In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality o...In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.展开更多
One of the recent advancements in the electrical power systems is the smart-grid technology.For the effective functioning of the smart grid,the process like monitoring and controlling have to be given importance.In th...One of the recent advancements in the electrical power systems is the smart-grid technology.For the effective functioning of the smart grid,the process like monitoring and controlling have to be given importance.In this paper,the Wireless Sensor Network(WSN)is utilized for tracking the power in smart grid applications.The smart grid is used to produce the electricity and it is connected with the sensor to transmit or receive the data.The data is transmitted quickly by using the Probabilistic Neural Network(PNN),which aids in identifying the shortest path of the nodes.While transmitting the data from the smart grid to the(Internet of Things)IoT web page,it is secured by introducing the secret keys between the neighbouring nodes through the process of key-management.In this method,the combination of Lagrange’s theorem and the Location Based Key(LBK)management is used for better security performance.This approach deli-vers optimal performance in terms of security,throughput,packet loss and delay,which are comparatively better than the existing methods.展开更多
Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysi...Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.展开更多
In recent decades,the cloud computing contributes a prominent role in health care sector as the patient health records are transferred and collected using cloud computing services.The doctors have switched to cloud co...In recent decades,the cloud computing contributes a prominent role in health care sector as the patient health records are transferred and collected using cloud computing services.The doctors have switched to cloud computing as it provides multiple advantageous measures including wide storage space and easy availability without any limitations.This necessitates the medical field to be redesigned by cloud technology to preserve information about patient’s critical diseases,electrocardiogram(ECG)reports,and payment details.The proposed work utilizes a hybrid cloud pattern to share Massachusetts Institute of Technology-Beth Israel Hospital(MIT-BIH)resources over the private and public cloud.The stored data are categorized as significant and non-significant by Artificial Neural Networks(ANN).The significant data undergoes encryption by Lagrange key management which automatically generates the key and stores it in the hidden layer.Upon receiving the request from a secondary user,the primary user verifies the authentication of the request and transmits the key via Gmail to the secondary user.Once the key matches the key in the hidden layer,the preserved information will be shared between the users.Due to the enhanced privacy preserving key generation,the proposed work prevents the tracking of keys by malicious users.The outcomes reveal that the introduced work provides improved success rate with reduced computational time.展开更多
We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural netw...We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.展开更多
A consistent focus in theoretical mechanics has been on how to apply Lagrange's equation to continuum mechanics.This paper uses the concept of a variational derivative and its laws of operation to investigate the ...A consistent focus in theoretical mechanics has been on how to apply Lagrange's equation to continuum mechanics.This paper uses the concept of a variational derivative and its laws of operation to investigate the derivation of Lagrange's equation,which is then applied to nonlinear elasto-dynamics.In accordance with the work-energy principle and the energy conservation law,kinetic and potential energies are proposed for rigid-elastic coupling dynamics,whose governing equation is established by manipulating Lagrange's equation.In addition,case studies are used to demonstrate the application of the proposed method to spacecraft dynamics.展开更多
To study a form invariance of Lagrange system, the form invariance of Lagrange equations under the infinitesimal transformations was used. The definition and criterion for the form invariance are given. The relati...To study a form invariance of Lagrange system, the form invariance of Lagrange equations under the infinitesimal transformations was used. The definition and criterion for the form invariance are given. The relation between the form invariance and the Noether symmetry was established.展开更多
An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） a...An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） are all real numbers has been presented here.展开更多
文摘In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.
文摘One of the recent advancements in the electrical power systems is the smart-grid technology.For the effective functioning of the smart grid,the process like monitoring and controlling have to be given importance.In this paper,the Wireless Sensor Network(WSN)is utilized for tracking the power in smart grid applications.The smart grid is used to produce the electricity and it is connected with the sensor to transmit or receive the data.The data is transmitted quickly by using the Probabilistic Neural Network(PNN),which aids in identifying the shortest path of the nodes.While transmitting the data from the smart grid to the(Internet of Things)IoT web page,it is secured by introducing the secret keys between the neighbouring nodes through the process of key-management.In this method,the combination of Lagrange’s theorem and the Location Based Key(LBK)management is used for better security performance.This approach deli-vers optimal performance in terms of security,throughput,packet loss and delay,which are comparatively better than the existing methods.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12272148 and 11772141).
文摘Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.
文摘In recent decades,the cloud computing contributes a prominent role in health care sector as the patient health records are transferred and collected using cloud computing services.The doctors have switched to cloud computing as it provides multiple advantageous measures including wide storage space and easy availability without any limitations.This necessitates the medical field to be redesigned by cloud technology to preserve information about patient’s critical diseases,electrocardiogram(ECG)reports,and payment details.The proposed work utilizes a hybrid cloud pattern to share Massachusetts Institute of Technology-Beth Israel Hospital(MIT-BIH)resources over the private and public cloud.The stored data are categorized as significant and non-significant by Artificial Neural Networks(ANN).The significant data undergoes encryption by Lagrange key management which automatically generates the key and stores it in the hidden layer.Upon receiving the request from a secondary user,the primary user verifies the authentication of the request and transmits the key via Gmail to the secondary user.Once the key matches the key in the hidden layer,the preserved information will be shared between the users.Due to the enhanced privacy preserving key generation,the proposed work prevents the tracking of keys by malicious users.The outcomes reveal that the introduced work provides improved success rate with reduced computational time.
基金supported by NSFC(No.11971296)National Key Research and Development Program of China(No.2021YFA1003004).
文摘We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.
基金supported by the National Natural Science Foundation of China(Grant No.10272034)
文摘A consistent focus in theoretical mechanics has been on how to apply Lagrange's equation to continuum mechanics.This paper uses the concept of a variational derivative and its laws of operation to investigate the derivation of Lagrange's equation,which is then applied to nonlinear elasto-dynamics.In accordance with the work-energy principle and the energy conservation law,kinetic and potential energies are proposed for rigid-elastic coupling dynamics,whose governing equation is established by manipulating Lagrange's equation.In addition,case studies are used to demonstrate the application of the proposed method to spacecraft dynamics.
文摘To study a form invariance of Lagrange system, the form invariance of Lagrange equations under the infinitesimal transformations was used. The definition and criterion for the form invariance are given. The relation between the form invariance and the Noether symmetry was established.
文摘An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） are all real numbers has been presented here.
