Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and repr...Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.展开更多
In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. ...In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.展开更多
A new application of Chebyshev polynomials of second kind Un(x) to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows ...A new application of Chebyshev polynomials of second kind Un(x) to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows to reduce powers and smooth functions of them to superpositions of the first N-1 powers of the considered operator in N-dimensional case. The method leads in two-dimensional case first to the recurrence relations for Chebyshev polynomials and due to initial conditions to the application of Chebyshev polynomials of second kind Un(x). Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind Un(x) comprising the known Generating function as special cases is constructed by means of a derived identity for operator functions f(A) of a general two-dimensional operator A. The basic results are Formulas (9.5) and (9.6) which are then specialized for different examples of functions f(x). The generalization of the theory for three-dimensional operators is started to attack and a partial problem connected with the eigenvalue problem and the Hamilton-Cayley identity is solved in an Appendix. A physical application of Chebyshev polynomials to a problem of relativistic kinematics of a uniformly accelerated system is solved. All operator calculations are made in coordinate-invariant form.展开更多
Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting proper...Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.展开更多
Based on compressive sensing and fractional discrete cosine transform(DCT)via polynomial interpolation(PI-FrDCT),an image encryption algorithm is proposed,in which the compression and encryption of an image are accomp...Based on compressive sensing and fractional discrete cosine transform(DCT)via polynomial interpolation(PI-FrDCT),an image encryption algorithm is proposed,in which the compression and encryption of an image are accomplished simultaneously.It can keep information secret more effectively with low data transmission.Three-dimensional piecewise and nonlinear chaotic maps are employed to obtain a generating sequence and the exclusive OR(XOR)matrix,which greatly enlarge the key space of the encryption system.Unlike many other fractional transforms,the output of PI-FrDCT is real,which facilitates the storage,transmission and display of the encrypted image.Due to the introduction of a plain-image-dependent disturbance factor,the initial values and system parameters of the encryption scheme are determined by cipher keys and plain-image.Thus,the proposed encryption scheme is very sensitive to the plain-image,which makes the encryption system more secure.Experimental results demonstrate the validity and the reliability of the proposed encryption algorithm.展开更多
The flow of a tetra-hybrid Casson nanofluid(Al_(2)O_(3)-CuO-TiO_(2)-Ag/H_(2)O)over a nonlinear stretching sheet is investigated.The Buongiorno model is used to account for thermophoresis and Brownian motion,while ther...The flow of a tetra-hybrid Casson nanofluid(Al_(2)O_(3)-CuO-TiO_(2)-Ag/H_(2)O)over a nonlinear stretching sheet is investigated.The Buongiorno model is used to account for thermophoresis and Brownian motion,while thermal radiation is incorporated to examine its influence on the thermal boundary layer.The governing partial differential equations(PDEs)are reduced to a system of nonlinear ordinary differential equations(ODEs)with fully non-dimensional similarity transformations involving all independent variables.To solve the obtained highly nonlinear system of differential equations,a novel Clique polynomial collocation method is applied.The analysis focuses on the effects of the Casson parameter,power index,radiation parameter,thermophoresis parameter,Brownian motion parameter,and Lewis number.The key findings show that thermal radiation intensifies the thermal boundary layer,the Casson parameter reduces the velocity,and the Lewis number suppresses the concentration with direct relevance to polymer processing,coating flows,electronic cooling,and biomedical applications.展开更多
Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet ...Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.展开更多
In this article we continue the consideration of geometrical constructions of regular n-gons for odd n by rhombic bicompasses and ruler used in [1] for the construction of the regular heptagon (n=7). We discuss the po...In this article we continue the consideration of geometrical constructions of regular n-gons for odd n by rhombic bicompasses and ruler used in [1] for the construction of the regular heptagon (n=7). We discuss the possible factorization of the cyclotomic polynomial in polynomial factors which contain not higher than quadratic radicals in the coefficients whereas usually the factorization of the cyclotomic polynomials is considered in products of irreducible factors with integer coefficients. In considering the regular heptagon we find a modified variant of its construction by rhombic bicompasses and ruler. In detail, supported by figures, we investigate the case of the regular tridecagon (n=13) which in addition to n=7 is the only candidate with low n (the next to this is n=769 ) for which such a construction by rhombic bicompasses and ruler seems to be possible. Besides the coordinate origin we find here two points to fix for the possible application of two bicompasses (or even four with the addition of the complex conjugate points to be fixed). With only one bicompass one has in addition the problem of the trisection of an angle which can be solved by a neusis construction that, however, is not in the spirit of constructions by compass and ruler and is difficult to realize during the action of bicompasses. As discussed it seems that to finish the construction by bicompasses the correlated action of two rhombic bicompasses must be applied in this case which avoids the disadvantages of the neusis construction. Single rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points on one circle determines the positions of all the other points on the second circle in unique way. The known case n=17 embedded in our method is discussed in detail.展开更多
Computational techniques are invaluable to the continued success and development of Magnetic Resonance Imaging (MRI) and to its widespread applications. New processing methods are essential for addressing issues at ea...Computational techniques are invaluable to the continued success and development of Magnetic Resonance Imaging (MRI) and to its widespread applications. New processing methods are essential for addressing issues at each stage of MRI techniques. In this study, we present new sets of non-exponential generating functions representing the NMR transverse magnetizations and signals which are mathematically designed based on the theory and dynamics of the Bloch NMR flow equations. These signals are functions of many spinning nuclei of materials and can be used to obtain information observed in all flow systems. The Bloch NMR flow equations are solved using the Boubaker polynomial expansion scheme (BPES) and analytically connect most of the experimentally valuable NMR parameters in a simplified way for general analyses of magnetic resonance imaging with adiabatic condition.展开更多
The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials.The fractional derivative is described in the Caputo sen...The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials.The fractional derivative is described in the Caputo sense.Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable.By solving the algebraic equations,the numerical solutions are acquired.The method in general is easy to implement and yields good results.Numerical examples are provided to demonstrate the validity and applicability of the method.展开更多
Approximation of finite population totals in the presence of auxiliary information is considered. A polynomial based on Lagrange polynomial is proposed. Like the local polynomial regression, Horvitz Thompson and ratio...Approximation of finite population totals in the presence of auxiliary information is considered. A polynomial based on Lagrange polynomial is proposed. Like the local polynomial regression, Horvitz Thompson and ratio estimators, this approximation technique is based on annual population total in order to fit in the best approximating polynomial within a given period of time (years) in this study. This proposed technique has shown to be unbiased under a linear polynomial. The use of real data indicated that the polynomial is efficient and can approximate properly even when the data is unevenly spaced.展开更多
We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix.
In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) i...In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) in C^(n).Let g be a convex function in U. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp.U^(n))when the mappings f are k-fold symmetric, k ∈ N. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.展开更多
A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symb...A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symbol second-order polynomial interpolation(SSPI)with cubature Kalman filter(CKF)to improve the precision and effectiveness of the data processing through using a three-stage processing approach of phase noise.First of all,the phase noise values in OFDM symbols are calculated by using pilot symbols.Then,second-order Newton interpolation(SNI)is used in second-order interpolation to acquire precise noise estimation.Afterwards,every OFDM symbol is partitioned into several sub-symbols,and second-order polynomial interpolation(SPI)is utilized in the time domain to enhance suppression accuracy and time resolution.Ultimately,CKF is employed to suppress the residual phase noise.The simulation results show that this method significantly suppresses the impact of the phase noise on the system,and the error floors can be decreased at the condition of 16 quadrature amplitude modulation(16QAM)and 32QAM.The proposed method can greatly improve the CO-OFDM system's ability to tolerate the wider laser linewidth.This method,compared to the linear interpolation sub-symbol common phase error compensation(LI-SCPEC)and Lagrange interpolation and extended Kalman filter(LRI-EKF)algorithms,has superior suppression effect.展开更多
In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs wit...In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.展开更多
The design synthesis is the key issue in the mechanical conceptual design to generate the design candidates that meet the design requirements.This paper devotes to propose a novel and computable synthesis approach of ...The design synthesis is the key issue in the mechanical conceptual design to generate the design candidates that meet the design requirements.This paper devotes to propose a novel and computable synthesis approach of mechanisms based on graph theory and polynomial operation.The graph framework of the synthesis approach is built firstly,and it involves:(1)the kinematic function units extracted from mechanisms;(2)the kinematic link graph that transforms the synthesis problem from mechanical domain into graph domain;(3)two graph representations,i.e.,walk representation and path representation,of design candidates;(4)a weighted matrix theorem that transforms the synthesis process into polynomial operation.Then,the formulas and algorithm to the polynomial operation are presented.Based on them,the computational flowchart to the synthesis approach is summarized.A design example is used to validate and illustrate the synthesis approach in detail.The proposed synthesis approach is not only supportive to enumerate the design candidates to the conceptual design of a mechanical system exhaustively and automatically,but also helpful to make that enumeration process computable.展开更多
In this article, we derive a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it f...In this article, we derive a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it for solving the non-stiff initial value problems, being the continuous interpolant derived and collocated at grid and off-grid points. Numerical examples of ordinary differential equations (ODEs) are solved using the proposed methods to show the validity and the accuracy of the introduced algorithms. A comparison with fourth-order Runge-Kutta method is given. The ob-tained numerical results reveal that the proposed method is efficient.展开更多
A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic...A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spherical but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (g<sub>e</sub> = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.展开更多
An M-metric least square method for polynomial analogy is presented. The relative normal eqUation is of diagonal form, such that the concise solution formula is explicit, and it is suitable to Parallel computation. On...An M-metric least square method for polynomial analogy is presented. The relative normal eqUation is of diagonal form, such that the concise solution formula is explicit, and it is suitable to Parallel computation. On the other hand, by error analysis of a typical example, we can see that the presented method is reliable.展开更多
文摘Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.
文摘In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.
文摘A new application of Chebyshev polynomials of second kind Un(x) to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows to reduce powers and smooth functions of them to superpositions of the first N-1 powers of the considered operator in N-dimensional case. The method leads in two-dimensional case first to the recurrence relations for Chebyshev polynomials and due to initial conditions to the application of Chebyshev polynomials of second kind Un(x). Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind Un(x) comprising the known Generating function as special cases is constructed by means of a derived identity for operator functions f(A) of a general two-dimensional operator A. The basic results are Formulas (9.5) and (9.6) which are then specialized for different examples of functions f(x). The generalization of the theory for three-dimensional operators is started to attack and a partial problem connected with the eigenvalue problem and the Hamilton-Cayley identity is solved in an Appendix. A physical application of Chebyshev polynomials to a problem of relativistic kinematics of a uniformly accelerated system is solved. All operator calculations are made in coordinate-invariant form.
文摘Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.
基金supported by National Natural Science Foundation of China(Nos.61662047 and 61462061).
文摘Based on compressive sensing and fractional discrete cosine transform(DCT)via polynomial interpolation(PI-FrDCT),an image encryption algorithm is proposed,in which the compression and encryption of an image are accomplished simultaneously.It can keep information secret more effectively with low data transmission.Three-dimensional piecewise and nonlinear chaotic maps are employed to obtain a generating sequence and the exclusive OR(XOR)matrix,which greatly enlarge the key space of the encryption system.Unlike many other fractional transforms,the output of PI-FrDCT is real,which facilitates the storage,transmission and display of the encrypted image.Due to the introduction of a plain-image-dependent disturbance factor,the initial values and system parameters of the encryption scheme are determined by cipher keys and plain-image.Thus,the proposed encryption scheme is very sensitive to the plain-image,which makes the encryption system more secure.Experimental results demonstrate the validity and the reliability of the proposed encryption algorithm.
基金the UGC,New Delhi,India for financial assistance via the UGC-Junior Research Fellowship(CSIR-UGC NET JULY 2024)(Student ID:241610090610)。
文摘The flow of a tetra-hybrid Casson nanofluid(Al_(2)O_(3)-CuO-TiO_(2)-Ag/H_(2)O)over a nonlinear stretching sheet is investigated.The Buongiorno model is used to account for thermophoresis and Brownian motion,while thermal radiation is incorporated to examine its influence on the thermal boundary layer.The governing partial differential equations(PDEs)are reduced to a system of nonlinear ordinary differential equations(ODEs)with fully non-dimensional similarity transformations involving all independent variables.To solve the obtained highly nonlinear system of differential equations,a novel Clique polynomial collocation method is applied.The analysis focuses on the effects of the Casson parameter,power index,radiation parameter,thermophoresis parameter,Brownian motion parameter,and Lewis number.The key findings show that thermal radiation intensifies the thermal boundary layer,the Casson parameter reduces the velocity,and the Lewis number suppresses the concentration with direct relevance to polymer processing,coating flows,electronic cooling,and biomedical applications.
基金the National Natural Science Foundation of China (Nos.11571238,11601332,91130014,11471312 and 91430216).
文摘Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.
文摘In this article we continue the consideration of geometrical constructions of regular n-gons for odd n by rhombic bicompasses and ruler used in [1] for the construction of the regular heptagon (n=7). We discuss the possible factorization of the cyclotomic polynomial in polynomial factors which contain not higher than quadratic radicals in the coefficients whereas usually the factorization of the cyclotomic polynomials is considered in products of irreducible factors with integer coefficients. In considering the regular heptagon we find a modified variant of its construction by rhombic bicompasses and ruler. In detail, supported by figures, we investigate the case of the regular tridecagon (n=13) which in addition to n=7 is the only candidate with low n (the next to this is n=769 ) for which such a construction by rhombic bicompasses and ruler seems to be possible. Besides the coordinate origin we find here two points to fix for the possible application of two bicompasses (or even four with the addition of the complex conjugate points to be fixed). With only one bicompass one has in addition the problem of the trisection of an angle which can be solved by a neusis construction that, however, is not in the spirit of constructions by compass and ruler and is difficult to realize during the action of bicompasses. As discussed it seems that to finish the construction by bicompasses the correlated action of two rhombic bicompasses must be applied in this case which avoids the disadvantages of the neusis construction. Single rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points on one circle determines the positions of all the other points on the second circle in unique way. The known case n=17 embedded in our method is discussed in detail.
文摘Computational techniques are invaluable to the continued success and development of Magnetic Resonance Imaging (MRI) and to its widespread applications. New processing methods are essential for addressing issues at each stage of MRI techniques. In this study, we present new sets of non-exponential generating functions representing the NMR transverse magnetizations and signals which are mathematically designed based on the theory and dynamics of the Bloch NMR flow equations. These signals are functions of many spinning nuclei of materials and can be used to obtain information observed in all flow systems. The Bloch NMR flow equations are solved using the Boubaker polynomial expansion scheme (BPES) and analytically connect most of the experimentally valuable NMR parameters in a simplified way for general analyses of magnetic resonance imaging with adiabatic condition.
文摘The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials.The fractional derivative is described in the Caputo sense.Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable.By solving the algebraic equations,the numerical solutions are acquired.The method in general is easy to implement and yields good results.Numerical examples are provided to demonstrate the validity and applicability of the method.
文摘Approximation of finite population totals in the presence of auxiliary information is considered. A polynomial based on Lagrange polynomial is proposed. Like the local polynomial regression, Horvitz Thompson and ratio estimators, this approximation technique is based on annual population total in order to fit in the best approximating polynomial within a given period of time (years) in this study. This proposed technique has shown to be unbiased under a linear polynomial. The use of real data indicated that the polynomial is efficient and can approximate properly even when the data is unevenly spaced.
文摘We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix.
基金supported by the National Natural Science Foundation of China(12061035)the Research Foundation of Jiangxi Science and Technology Normal University of China(2021QNBJRC003)supported by the Graduate Innovation Fund of Jiangxi Science and Technology Normal University(YC2024-X10).
文摘In this paper,the class of starlike functions of complex order γ(γ∈ℂ−{0})is extended from the case on unit disk U=(z∈C:|z|<1)to the case on the unit ball B in a complex Banach space or the unit polydisk U^(n) in C^(n).Let g be a convex function in U. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of g-parametric starlike mappings of complex order γ on B (resp.U^(n))when the mappings f are k-fold symmetric, k ∈ N. Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.
基金supported by the National Natural Science Foundation of China(Nos.U21A20447 and 61971079)。
文摘A novel suppression method of the phase noise is proposed to reduce the negative impacts of phase noise in coherent optical orthogonal frequency division multiplexing(CO-OFDM)systems.The method integrates the sub-symbol second-order polynomial interpolation(SSPI)with cubature Kalman filter(CKF)to improve the precision and effectiveness of the data processing through using a three-stage processing approach of phase noise.First of all,the phase noise values in OFDM symbols are calculated by using pilot symbols.Then,second-order Newton interpolation(SNI)is used in second-order interpolation to acquire precise noise estimation.Afterwards,every OFDM symbol is partitioned into several sub-symbols,and second-order polynomial interpolation(SPI)is utilized in the time domain to enhance suppression accuracy and time resolution.Ultimately,CKF is employed to suppress the residual phase noise.The simulation results show that this method significantly suppresses the impact of the phase noise on the system,and the error floors can be decreased at the condition of 16 quadrature amplitude modulation(16QAM)and 32QAM.The proposed method can greatly improve the CO-OFDM system's ability to tolerate the wider laser linewidth.This method,compared to the linear interpolation sub-symbol common phase error compensation(LI-SCPEC)and Lagrange interpolation and extended Kalman filter(LRI-EKF)algorithms,has superior suppression effect.
基金This work was supported by the Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment,Postdoctoral Startup Fund of Taiyuan University of Science and Technology(20152034)the Natural Science Foundation of Shanxi Province(201701D221135)National College Students Innovation and Entrepreneurship Project(201710109003)and(201610109007).
文摘In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.
基金Supported by State Key Program of National Natural Science Foundation of China(Grant No.51535009)111 Project of China(Grant No.B13044).
文摘The design synthesis is the key issue in the mechanical conceptual design to generate the design candidates that meet the design requirements.This paper devotes to propose a novel and computable synthesis approach of mechanisms based on graph theory and polynomial operation.The graph framework of the synthesis approach is built firstly,and it involves:(1)the kinematic function units extracted from mechanisms;(2)the kinematic link graph that transforms the synthesis problem from mechanical domain into graph domain;(3)two graph representations,i.e.,walk representation and path representation,of design candidates;(4)a weighted matrix theorem that transforms the synthesis process into polynomial operation.Then,the formulas and algorithm to the polynomial operation are presented.Based on them,the computational flowchart to the synthesis approach is summarized.A design example is used to validate and illustrate the synthesis approach in detail.The proposed synthesis approach is not only supportive to enumerate the design candidates to the conceptual design of a mechanical system exhaustively and automatically,but also helpful to make that enumeration process computable.
文摘In this article, we derive a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it for solving the non-stiff initial value problems, being the continuous interpolant derived and collocated at grid and off-grid points. Numerical examples of ordinary differential equations (ODEs) are solved using the proposed methods to show the validity and the accuracy of the introduced algorithms. A comparison with fourth-order Runge-Kutta method is given. The ob-tained numerical results reveal that the proposed method is efficient.
文摘A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spherical but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (g<sub>e</sub> = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.
文摘An M-metric least square method for polynomial analogy is presented. The relative normal eqUation is of diagonal form, such that the concise solution formula is explicit, and it is suitable to Parallel computation. On the other hand, by error analysis of a typical example, we can see that the presented method is reliable.
