gambling. In general gambling consumption is increasing every year, but the distribution of consumption has radically changed from Land Based gambling to Remote gambling. The purpose of this article is to present an overview of the world gambling industry and a specific overview in Austria, Croatia, Italy and Slovenia in order to find some main similarities and differences in observed period. Design/Methodology/Approach – The main research question is How important is gambling for the involved countries and what proportion of the national GDP does the gambling revenue account for? This paper presents the analysis of five statistical databases for the last sixteen years in order to find out some patterns, cyclical or seasonal features or other significant information that allows us to do forecasting of the future revenue with a certain degree of accuracy. We have systematically searched and collected data from the World Bank and the National Statistical Offices websites of the given countries. Statistical methods were used for benchmark analysis, while Box and Jenkins approach and ARIMA modelling were used for forecasting. Findings – The smallest increase was recorded in Slovenia and the largest in Italy. The same effects were also observed in the GDP of these countries. Thus, the state budgets of Croatia and Italy are increasingly dependent on gambling taxes. It also has negative wages. The gambling addictions among the locals have become more frequent as well. Originality of the research – The article shows the forecasts of the gambling revenue and its share in the GDP by 2027. We want to alert decision makers to adopt appropriate policies. States need to rethink their views on gambling and the excessive dependence of the state budget on gambling taxes. This is the first time a single comparative analysis of these countries and the above mentioned forecast has been conducted.

scholarly journals Axiological Vectors in the American and Russian Presidental Discourse: The State of the Union Address and the Address to the Russian Federal Assembly Compared

2017 ◽  
Vol 1 (2(2)) ◽  
pp. 121-150
Author(s):  
Ruslan Saduov
Keyword(s):  
The State ◽  
The Political ◽  
Political Leaders ◽  
Cultural Economic ◽  
State Of The Union ◽  
The Usa ◽  
Presidential Discourse ◽  
The Given

Presidential discourse is an indicative of axiological and other developmental vectors of a linguocultural community. It informs one about the main social, cultural, economic, and political changes in a country. In this respect, the annual State of the Union Address in the USA and the Address to the Federal Assembly in Russia are seen as the highlights of the political calendar in both countries, as these statements summarise the most relevant issues and enable their respective leaders to elaborate on their vision of their nation’s future. This paper aims to analise and compare the axiological vectors developed in the given presidential addresses in both Russia and the USA in the period from 2009 to 2015. It traces not only the most relevant values promoted by the political leaders, but also any axiological changes that occurred in the eventful years under investigation. The results of the research inform one about the current axiological identities of the linguocultural communities in question and the changing vectors of their development.

scholarly journals State estimation for bilinear impulsive control systems under uncertainties

2018 ◽  
pp. 35-41 ◽  
Author(s):  
Oxana G. Matviychuk
Keyword(s):  
Differential Equations ◽  
State Estimation ◽  
Control Systems ◽  
Uncertain Systems ◽  
Impulsive Control ◽  
Compact Set ◽  
The Matrix ◽  
Initial States ◽  
Impulsive Control Systems ◽  
The Given

The state estimation problem for uncertain impulsive control systems with a special structure is considered. The initial states are taken to be unknown but bounded with given bounds. We assume here that the coefficients of the matrix included in the differential equations are not exactly known, but belong to the given compact set in the corresponding space. We present here algorithms that allow to find the external ellipsoidal estimates of reachable sets for such bilinear impulsive uncertain systems.

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