Search results for: trigonometric functions

Authors: Pablo Martin, Jorge Olivares, Fernando Maass

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The use of fractional derivatives to different problems in Engineering and Physics has been increasing in the last decade. For this reason, we have here considered partial derivatives when the integral is a spherical Bessel function of the first kind in both regular and modified ones simple initial conditions have been also considered. In this way, the solution has been found as a combination of hypergeometric functions. The case of a general rational value for α of the fractional derivative α has been solved in a general way for alpha between zero and two. The modified spherical Bessel functions of the first kind have been also considered and how to go from the regular case to the modified one will be also shown.

Keywords: caputo fractional derivatives, hypergeometric functions, linear differential equations, spherical Bessel functions

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Authors: Soumitr Sanjay Dubey, Shubhajit Roy Chowdhury, Rahul Shrestha

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In this paper, we have presented a novel approach of calculating eigenvalues of any matrix for the first time on Field Programmable Gate Array (FPGA) using Triangular Systolic Arra (TSA) architecture. Conventionally, additional computation unit is required in the architecture which is compliant to the algorithm for determining the eigenvalues and this in return enhances the delay and power consumption. However, recently reported works are only dedicated for symmetric matrices or some specific case of matrix. This works presents an architecture to calculate eigenvalues of any matrix based on QR algorithm which is fully implementable on FPGA. For the implementation of QR algorithm we have used TSA architecture, which is further utilising CORDIC (CO-ordinate Rotation DIgital Computer) algorithm, to calculate various trigonometric and arithmetic functions involved in the procedure. The proposed architecture gives an error in the range of 10−4. Power consumption by the design is 0.598W. It can work at the frequency of 900 MHz.

Keywords: coordinate rotation digital computer, three angle complex rotation, triangular systolic array, QR algorithm

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Authors: Haoshu Li, Shaolong Wan

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The recent focus has been on directional signal amplification of a signal input at one end of a one-dimensional chain and measured at the other end. The amplification rate is given by the end-to-end Green’s functions of the system. In this work, we derive the exact formulas for the end-to-end Green's functions of non-Hermitian single-band systems. While in the bulk region, it is found that the Green's functions are displaced from the prior established integral formula by O(e⁻ᵇᴸ). The results confirm the correspondence between the signal amplification and the non-Hermitian skin effect.

Keywords: non-Hermitian, Green's function, non-Hermitian skin effect, signal amplification

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Authors: Kassim A. Jassim

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In the present paper, we introduce and discuss a new class Kp(λ,α) of meromorphic multivalent functions in the punctured unit disk U*={z∈¢:0<|z|<1} defined by Ruscheweyh derivative. We obtain some sufficient conditions for the functions belonging to the class Kp(λ,α).

Keywords: meromorphic multivalent function, Ruscheweyh derivative, hadamard product

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

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Fractional derivatives have become very important in several areas of Engineering, however, the solutions of simple differential equations are not known. Here we are considering the simplest first order nonhomogeneous differential equations with Bessel regular functions of the first kind, in this way the solutions have been found which are hypergeometric solutions for any fractional derivative of order α, where α is rational number α=m/p, between zero and one. The way to find this result is by using Laplace transform and the Caputo definitions of fractional derivatives. This method is for values longer than one. However for α entire number the hypergeometric functions are Kumer type, no integer values of alpha, the hypergeometric function is more complicated is type ₂F₃(a,b,c, t2/2). The argument of the hypergeometric changes sign when we go from the regular Bessel functions to the modified Bessel functions of the first kind, however it integer seems that using precise values of α and considering no integers values of α, a solution can be obtained in terms of two hypergeometric functions. Further research is required for future papers in order to obtain the general solution for any rational value of α.

Keywords: Caputo, fractional calculation, hypergeometric, linear differential equations

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Authors: Artion Kashuri, Rozana Liko

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In this paper, the authors establish an integral identity using delta differentiable functions. By applying this identity, some new results via a general class of convex functions with respect to two nonnegative functions on a time scale are given. Also, for suitable choices of nonnegative functions, some special cases are deduced. Finally, in order to illustrate the efficiency of our main results, some applications to special means are obtained as well. We hope that current work using our idea and technique will attract the attention of researchers working in mathematical analysis, mathematical inequalities, numerical analysis, special functions, fractional calculus, quantum mechanics, quantum calculus, physics, probability and statistics, differential and difference equations, optimization theory, and other related fields in pure and applied sciences.

Keywords: convex functions, Hermite-Hadamard inequality, special means, time scale

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Authors: Meraj Ali Khan, Falleh R. Al-Solamy

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In the present paper, second order duality for multiobjective nonlinear programming are investigated under the second order generalized (F, b, φ, ρ, θ)− univex functions. The weak, strong and converse duality theorems are proved. Further, we also illustrated an example of (F, b, φ, ρ, θ)− univex functions. Results obtained in this paper extend some previously known results of multiobjective nonlinear programming in the literature.

Keywords: duality, multiobjective programming, univex functions, univex

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Authors: Fariha Asif

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The main purpose of this study was to investigate the effectiveness of eclectic approach in teaching of communicative functions. The objectives of the study were to get the information about the use of communicative functions through eclectic approach and to point out the most effective way of teaching functional communication and social interaction with the help of communicative activities through eclectic approach. The next step was to select sample from the selected population. As the research was descriptive so a questionnaire was developed on the basis of hypothesis and distributed to different selected schools of Lahore, Pakistan. Then data was tabulated, analyzed and interpreted through computer by finding percentages of different responses given by teachers to see the results. It was concluded that eclectic approach is effective in teaching communicative functions and communicative functions are better when taught through eclectic approach and communicative activities are more appropriate way of teaching communicative functions. It was found those teachers who were qualified in ELT gave better opinions as compare to those who did not have this degree. Techniques like presentations, dialogues and roleplay proved to be effective for teaching functional communication through communicative activities and also motivate the students not only in learning rules but also in using them to communicate with others.

Keywords: methodology, functions, teaching, ESP

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Authors: Mohamed Rahal, Djaouida Guetta

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In this paper, the one-dimensional unconstrained global optimization problem of continuous functions satifying a Hölder condition is considered. We extend the algorithm of sequential covering SCA for Lipschitz functions to a large class of Hölder functions. The convergence of the method is studied and the algorithm can be applied to systems of nonlinear equations. Finally, some numerical examples are presented and illustrate the efficiency of the present approach.

Keywords: global optimization, Hölder functions, sequential covering method, systems of nonlinear equations

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Authors: Mohammad Arif Khan

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This paper analyzes and presents the development of Artificial Neural Network based controller of space vector modulation (ANN-SVPWM) for a seven-phase voltage source inverter. At first, the conventional method of producing sinusoidal output voltage by utilizing six active and one zero space vectors are used to synthesize the input reference, is elaborated and then new PWM scheme called Artificial Neural Network Based PWM is presented. The ANN based controller has the advantage of the very fast implementation and analyzing the algorithms and avoids the direct computation of trigonometric and non-linear functions. The ANN controller uses the individual training strategy with the fixed weight and supervised models. A computer simulation program has been developed using Matlab/Simulink together with the neural network toolbox for training the ANN-controller. A comparison of the proposed scheme with the conventional scheme is presented based on various performance indices. Extensive Simulation results are provided to validate the findings.

Keywords: space vector PWM, total harmonic distortion, seven-phase, voltage source inverter, multi-phase, artificial neural network

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Authors: Kenichiro Hida, Shin-Ichi Kuribayashi

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In a conventional network, most network devices, such as routers, are dedicated devices that do not have much variation in capacity. In recent years, a new concept of network functions virtualisation (NFV) has come into use. The intention is to implement a variety of network functions with software on general-purpose servers and this allows the network operator to select their capacities and locations without any constraints. This paper focuses on the allocation of NFV-based routing functions which are one of critical network functions, and presents the virtual routing function allocation algorithm that minimizes the total power consumption. In addition, this study presents the useful allocation policy of virtual routing functions, based on an evaluation with a ladder-shaped network model. This policy takes the ratio of the power consumption of a routing function to that of a circuit and traffic distribution between areas into consideration. Furthermore, the present paper shows that there are cases where the use of NFV-based routing functions makes it possible to reduce the total power consumption dramatically, in comparison to a conventional network, in which it is not economically viable to distribute small-capacity routing functions.

Keywords: NFV, resource allocation, virtual routing function, minimum power consumption

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Authors: Yamin Sayyari

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In this paper, we generalized the Jensen's inequality for m-convex functions and also we present a correction of Jensen's inequality which is a better than the generalization of this inequality for m-convex functions. Finally, we have found new lower and new upper bounds for Jensen's discrete inequality.

Keywords: Jensen's inequality, m-convex function, Convex function, Inequality

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Authors: Prabhsharan Singh, Rahul Sindhu, Piyush Sikka

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The steering system is an integral part and medium through which the driver communicates with the vehicle and terrain, hence the most suitable steering geometry as per requirements must be chosen. The function of the chosen steering geometry of an All-Terrain Vehicle (ATV) is to provide the desired understeer gradient, minimum tire slippage, expected weight transfer during turning as these are requirements for a good steering geometry of a BAJA ATV. This research paper focuses on choosing the best suitable steering geometry for BAJA ATV tracks by reasoning the working principle and using fundamental trigonometric functions for obtaining these geometries on the same vehicle itself, namely Ackermann, Anti- Ackermann, Parallel Ackermann. Full vehicle analysis was carried out on Adams Car Analysis software, and graphical results were obtained for various parameters. Steering geometries were achieved by using a single versatile knuckle for frontward and rearward tie-rod placement and were practically tested with the help of data acquisition systems set up on the ATV. Each was having certain characteristics, setup, and parameters were observed for the BAJA ATV, and correlations were created between analytical and practical values.

Keywords: all-terrain vehicle, Ackermann, Adams car, Baja Sae, steering geometry, steering system, tire slip, traction, understeer gradient

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Authors: Mojtaba Hakimi-Moghaddam

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Positive real and strictly positive real transfer functions are important concepts in the control theory. In this paper, the results of researches in these areas are summarized. Definitions together with their graphical interpretations are mentioned. The equivalent conditions in the frequency domain and state space representations are reviewed. Their equivalent electrical networks are explained. Also, a comprehensive discussion about a difference between behavior of real part of positive real and strictly positive real transfer functions in high frequencies is presented. Furthermore, several illustrative examples are given.

Keywords: real rational transfer functions, positive realness property, strictly positive realness property, equivalent conditions

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Authors: Edamana Krishnan, Khalil Al-Ghafri

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In this paper, modified K(n,n) and K(n+1,n+1) equations have been solved using mapping methods which give a variety of solutions in terms of Jacobi elliptic functions. The solutions when m approaches 0 and 1, with m as the modulus of the JEFs have also been deduced. The role of constraint conditions has been discussed.

Keywords: travelling wave solutions, solitary wave solutions, compactons, Jacobi elliptic functions, mapping methods

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Authors: Jose Juan Peña, J. Morales, J. García-Ravelo

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In the search of potential models applied in the theoretical treatment of diatomic molecules, some of them have been constructed by using standard hyperbolic functions as well as from the so-called q-deformed hyperbolic functions (sc q-dhf) for displacing and modifying the shape of the potential under study. In order to transcend the scope of hyperbolic functions, in this work, a kind of generalized q-deformed hyperbolic functions (g q-dhf) is presented. By a suitable transformation, through the q deformation parameter, it is shown that these g q-dhf can be expressed in terms of their corresponding standard ones besides they can be reduced to the sc q-dhf. As a useful application of the proposed approach, and considering a class of exactly solvable multi-parameter exponential-type potentials, some new q-deformed quantum interactions models that can be used as interesting alternative in quantum physics and quantum states are presented. Furthermore, due that quantum potential models are conditioned on the q-dependence of the parameters that characterize to the exponential-type potentials, it is shown that many specific cases of q-deformed potentials are obtained as particular cases from the proposal.

Keywords: diatomic molecules, exponential-type potentials, hyperbolic functions, q-deformed potentials

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Authors: Jorge Olivares, Fernando Maass, Pablo Martin

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Fractional derivatives have been considered recently as a way to solve different problems in Engineering. In this way, second kind modified Bessel functions are considered here. The order α fractional differential equations of second kind Bessel functions, Kᵥ(x), are studied with simple initial conditions. The Laplace transform and Caputo definition of fractional derivatives are considered. Solutions have been found for ν=1/3, 1/2, 2/3, -1/3, -1/2 and (-2/3). In these cases, the solutions are the sum of two hypergeometric functions. The α fractional derivatives have been for α=1/3, 1/2 and 2/3, and the above values of ν. No convergence has been found for the integer values of ν Furthermore when α has been considered as a rational found m/p, no general solution has been found. Clearly, this case is more difficult to treat than those of first kind Bessel Function.

Keywords: Caputo, modified Bessel functions, hypergeometric, linear fractional differential equations, transform Laplace

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Authors: Tarul Garg, P. N. Agrawal

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We propose to define the q-analogue of the α-Bernstein Kantorovich operators and then introduce the q-bivariate generalization of these operators to study the approximation of functions of two variables. We obtain the rate of convergence of these bivariate operators by means of the total modulus of continuity, partial modulus of continuity and the Peetre’s K-functional for continuous functions. Further, in order to study the approximation of functions of two variables in a space bigger than the space of continuous functions, i.e. Bögel space; the GBS (Generalized Boolean Sum) of the q-bivariate operators is considered and degree of approximation is discussed for the Bögel continuous and Bögel differentiable functions with the aid of the Lipschitz class and the mixed modulus of smoothness.

Keywords: Bögel continuous, Bögel differentiable, generalized Boolean sum, K-functional, mixed modulus of smoothness

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Authors: Sunyoung Park

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Context: Talent management is a strategic approach that has received considerable attention in recent years to improve employee competency and organizational performance in many organizations. The implementation of talent management involves identifying objectives and positions within the organization, developing a pool of high-potential employees, and establishing appropriate HR functions to promote high employee and organizational performance. This study aims to investigate the relationship between talent management, HR functions, employee competency, and organizational performance in the South Korean context. Research Aim: The main objective of this study is to investigate the structural relationships among talent management, human resources (HR) functions, employee competency, and organizational performance. Methodology: To achieve the research aim, this study used a quantitative research method. Specifically, a total of 1,478 responses were analyzed using structural equation modeling based on data obtained from the Human Capital Corporate Panel (HCCP) survey in South Korea. Findings: The study revealed that talent management has a positive influence on HR functions and employee competency. Additionally, HR functions directly affect employee competency and organizational performance. Employee competency was found to be related to organizational performance. Moreover, talent management and HR functions indirectly affect organizational performance through employee competency. Theoretical Importance: This study provides empirical evidence of the relationship between talent management, HR functions, employee competency, and organizational performance in the South Korean context. The findings suggest that organizations should focus on developing appropriate talent management and HR functions to improve employee competency, which, in turn, will lead to better organizational performance. Moreover, the study contributes to the existing literature by emphasizing the importance of the relationship between talent management and HR functions in improving organizational performance.

Keywords: employee competency, HR functions, organizational performance, talent management

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Authors: Nicolae Pascu

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Aside from being a fundamental tool in Complex analysis, Schwarz Lemma-which was finalized in its most complete form at the beginning of the last century-generated an important area of research in various fields of mathematics, which continues to advance even today. We present some properties of analytic functions in the half-plane which satisfy the conditions of the classical Schwarz Lemma (Carathéodory functions) and obtain a generalization of the well-known Aleksandrov-Sobolev Lemma for analytic functions in the half-plane (the correspondent of Schwarz-Pick Lemma from the unit disk). Using this Schwarz-type lemma, we obtain a characterization for the entire class of Carathéodory functions, which might be of independent interest. We prove two monotonicity properties for Carathéodory functions that do not depend upon their normalization at infinity (the hydrodynamic normalization). The method is based on conformal mapping arguments for analytic functions in the half-plane satisfying appropriate conditions, in the spirit of Schwarz lemma. According to the research findings in this paper, our main results give estimates for the modulus and the argument for the entire class of Carathéodory functions. As applications, we give several extensions of Julia-Wolf-Carathéodory Lemma in a half-strip and show that our results are sharp.

Keywords: schwarz lemma, Julia-wolf-caratéodory lemma, analytic function, normalization condition, caratéodory function

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Authors: Madina Hamiane

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Modelling and simulation of an analogy function generator is presented based on a polynomial expansion model. The proposed function generator model is based on a 10th order polynomial approximation of any of the required functions. The polynomial approximations of these functions can then be implemented using basic CMOS circuit blocks. In this paper, a circuit model is proposed that can simultaneously generate many different mathematical functions. The circuit model is designed and simulated with HSPICE and its performance is demonstrated through the simulation of a number of non-linear functions.

Keywords: modelling and simulation, analog function generator, polynomial approximation, CMOS transistors

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Authors: Shuai Yuan, Minxia Shi, Xu Zhang, Jianzhi Yang, Kangqi Tian, Yuzheng Ma

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The precise measurement of muscle activities is essential for understanding the function of various body movements. This work aims to develop a muscle magnetic field signal detection system based on mathematical analysis. Medical research has underscored that early detection of muscle atrophy, coupled with lifestyle adjustments such as dietary control and increased exercise, can significantly enhance muscle-related diseases. Currently, surface electromyography (sEMG) is widely employed in research as an early predictor of muscle atrophy. Nonetheless, the primary limitation of using sEMG to forecast muscle strength is its inability to directly measure the signals generated by muscles. Challenges arise from potential skin-electrode contact issues due to perspiration, leading to inaccurate signals or even signal loss. Additionally, resistance and phase are significantly impacted by adipose layers. The recent emergence of optically pumped magnetometers introduces a fresh avenue for bio-magnetic field measurement techniques. These magnetometers possess high sensitivity and obviate the need for a cryogenic environment unlike superconducting quantum interference devices (SQUIDs). They detect muscle magnetic field signals in the range of tens to thousands of femtoteslas (fT). The utilization of magnetometers for capturing muscle magnetic field signals remains unaffected by issues of perspiration and adipose layers. Since their introduction, optically pumped atomic magnetometers have found extensive application in exploring the magnetic fields of organs such as cardiac and brain magnetism. The optimal operation of these magnetometers necessitates an environment with an ultra-weak magnetic field. To achieve such an environment, researchers usually utilize a combination of active magnetic compensation technology with passive magnetic shielding technology. Passive magnetic shielding technology uses a magnetic shielding device built with high permeability materials to attenuate the external magnetic field to a few nT. Compared with more layers, the coils that can generate a reverse magnetic field to precisely compensate for the residual magnetic fields are cheaper and more flexible. To attain even lower magnetic fields, compensation coils designed by Biot-Savart law are involved to generate a counteractive magnetic field to eliminate residual magnetic fields. By solving the magnetic field expression of discrete points in the target region, the parameters that determine the current density distribution on the plane can be obtained through the conventional target field method. The current density is obtained from the partial derivative of the stream function, which can be represented by the combination of trigonometric functions. Optimization algorithms in mathematics are introduced into coil design to obtain the optimal current density distribution. A one-dimensional linear regression analysis was performed on the collected data, obtaining a coefficient of determination R2 of 0.9349 with a p-value of 0. This statistical result indicates a stable relationship between the peak-to-peak value (PPV) of the muscle magnetic field signal and the magnitude of grip strength. This system is expected to be a widely used tool for healthcare professionals to gain deeper insights into the muscle health of their patients.

Keywords: muscle magnetic signal, magnetic shielding, compensation coils, trigonometric functions.

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Authors: F. Maass, P. Martin, J. Olivares

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The aim of this work is the application of the program GeoGebra in teaching the study of nonhomogeneous linear second order differential equations with constant coefficients. Different kind of functions or forces will be considered in the right hand side of the differential equations, in particular, the emphasis will be placed in the case of trigonometrical functions producing the resonance phenomena. In order to obtain this, the frequencies of the trigonometrical functions will be changed. Once the resonances appear, these have to be correlationated with the roots of the second order algebraic equation determined by the coefficients of the differential equation. In this way, the physics and engineering students will understand resonance effects and its consequences in the simplest way. A large variety of examples will be shown, using different kind of functions for the nonhomogeneous part of the differential equations.

Keywords: education, geogebra, ordinary differential equations, resonance

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Authors: Jean-Marie Vilaire, Ricardo Abreu-Blaya, Juan Bory-Reyes

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The classical Beltrami system of elliptic equations generalizes the Cauchy Riemann equation in the complex plane and offers the possibility to consider homogeneous system with no terms of zero order. The theory of Douglis-valued functions, called Hyper-analytic functions, is special case of the above situation. In this note, we prove an analogue of the N. Davydov theorem in the framework of the theory of hyperanalytic functions. The used methodology contemplates characteristic methods of the hypercomplex analysis as well as the singular integral operators and elliptic systems of the partial differential equations theories.

Keywords: Beltrami equation, Douglis algebra-valued function, Hypercomplex Cauchy type integral, Sokhotski-Plemelj formulae

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Authors: Shun Horikoshi, Tohru Kawabe

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This study presents performance analysis results of SMC (Sliding mode control) with changing the chattering functions applied to slip suppression problem of electric vehicles (EVs). In SMC, chattering phenomenon always occurs through high frequency switching of the control inputs. It is undesirable phenomenon and degrade the control performance, since it causes the oscillations of the control inputs. Several studies have been conducted on this problem by introducing some general saturation function. However, study about whether saturation function was really best and the performance analysis when using the other functions, weren’t being done so much. Therefore, in this paper, several candidate functions for SMC are selected and control performance of candidate functions is analyzed. In the analysis, evaluation function based on the trade-off between slip suppression performance and chattering reduction performance is proposed. The analyses are conducted in several numerical simulations of slip suppression problem of EVs. Then, we can see that there is no difference of employed candidate functions in chattering reduction performance. On the other hand, in slip suppression performance, the saturation function is excellent overall. So, we conclude the saturation function is most suitable for slip suppression sliding mode control.

Keywords: sliding mode control, chattering function, electric vehicle, slip suppression, performance analysis

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Authors: Chantima Wangsomchok

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The main purposes of this study are (1) to test the students’ communicative competence within six main functions: greeting, parting, thanking, offering, requesting and suggesting, (2) to employ error analysis in the students’ communicative competence within those functions, and (3) to compare the characteristics of the error found from the investigation. The subjects of the study is 328 first-year undergraduates taking the Foundation English course in the first semester of the 2008 academic year at Suan Sunandha Rajabhat University. This study found that while the subjects showed high communicative competence in the use of the following three functions: greeting, thanking, and offering, they seemed to show poor communicative competence in suggesting, requesting and parting instead. In addition, this study found that the grammatical errors were likely to be most frequently found in the parting function. In the same way, the type of errors which were less frequently found was in the functions of thanking and requesting respectively. Instead, the students tended to have high pragmatic failure in the use of greeting and suggesting functions.

Keywords: error analysis, functions of English language, communicative competence, cognitive science

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Authors: A. Zerarka, W. Djoudi

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The method developed in this work uses a generalised coth and csch funtions method to construct new exact travelling solutions to the nonlinear lattice equation. The technique of the homogeneous balance method is used to handle the appropriated solutions.

Keywords: coth functions, csch functions, nonlinear partial differential equation, travelling wave solutions

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Authors: Deepa, Yashbir Singh, Shi Yi Wu, Michael Friebe, Joao Manuel R. S. Tavares, Hu Wei-Chih

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Cardiovascular disease can be detected by measuring the regional and global wall motion of the left ventricle (LV) of the heart; In this study, we designed a dynamic simulation tool using Computed Tomography (CT) images to assess the difference between actual and simulated left ventricular functions. Thirteen healthy subjects were involved in the study with actual and simulated left ventricular functions. In this research, we found the high correlation between actual left ventricular wall motion and simulated left ventricular wall motion. Our results confirm that our simulation tool is feasible for simulating left ventricular motion.

Keywords: cardiac imaging, left-ventricular remodeling, cardiac wall motion, myocardial functions

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Authors: M. Saeed, A. Shahidzadeh

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Interaction between teachers and learners requires applying a set of samples (examples) which helps to create coordination between the goals and methods. The main result and achievement and application of samples (examples) are that they can bring the teacher and learner to a shared understanding of the concept. mathematical concepts, and also one of the challenging issues in the discussion of the function is to find the collision points of functions of and, regarding that the example space of teachers is different in this issue, this paper aims to present an example space including several problems of the secondary school with the help of intuition and drawing various graphs of functions of and for more familiarity of teachers.

Keywords: inverse function, educational example, Mathematic example, example space

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Authors: Tareq Hamadneh, Jochen Merker, Hassan Al-Zoubi

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Bernstein's expansion for exponentials in Taylor functions provides lower and upper optimization values for the range of its original function. these values converge to the original functions if the degree is elevated or the domain subdivided. Taylor polynomial can be applied so that the exponential is a polynomial of finite degree over a given domain. Bernstein's basis has two main properties: its sum equals 1, and positive for all x 2 (0; 1). In this work, we prove the existence of fixed points for exponential functions in a given domain using the optimization values of Bernstein. The Bernstein basis of finite degree T over a domain D is defined non-negatively. Any polynomial p of degree t can be expanded into the Bernstein form of maximum degree t ≤ T, where we only need to compute the coefficients of Bernstein in order to optimize the original polynomial. The main property is that p(x) is approximated by the minimum and maximum Bernstein coefficients (Bernstein bound). If the bound is contained in the given domain, then we say that p(x) has fixed points in the same domain.

Keywords: Bernstein polynomials, Stability of control functions, numerical optimization, Taylor function

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