We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficien...We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.展开更多
In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to ...In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.展开更多
In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.? The usefu...In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.? The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.展开更多
In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivisio...In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivision curve C0 to C3 continuity and convergence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed. The systems of curve and surface design based on our scheme have been developed successfully in garment CAD especially for clothes modelling.展开更多
A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to in...A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to investigate the continuity of the scheme. The variety of effects can be achieved in correspondence for different values of parameter. The applications of the proposed scheme are illustrated in comparison with the established subdivision schemes.展开更多
A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Four...A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum展开更多
In this paper, we present an algorithm for reconstruction of B-spline surface such that it interpolates the four given bound- ary curves and simultaneously approximates some given inner points. The main idea of our me...In this paper, we present an algorithm for reconstruction of B-spline surface such that it interpolates the four given bound- ary curves and simultaneously approximates some given inner points. The main idea of our method is： first, we construct an initial surface which interpolates the four given boundary curves; then, while keeping the boundary control points of the initial surface un- changed, we reposition the inner control points of the surface with energy optimization method. Examples show that our algorithm is practicable and effective.展开更多
This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, ...This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, is obtained, where n is the number of parameters needed in the approximation. By means of the approximation, a learning rate of regularized least square algorithm with the Lipschitz kernel on the sphere is also deduced.展开更多
This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximat...This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximation effects than those of prevailing methods.In principle,the IBM can be applied for bounding more bounded smooth functions and their integrals as well,and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.展开更多
We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function and its Fourier transform. We show that this problem always has a solution within a spec...We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function and its Fourier transform. We show that this problem always has a solution within a specified degree of accuracy, provided the distributions satisfy the necessary regularity conditions. We describe the algorithm and construction of and provide examples of approximating several pairs of distributions using the algorithm.展开更多
In this paper, the author presents a class of stationary ternary 4-point approximating symmetrical subdivision algorithm that reproduces cubic polynomials. By these subdivision algorithms at each refinement step, new ...In this paper, the author presents a class of stationary ternary 4-point approximating symmetrical subdivision algorithm that reproduces cubic polynomials. By these subdivision algorithms at each refinement step, new insertion control points on a finer grid are computed by weighted sums of already existing control points. In the limit of the recursive process, data is defined on a dense set of point, The objective is to find an improved subdivision approximating algorithm which has a smaller support and a higher approximating order. The author chooses a ternary scheme because the best way to get a smaller support is to pass from the binary to ternary or complex algorithm and uses polynomial reproducing propriety to get higher approximation order. Using the cardinal Lagrange polynomials the author has proposed a 4-point approximating ternary subdivision algorithm and found that a higher regularity of limit function does not guarantee a higher approximating order. The proposed 4-point ternary approximation subdivision family algorithms with the mask a have the limit function in C2 and have approximation order 4. Also the author has demonstrated that in this class there is no algorithm whose limit function is in C3. It can be seen that this stationary ternary 4-point approximating symmetrical subdivision algorithm has a lower computational cost than the 6-point binary approximation subdivision algorithm for a greater range of points.展开更多
We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law.The dynamical systems under consideration ...We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law.The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations(PDE),which means that the Bellman equation suffers from the curse of dimensionality.Its non linearity is handled by the Policy Iteration algorithm,where the problem is reduced to a sequence of linear equations,which remain the computational bottleneck due to their high dimensions.We reformulate the linearized Bellman equations via the Koopman operator into an operator equation,that is solved using a minimal residual method.Using the Koopman operator we identify a preconditioner for operator equation,which deems essential in our numerical tests.To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats,in particular tensor trains(TT tensors)and multi-polynomials,together with high-dimensional quadrature,e.g.Monte-Carlo.By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.展开更多
Recently,one of the main challenges facing the smart grid is insufficient computing resources and intermittent energy supply for various distributed components(such as monitoring systems for renewable energy power sta...Recently,one of the main challenges facing the smart grid is insufficient computing resources and intermittent energy supply for various distributed components(such as monitoring systems for renewable energy power stations).To solve the problem,we propose an energy harvesting based task scheduling and resource management framework to provide robust and low-cost edge computing services for smart grid.First,we formulate an energy consumption minimization problem with regard to task offloading,time switching,and resource allocation for mobile devices,which can be decoupled and transformed into a typical knapsack problem.Then,solutions are derived by two different algorithms.Furthermore,we deploy renewable energy and energy storage units at edge servers to tackle intermittency and instability problems.Finally,we design an energy management algorithm based on sampling average approximation for edge computing servers to derive the optimal charging/discharging strategies,number of energy storage units,and renewable energy utilization.The simulation results show the efficiency and superiority of our proposed framework.展开更多
Wigner-Ville distribution(WVD)is widely used in the field of signal processing due to its excellent time-frequency(TF)concentration.However,WVD is severely limited by the cross-term when working with multicomponent si...Wigner-Ville distribution(WVD)is widely used in the field of signal processing due to its excellent time-frequency(TF)concentration.However,WVD is severely limited by the cross-term when working with multicomponent signals.In this paper,we analyze the property differences between auto-term and cross-term in the one-dimensional sequence and the two-dimensional plane and approximate entropy and Rényi entropy are employed to describe them,respectively.Based on this information,we propose a new method to achieve adaptive cross-term removal by combining seeded region growing.Compared to other methods,the new method can achieve cross-term removal without decreasing the TF concentration of the auto-term.Simulation and experimental data processing results show that the method is adaptive and is not constrained by the type or distribution of signals.And it performs well in low signal-to-noise ratio environments.展开更多
This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces.Moreov...This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces.Moreover,we establish the controllability of the considered system.To do so,first,we investigate the approximate controllability of the corresponding linear system.Subsequently,we prove the nonlinear system is approximately controllable if the corresponding linear system is approximately controllable.To reach the conclusions,the theory of resolvent operators,the Banach contraction mapping principle,and fixed point theorems are used.While concluding,some examples are given to demonstrate the efficacy of the proposed results.展开更多
This study directs the discussion of HIV disease with a novel kind of complex dynamical generalized and piecewise operator in the sense of classical and Atangana Baleanu(AB)derivatives having arbitrary order.The HIV i...This study directs the discussion of HIV disease with a novel kind of complex dynamical generalized and piecewise operator in the sense of classical and Atangana Baleanu(AB)derivatives having arbitrary order.The HIV infection model has a susceptible class,a recovered class,along with a case of infection divided into three sub-different levels or categories and the recovered class.The total time interval is converted into two,which are further investigated for ordinary and fractional order operators of the AB derivative,respectively.The proposed model is tested separately for unique solutions and existence on bi intervals.The numerical solution of the proposed model is treated by the piece-wise numerical iterative scheme of Newtons Polynomial.The proposed method is established for piece-wise derivatives under natural order and non-singular Mittag-Leffler Law.The cross-over or bending characteristics in the dynamical system of HIV are easily examined by the aspect of this research having a memory effect for controlling the said disease.This study uses the neural network(NN)technique to obtain a better set of weights with low residual errors,and the epochs number is considered 1000.The obtained figures represent the approximate solution and absolute error which are tested with NN to train the data accurately.展开更多
Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous ...Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous quantitative analyses often simplified the chorus dispersion relation by using the cold plasma assumption.However,the applicability of the cold plasma assumption is doubtful,especially during geomagnetic disturbances.We here present a systematic statistical analysis on the validity of the cold plasma dispersion relation of chorus waves based on observations from the Van Allen Probes over the period from 2012 to 2018.The statistical results show that the observed magnetic field intensities deviate substantially from those calculated from the cold plasma dispersion relation and that they become more pronounced with an increase in geomagnetic activity or a decrease in background plasma density.The region with large deviations is mainly concentrated in the nightside and expands in both the radial and azimuthal directions as the geomagnetic activity increases or the background plasma density decreases.In addition,the bounce-averaged electron scattering rates are computed by using the observed and cold plasma dispersion relation of chorus waves.Compared with usage of the cold plasma dispersion relation,usage of the observed dispersion relation considerably lowers the minimum resonant energy of electrons and lowers the scattering rates of electrons above tens of kiloelectronvolts but enhances those below.Furthermore,these differences are more pronounced with the enhancement of geomagnetic activity or the decrease in background plasma density.展开更多
In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to...In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.展开更多
LetΩbe homogeneous of degree zero,integrable on S^(d−1) and have vanishing moment of order one,a be a function on R^(d) such that ∇a∈L^(∞)(R^(d)).Let T*_(Ω,a) be the maximaloperator associated with the d-dimensional...LetΩbe homogeneous of degree zero,integrable on S^(d−1) and have vanishing moment of order one,a be a function on R^(d) such that ∇a∈L^(∞)(R^(d)).Let T*_(Ω,a) be the maximaloperator associated with the d-dimensional Calder´on commutator defined by T*_(Ωa)f(x):=sup_(ε>0)|∫_(|x-y|>ε)^Ω(x-y)/|x-y|^(d+1)(a(x)-a(y))f(y)dy.In this paper,the authors establish bilinear sparse domination for T*_(Ω,a) under the assumption Ω∈L∞(Sd−1).As applications,some quantitative weighted bounds for T*_(Ω,a) are obtained.展开更多
Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).展开更多
文摘We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.
文摘In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.
文摘In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.? The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.
基金Supported by the Indigenous PhD Scholarship Scheme of Higher Education Commission (HEC) Pakistan
文摘In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivision curve C0 to C3 continuity and convergence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed. The systems of curve and surface design based on our scheme have been developed successfully in garment CAD especially for clothes modelling.
文摘A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to investigate the continuity of the scheme. The variety of effects can be achieved in correspondence for different values of parameter. The applications of the proposed scheme are illustrated in comparison with the established subdivision schemes.
文摘A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum
基金Supported by the Natural Science Foundation of Hebei Province
文摘In this paper, we present an algorithm for reconstruction of B-spline surface such that it interpolates the four given bound- ary curves and simultaneously approximates some given inner points. The main idea of our method is： first, we construct an initial surface which interpolates the four given boundary curves; then, while keeping the boundary control points of the initial surface un- changed, we reposition the inner control points of the surface with energy optimization method. Examples show that our algorithm is practicable and effective.
基金Supported by the National Natural Science Foundation of China(61272023,91330118)
文摘This paper investigates some approximation properties and learning rates of Lipschitz kernel on the sphere. A perfect convergence rate on the shifts of Lipschitz kernel on the sphere, which is faster than O（n-1/2）, is obtained, where n is the number of parameters needed in the approximation. By means of the approximation, a learning rate of regularized least square algorithm with the Lipschitz kernel on the sphere is also deduced.
基金Supported by the National Natural Science Foundation of China(61672009,61502130).
文摘This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximation effects than those of prevailing methods.In principle,the IBM can be applied for bounding more bounded smooth functions and their integrals as well,and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.
文摘We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function and its Fourier transform. We show that this problem always has a solution within a specified degree of accuracy, provided the distributions satisfy the necessary regularity conditions. We describe the algorithm and construction of and provide examples of approximating several pairs of distributions using the algorithm.
文摘In this paper, the author presents a class of stationary ternary 4-point approximating symmetrical subdivision algorithm that reproduces cubic polynomials. By these subdivision algorithms at each refinement step, new insertion control points on a finer grid are computed by weighted sums of already existing control points. In the limit of the recursive process, data is defined on a dense set of point, The objective is to find an improved subdivision approximating algorithm which has a smaller support and a higher approximating order. The author chooses a ternary scheme because the best way to get a smaller support is to pass from the binary to ternary or complex algorithm and uses polynomial reproducing propriety to get higher approximation order. Using the cardinal Lagrange polynomials the author has proposed a 4-point approximating ternary subdivision algorithm and found that a higher regularity of limit function does not guarantee a higher approximating order. The proposed 4-point ternary approximation subdivision family algorithms with the mask a have the limit function in C2 and have approximation order 4. Also the author has demonstrated that in this class there is no algorithm whose limit function is in C3. It can be seen that this stationary ternary 4-point approximating symmetrical subdivision algorithm has a lower computational cost than the 6-point binary approximation subdivision algorithm for a greater range of points.
基金support from the Research Training Group“Differential Equation-and Data-driven Models in Life Sciences and Fluid Dynamics:An Interdisciplinary Research Training Group(DAEDALUS)”(GRK 2433)funded by the German Research Foundation(DFG).
文摘We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law.The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations(PDE),which means that the Bellman equation suffers from the curse of dimensionality.Its non linearity is handled by the Policy Iteration algorithm,where the problem is reduced to a sequence of linear equations,which remain the computational bottleneck due to their high dimensions.We reformulate the linearized Bellman equations via the Koopman operator into an operator equation,that is solved using a minimal residual method.Using the Koopman operator we identify a preconditioner for operator equation,which deems essential in our numerical tests.To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats,in particular tensor trains(TT tensors)and multi-polynomials,together with high-dimensional quadrature,e.g.Monte-Carlo.By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.
基金supported in part by the National Natural Science Foundation of China under Grant No.61473066in part by the Natural Science Foundation of Hebei Province under Grant No.F2021501020+2 种基金in part by the S&T Program of Qinhuangdao under Grant No.202401A195in part by the Science Research Project of Hebei Education Department under Grant No.QN2025008in part by the Innovation Capability Improvement Plan Project of Hebei Province under Grant No.22567637H
文摘Recently,one of the main challenges facing the smart grid is insufficient computing resources and intermittent energy supply for various distributed components(such as monitoring systems for renewable energy power stations).To solve the problem,we propose an energy harvesting based task scheduling and resource management framework to provide robust and low-cost edge computing services for smart grid.First,we formulate an energy consumption minimization problem with regard to task offloading,time switching,and resource allocation for mobile devices,which can be decoupled and transformed into a typical knapsack problem.Then,solutions are derived by two different algorithms.Furthermore,we deploy renewable energy and energy storage units at edge servers to tackle intermittency and instability problems.Finally,we design an energy management algorithm based on sampling average approximation for edge computing servers to derive the optimal charging/discharging strategies,number of energy storage units,and renewable energy utilization.The simulation results show the efficiency and superiority of our proposed framework.
基金Supported by the National Natural Science Foundation of China(62201171).
文摘Wigner-Ville distribution(WVD)is widely used in the field of signal processing due to its excellent time-frequency(TF)concentration.However,WVD is severely limited by the cross-term when working with multicomponent signals.In this paper,we analyze the property differences between auto-term and cross-term in the one-dimensional sequence and the two-dimensional plane and approximate entropy and Rényi entropy are employed to describe them,respectively.Based on this information,we propose a new method to achieve adaptive cross-term removal by combining seeded region growing.Compared to other methods,the new method can achieve cross-term removal without decreasing the TF concentration of the auto-term.Simulation and experimental data processing results show that the method is adaptive and is not constrained by the type or distribution of signals.And it performs well in low signal-to-noise ratio environments.
文摘This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces.Moreover,we establish the controllability of the considered system.To do so,first,we investigate the approximate controllability of the corresponding linear system.Subsequently,we prove the nonlinear system is approximately controllable if the corresponding linear system is approximately controllable.To reach the conclusions,the theory of resolvent operators,the Banach contraction mapping principle,and fixed point theorems are used.While concluding,some examples are given to demonstrate the efficacy of the proposed results.
基金supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University(IMSIU)(grant number IMSIU-RP23066).
文摘This study directs the discussion of HIV disease with a novel kind of complex dynamical generalized and piecewise operator in the sense of classical and Atangana Baleanu(AB)derivatives having arbitrary order.The HIV infection model has a susceptible class,a recovered class,along with a case of infection divided into three sub-different levels or categories and the recovered class.The total time interval is converted into two,which are further investigated for ordinary and fractional order operators of the AB derivative,respectively.The proposed model is tested separately for unique solutions and existence on bi intervals.The numerical solution of the proposed model is treated by the piece-wise numerical iterative scheme of Newtons Polynomial.The proposed method is established for piece-wise derivatives under natural order and non-singular Mittag-Leffler Law.The cross-over or bending characteristics in the dynamical system of HIV are easily examined by the aspect of this research having a memory effect for controlling the said disease.This study uses the neural network(NN)technique to obtain a better set of weights with low residual errors,and the epochs number is considered 1000.The obtained figures represent the approximate solution and absolute error which are tested with NN to train the data accurately.
基金supported by the National Natural Science Foundation of China (NSFC) through Grant Number 42074193
文摘Theoretical analysis has demonstrated that the dispersion relation of chorus waves plays an essential role in the resonant interaction and energy transformation between the waves and magnetospheric electrons.Previous quantitative analyses often simplified the chorus dispersion relation by using the cold plasma assumption.However,the applicability of the cold plasma assumption is doubtful,especially during geomagnetic disturbances.We here present a systematic statistical analysis on the validity of the cold plasma dispersion relation of chorus waves based on observations from the Van Allen Probes over the period from 2012 to 2018.The statistical results show that the observed magnetic field intensities deviate substantially from those calculated from the cold plasma dispersion relation and that they become more pronounced with an increase in geomagnetic activity or a decrease in background plasma density.The region with large deviations is mainly concentrated in the nightside and expands in both the radial and azimuthal directions as the geomagnetic activity increases or the background plasma density decreases.In addition,the bounce-averaged electron scattering rates are computed by using the observed and cold plasma dispersion relation of chorus waves.Compared with usage of the cold plasma dispersion relation,usage of the observed dispersion relation considerably lowers the minimum resonant energy of electrons and lowers the scattering rates of electrons above tens of kiloelectronvolts but enhances those below.Furthermore,these differences are more pronounced with the enhancement of geomagnetic activity or the decrease in background plasma density.
文摘In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.
文摘LetΩbe homogeneous of degree zero,integrable on S^(d−1) and have vanishing moment of order one,a be a function on R^(d) such that ∇a∈L^(∞)(R^(d)).Let T*_(Ω,a) be the maximaloperator associated with the d-dimensional Calder´on commutator defined by T*_(Ωa)f(x):=sup_(ε>0)|∫_(|x-y|>ε)^Ω(x-y)/|x-y|^(d+1)(a(x)-a(y))f(y)dy.In this paper,the authors establish bilinear sparse domination for T*_(Ω,a) under the assumption Ω∈L∞(Sd−1).As applications,some quantitative weighted bounds for T*_(Ω,a) are obtained.
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
