Let A be a subalgebra of B（X） containing the identity operator I and an idempotent P. Suppose that α,β： A →A are ring epimorphisms and there exists some nest N on 2（ such that α（P）（X） and β（P）（X） are ...Let A be a subalgebra of B（X） containing the identity operator I and an idempotent P. Suppose that α,β： A →A are ring epimorphisms and there exists some nest N on 2（ such that α（P）（X） and β（P）（X） are non-trivial elements of N. Let A contain all rank one operators in AlgN and δ ： A→ B（X） be an additive mapping. It is shown that, if δ is （α, β）-derivable at zero point, then there exists an additive （α, β）-derivation τ ： A →β（X） such that δ（A） =τ（A） ＋ α（A）δ（I） for all A∈A. It is also shown that if δ is generalized （α,β）-derivable at zero point, then δ is an additive generalized （α, β）-derivation. Moreover, by use of this result, the additive maps （generalized） （α,β）-derivable at zero point on several nest algebras, are also characterized.展开更多
Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained. These relatio...Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained. These relations and expressions play important roles in the meshless finite point method.展开更多
Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problem...Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC(q, N) is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization(PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC(q, N) model is used to predict the municipal solid waste(MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model.展开更多
Cyber-physical systems(CPSs) are man-made complex systems coupled with natural processes that, as a whole,should be described by distributed parameter systems(DPSs) in general forms. This paper presents three such gen...Cyber-physical systems(CPSs) are man-made complex systems coupled with natural processes that, as a whole,should be described by distributed parameter systems(DPSs) in general forms. This paper presents three such general models for generalized DPSs that can be used to characterize complex CPSs. These three different types of fractional operators based DPS models are: fractional Laplacian operator, fractional power of operator or fractional derivative. This research investigation is motivated by many fractional order models describing natural, physical, and anomalous phenomena, such as sub-diffusion process or super-diffusion process. The relationships among these three different operators are explored and explained. Several potential future research opportunities are then articulated followed by some conclusions and remarks.展开更多
This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries(KdV)equa-tion has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acous...This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries(KdV)equa-tion has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acoustic waves in plasma.Indeed,the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method(HPTM)to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness.This approach is connected with the fuzzy generalized integral transform and HPTM.Besides that,two novel results for fuzzy generalized integral transforma-tion concerning fuzzy partial gH-derivatives are presented.Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method.Furthermore,2D and 3D simulations de-pict the comparison analysis between two fractional derivative operators(Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense)under generalized gH-differentiability.The projected method(GHPTM)demonstrates a diverse spectrum of applications for dealing with nonlinear wave equa-tions in scientific domains.The current work,as a novel use of GHPTM,demonstrates some key differ-ences from existing similar methods.展开更多
After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organiz...After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organize homogeneous divisions of the limited development of the exponential function, that is opening the way to the use of a whole bunch of new primary functions in Differential Calculus. He then shows how new supercomplex products in dimension 3 make it possible to calculate fractals whose connexity depends on the product considered. We recall the geometry of convex polygons and regular polygons.展开更多
文摘Let A be a subalgebra of B（X） containing the identity operator I and an idempotent P. Suppose that α,β： A →A are ring epimorphisms and there exists some nest N on 2（ such that α（P）（X） and β（P）（X） are non-trivial elements of N. Let A contain all rank one operators in AlgN and δ ： A→ B（X） be an additive mapping. It is shown that, if δ is （α, β）-derivable at zero point, then there exists an additive （α, β）-derivation τ ： A →β（X） such that δ（A） =τ（A） ＋ α（A）δ（I） for all A∈A. It is also shown that if δ is generalized （α,β）-derivable at zero point, then δ is an additive generalized （α, β）-derivation. Moreover, by use of this result, the additive maps （generalized） （α,β）-derivable at zero point on several nest algebras, are also characterized.
基金Supported by the National Natural Science Foundation of China 1060100910701014+1 种基金10871029)the Foundation of China Academy of Engineering Physics (2007B09008)
文摘Relations of the 3D multi-directional derivatives are studied in this paper. These relations are applied to a geeral second-order linear elliptical operator and the corresponding expression are obtained. These relations and expressions play important roles in the meshless finite point method.
基金supported by the National Natural Science Foundation of China (51479151,61403288)。
文摘Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC(q, N) is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization(PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC(q, N) model is used to predict the municipal solid waste(MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model.
基金supported by Chinese Universities Scientific Fund(CUSF-DHD-2014061)Natural Science Foundation of Shanghai(15ZR1400800)
文摘Cyber-physical systems(CPSs) are man-made complex systems coupled with natural processes that, as a whole,should be described by distributed parameter systems(DPSs) in general forms. This paper presents three such general models for generalized DPSs that can be used to characterize complex CPSs. These three different types of fractional operators based DPS models are: fractional Laplacian operator, fractional power of operator or fractional derivative. This research investigation is motivated by many fractional order models describing natural, physical, and anomalous phenomena, such as sub-diffusion process or super-diffusion process. The relationships among these three different operators are explored and explained. Several potential future research opportunities are then articulated followed by some conclusions and remarks.
文摘This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries(KdV)equa-tion has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acoustic waves in plasma.Indeed,the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method(HPTM)to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness.This approach is connected with the fuzzy generalized integral transform and HPTM.Besides that,two novel results for fuzzy generalized integral transforma-tion concerning fuzzy partial gH-derivatives are presented.Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method.Furthermore,2D and 3D simulations de-pict the comparison analysis between two fractional derivative operators(Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense)under generalized gH-differentiability.The projected method(GHPTM)demonstrates a diverse spectrum of applications for dealing with nonlinear wave equa-tions in scientific domains.The current work,as a novel use of GHPTM,demonstrates some key differ-ences from existing similar methods.
文摘After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organize homogeneous divisions of the limited development of the exponential function, that is opening the way to the use of a whole bunch of new primary functions in Differential Calculus. He then shows how new supercomplex products in dimension 3 make it possible to calculate fractals whose connexity depends on the product considered. We recall the geometry of convex polygons and regular polygons.
