Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Alg...Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Algorithm to prove the stability and convergence of the approximations.展开更多
This article, in allusion to the limitation of conventional stellar horizon atmospheric refraction based on orbital dynamics model and nonlinear Kalman filter in practical applications, proposes a new celestial analyt...This article, in allusion to the limitation of conventional stellar horizon atmospheric refraction based on orbital dynamics model and nonlinear Kalman filter in practical applications, proposes a new celestial analytic positioning method by stellar horizon atmospheric refraction for high-altitude flight vehicles, such as spacecraft, airplanes and ballistic missiles. First, by setting up the geometric connexion among the flight vehicle, the Earth and the altitude of starlight refraction, an expression for t...展开更多
Nonlinear Jacobi iteration and nonlinear Gauss-Seidel iteration are proposed to solve the famous Numerov finite difference scheme for nonlinear two-points boundary value problem. The concept of supersolutions and subs...Nonlinear Jacobi iteration and nonlinear Gauss-Seidel iteration are proposed to solve the famous Numerov finite difference scheme for nonlinear two-points boundary value problem. The concept of supersolutions and subsolutions for nonlinear algebraic systems are introduced. By taking such solutions as initial values, the above two iterations provide monotone sequences, which fend to the solutions of Numerov scheme at geometric convergence rates. The global existence and uniqueness of solution of Numerov scheme are discussed also. The numerical results show the advantages of these two iterations.展开更多
The normal graded approximation and variable precision approximation are defined in approximate space. The relationship between graded approximation and variable precision approximation is studied, and an important fo...The normal graded approximation and variable precision approximation are defined in approximate space. The relationship between graded approximation and variable precision approximation is studied, and an important formula of conversion between them is achieved The product approximation of grade and precision is defined and its basic properties are studied.展开更多
Deep Neural Networks(DNNs)have become the tool of choice for machine learning practitioners today.One important aspect of designing a neural network is the choice of the activation function to be used at the neurons o...Deep Neural Networks(DNNs)have become the tool of choice for machine learning practitioners today.One important aspect of designing a neural network is the choice of the activation function to be used at the neurons of the different layers.In this work,we introduce a four-output activation function called the Reflected Rectified Linear Unit(RRe LU)activation which considers both a feature and its negation during computation.Our activation function is"sparse",in that only two of the four possible outputs are active at a given time.We test our activation function on the standard MNIST and CIFAR-10 datasets,which are classification problems,as well as on a novel Computational Fluid Dynamics(CFD)dataset which is posed as a regression problem.On the baseline network for the MNIST dataset,having two hidden layers,our activation function improves the validation accuracy from 0.09 to 0.97 compared to the well-known Re LU activation.For the CIFAR-10 dataset,we use a deep baseline network that achieves 0.78 validation accuracy with 20 epochs but overfits the data.Using the RRe LU activation,we can achieve the same accuracy without overfitting the data.For the CFD dataset,we show that the RRe LU activation can reduce the number of epochs from 100(using Re LU)to 10 while obtaining the same levels of performance.展开更多
L-band digital aeronautical communication system 1(L-DACS1) is a promising candidate data-link for future air-ground communication, but it is severely interfered by the pulse pairs(PPs) generated by distance measure e...L-band digital aeronautical communication system 1(L-DACS1) is a promising candidate data-link for future air-ground communication, but it is severely interfered by the pulse pairs(PPs) generated by distance measure equipment. A novel PP mitigation approach is proposed in this paper. Firstly, a deformed PP detection(DPPD) method that combines a filter bank, correlation detection, and rescanning is proposed to detect the deformed PPs(DPPs) which are caused by multiple filters in the receiver. Secondly, a finite impulse response(FIR) model is used to approximate the overall characteristic of filters, and then the waveform of DPP can be acquired by the original waveform of PP and the FIR model. Finally, sparse representation is used to estimate the position and amplitude of each DPP, and then reconstruct each DPP. The reconstructed DPPs will be subtracted from the contaminated signal to mitigate interference. Numerical experiments show that the bit error rate performance of our approach is about 5 dB better than that of recent works and is closer to interference-free environment.展开更多
The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation metho...The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.展开更多
The optical properties of α-BeH2 in an Orthorhombic crystal structure with the space group (Ibam) are investigated. We have calculated the optical properties including dielelectric function, refractive index and exti...The optical properties of α-BeH2 in an Orthorhombic crystal structure with the space group (Ibam) are investigated. We have calculated the optical properties including dielelectric function, refractive index and extinction coefficient, using density functional approach. A theoretical explanation of the relationship between the dielectric function and other optical constants has been provided. Furthermore, the real and imaginary components of the dielectric function have been examined. The effects of the exchange-correlation potentials (GGA and GGA + U) applied on this compound’s absorption peaks and edges have also been investigated. It was found that using the GGA + U approximation caused the conduction band to shift, which in turn caused the initial absorption peak to shift.展开更多
We analyze mathematical models governing planar flow of chemical reaction from unburnt gases to burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction are significant....We analyze mathematical models governing planar flow of chemical reaction from unburnt gases to burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction are significant. These models can be then formulated as the Navier-Stokes equations for exothermically reacting compressible fluids. We first establish the existence and dynamic behavior, including stability, regularity, and large-time behavior, of global discontinuous solutions of large oscillation to the Navier-Stokes equations with constant adiabatic exponent γ and specific heat Cv. Our approach for the existence and regularity is to combine the difference approximation techniques with the energy methods, total variation estimates, and weak convergence arguments to deal with large jump discontinuities; and for large-time behavior is an a posteriori argument directly from the weak form of the equations. The approach and ideas we develop here can be applied to solving a more complicated model where γand cv vary as the phase changes; and we then describe this model in detail and contrast the results on the asymptotic behavior of the solutions of these two different models. We also discuss other physical models describing dynamic combustion.展开更多
A new method for shape modification of non-uniform rational B-splines (NURBS) curves was presented, which was based on constrained optimization by means of altering the corresponding weights of their control points. U...A new method for shape modification of non-uniform rational B-splines (NURBS) curves was presented, which was based on constrained optimization by means of altering the corresponding weights of their control points. Using this method, the original NURBS curve was modified to satisfy the specified geometric constraints, including single point and multi-point constraints. With the introduction of free parameters, the shapes of modified NURBS curves could be further controlled by users without destroying geometric constraints and seem more naturally. Since explicit formulae were derived to compute new weights of the modified curve, the method was simple and easy to program. Practical examples showed that the method was applicable for computer aided design (CAD) system.展开更多
We use a combination of both algebraic and numerical techniques to construct a C-1-continuous, piecewise (m, n) rational epsilon-approximation of a real algebraic plane curve of degree d. At singular points we use the...We use a combination of both algebraic and numerical techniques to construct a C-1-continuous, piecewise (m, n) rational epsilon-approximation of a real algebraic plane curve of degree d. At singular points we use the classical Weierstrass Preparation Theorem and Newton power series factorizations, based on the technique of Hensel lifting. These, together with modified rational Pade approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve. Besides singular points we obtain an adaptive selection of simple points about which the curve approximations yield a small number of pieces yet achieve C-1 continuity between pieces. The simpler cases of C-1 and C-0 continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.展开更多
Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, co...Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.展开更多
The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R~2 and it is well-known that by using linear combinations of these...The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R~2 and it is well-known that by using linear combinations of these basic estimates,modern extrapolation techniques can greatly speed up the approximation process.Similarly,when n vertices are randomly selected on the circle,the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to π almost surely as n→∞,and by further applying extrapolation processes,faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons.In this paper,we further develop nonlinear extrapolation methods for approximating π through certain nonlinear functions of the semiperimeter and area of such polygons.We focus on two types of extrapolation estimates of the forms χ_n=S_n~αA_n~β and Y_n(p)=(αS_n~p+βA_n~p)~(1/p) where α+β=1,p≠0,and Sn and An respectively represents the semiperimeter and area of a random n-gon inscribed in the unit circle in R~2,and Xn may be viewed as the limit of Y_n(p) when p→0.By deriving probabilistic asymptotic expansions with carefully controlled error estimates for Xn and Y_n(p),we show that the choice α=4/3,β=-1/3 minimizes the approximation error in both cases,and their distributions are also asymptotically normal.展开更多
This paper studies a class of impulsive neutral stochastic partial differential equations in real Hilbert spaces.The main goal here is to consider the Trotter-Kato approximations of mild solutions of such equations in...This paper studies a class of impulsive neutral stochastic partial differential equations in real Hilbert spaces.The main goal here is to consider the Trotter-Kato approximations of mild solutions of such equations in the pth-mean(p≥2).As an application,a classical limit theorem on the dependence of such equations on a parameter is obtained.The novelty of this paper is that the combination of this approximating system and such equations has not been considered before.展开更多
文摘Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Algorithm to prove the stability and convergence of the approximations.
文摘This article, in allusion to the limitation of conventional stellar horizon atmospheric refraction based on orbital dynamics model and nonlinear Kalman filter in practical applications, proposes a new celestial analytic positioning method by stellar horizon atmospheric refraction for high-altitude flight vehicles, such as spacecraft, airplanes and ballistic missiles. First, by setting up the geometric connexion among the flight vehicle, the Earth and the altitude of starlight refraction, an expression for t...
文摘Nonlinear Jacobi iteration and nonlinear Gauss-Seidel iteration are proposed to solve the famous Numerov finite difference scheme for nonlinear two-points boundary value problem. The concept of supersolutions and subsolutions for nonlinear algebraic systems are introduced. By taking such solutions as initial values, the above two iterations provide monotone sequences, which fend to the solutions of Numerov scheme at geometric convergence rates. The global existence and uniqueness of solution of Numerov scheme are discussed also. The numerical results show the advantages of these two iterations.
基金supported by National Natural Science Foundation of China (Grant No.10971072)Guangdong Provincial Natural Science Foundation (Grant No.8151027501000114)
文摘Let a≥1 be an integer.In this paper,we will prove the equation in the title has at most three positive integer solutions.
基金Supported by the National Natural Science Foundation of China (No. 69803007)
文摘The normal graded approximation and variable precision approximation are defined in approximate space. The relationship between graded approximation and variable precision approximation is studied, and an important formula of conversion between them is achieved The product approximation of grade and precision is defined and its basic properties are studied.
文摘Deep Neural Networks(DNNs)have become the tool of choice for machine learning practitioners today.One important aspect of designing a neural network is the choice of the activation function to be used at the neurons of the different layers.In this work,we introduce a four-output activation function called the Reflected Rectified Linear Unit(RRe LU)activation which considers both a feature and its negation during computation.Our activation function is"sparse",in that only two of the four possible outputs are active at a given time.We test our activation function on the standard MNIST and CIFAR-10 datasets,which are classification problems,as well as on a novel Computational Fluid Dynamics(CFD)dataset which is posed as a regression problem.On the baseline network for the MNIST dataset,having two hidden layers,our activation function improves the validation accuracy from 0.09 to 0.97 compared to the well-known Re LU activation.For the CIFAR-10 dataset,we use a deep baseline network that achieves 0.78 validation accuracy with 20 epochs but overfits the data.Using the RRe LU activation,we can achieve the same accuracy without overfitting the data.For the CFD dataset,we show that the RRe LU activation can reduce the number of epochs from 100(using Re LU)to 10 while obtaining the same levels of performance.
基金supported in part by the National Natural Science Foundation (Nos. U1533107 and U1433105)the Civil Aviation Science and Technology Innovation Foundation (No. MHRD20130217)the Fundamental Research Funds for the Central Universities of CAUC (No. 3122016D003)
文摘L-band digital aeronautical communication system 1(L-DACS1) is a promising candidate data-link for future air-ground communication, but it is severely interfered by the pulse pairs(PPs) generated by distance measure equipment. A novel PP mitigation approach is proposed in this paper. Firstly, a deformed PP detection(DPPD) method that combines a filter bank, correlation detection, and rescanning is proposed to detect the deformed PPs(DPPs) which are caused by multiple filters in the receiver. Secondly, a finite impulse response(FIR) model is used to approximate the overall characteristic of filters, and then the waveform of DPP can be acquired by the original waveform of PP and the FIR model. Finally, sparse representation is used to estimate the position and amplitude of each DPP, and then reconstruct each DPP. The reconstructed DPPs will be subtracted from the contaminated signal to mitigate interference. Numerical experiments show that the bit error rate performance of our approach is about 5 dB better than that of recent works and is closer to interference-free environment.
文摘The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.
文摘The optical properties of α-BeH2 in an Orthorhombic crystal structure with the space group (Ibam) are investigated. We have calculated the optical properties including dielelectric function, refractive index and extinction coefficient, using density functional approach. A theoretical explanation of the relationship between the dielectric function and other optical constants has been provided. Furthermore, the real and imaginary components of the dielectric function have been examined. The effects of the exchange-correlation potentials (GGA and GGA + U) applied on this compound’s absorption peaks and edges have also been investigated. It was found that using the GGA + U approximation caused the conduction band to shift, which in turn caused the initial absorption peak to shift.
基金Supported in part by the National Science Foundation under Grants DMS-9971793, INT-9987378,and INT-9726215.Supported in part by the National Science Foundation under Grant DMS-9703703.Supported in part by the National Science Foundation under Grants
文摘We analyze mathematical models governing planar flow of chemical reaction from unburnt gases to burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction are significant. These models can be then formulated as the Navier-Stokes equations for exothermically reacting compressible fluids. We first establish the existence and dynamic behavior, including stability, regularity, and large-time behavior, of global discontinuous solutions of large oscillation to the Navier-Stokes equations with constant adiabatic exponent γ and specific heat Cv. Our approach for the existence and regularity is to combine the difference approximation techniques with the energy methods, total variation estimates, and weak convergence arguments to deal with large jump discontinuities; and for large-time behavior is an a posteriori argument directly from the weak form of the equations. The approach and ideas we develop here can be applied to solving a more complicated model where γand cv vary as the phase changes; and we then describe this model in detail and contrast the results on the asymptotic behavior of the solutions of these two different models. We also discuss other physical models describing dynamic combustion.
文摘A new method for shape modification of non-uniform rational B-splines (NURBS) curves was presented, which was based on constrained optimization by means of altering the corresponding weights of their control points. Using this method, the original NURBS curve was modified to satisfy the specified geometric constraints, including single point and multi-point constraints. With the introduction of free parameters, the shapes of modified NURBS curves could be further controlled by users without destroying geometric constraints and seem more naturally. Since explicit formulae were derived to compute new weights of the modified curve, the method was simple and easy to program. Practical examples showed that the method was applicable for computer aided design (CAD) system.
文摘We use a combination of both algebraic and numerical techniques to construct a C-1-continuous, piecewise (m, n) rational epsilon-approximation of a real algebraic plane curve of degree d. At singular points we use the classical Weierstrass Preparation Theorem and Newton power series factorizations, based on the technique of Hensel lifting. These, together with modified rational Pade approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve. Besides singular points we obtain an adaptive selection of simple points about which the curve approximations yield a small number of pieces yet achieve C-1 continuity between pieces. The simpler cases of C-1 and C-0 continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.
基金Supported by the National Natural Science Foundation of China (No.69803007)
文摘Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.
基金supported in part by the National Natural Science Foundation of China (No.12131003)。
文摘The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R~2 and it is well-known that by using linear combinations of these basic estimates,modern extrapolation techniques can greatly speed up the approximation process.Similarly,when n vertices are randomly selected on the circle,the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to π almost surely as n→∞,and by further applying extrapolation processes,faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons.In this paper,we further develop nonlinear extrapolation methods for approximating π through certain nonlinear functions of the semiperimeter and area of such polygons.We focus on two types of extrapolation estimates of the forms χ_n=S_n~αA_n~β and Y_n(p)=(αS_n~p+βA_n~p)~(1/p) where α+β=1,p≠0,and Sn and An respectively represents the semiperimeter and area of a random n-gon inscribed in the unit circle in R~2,and Xn may be viewed as the limit of Y_n(p) when p→0.By deriving probabilistic asymptotic expansions with carefully controlled error estimates for Xn and Y_n(p),we show that the choice α=4/3,β=-1/3 minimizes the approximation error in both cases,and their distributions are also asymptotically normal.
基金Supported by the National Natural Science Foundation of China(Grant No.12171361)the Humanity and Social Science Youth foundation of Ministry of Education(Grant No.20YJC790174)。
文摘This paper studies a class of impulsive neutral stochastic partial differential equations in real Hilbert spaces.The main goal here is to consider the Trotter-Kato approximations of mild solutions of such equations in the pth-mean(p≥2).As an application,a classical limit theorem on the dependence of such equations on a parameter is obtained.The novelty of this paper is that the combination of this approximating system and such equations has not been considered before.
