As the development of smart grid and energy internet, this leads to a significantincrease in the amount of data transmitted in real time. Due to the mismatch withcommunication networks that were not designed to carry ...As the development of smart grid and energy internet, this leads to a significantincrease in the amount of data transmitted in real time. Due to the mismatch withcommunication networks that were not designed to carry high-speed and real time data,data losses and data quality degradation may happen constantly. For this problem,according to the strong spatial and temporal correlation of electricity data which isgenerated by human’s actions and feelings, we build a low-rank electricity data matrixwhere the row is time and the column is user. Inspired by matrix decomposition, we dividethe low-rank electricity data matrix into the multiply of two small matrices and use theknown data to approximate the low-rank electricity data matrix and recover the missedelectrical data. Based on the real electricity data, we analyze the low-rankness of theelectricity data matrix and perform the Matrix Decomposition-based method on the realdata. The experimental results verify the efficiency and efficiency of the proposed scheme.展开更多
QR and LU decompositions are the most important matrix decomposition algorithms. Many studies work on accelerating these algorithms by FPGA or ASIC in a case by case style. In this paper, we propose a unified framewor...QR and LU decompositions are the most important matrix decomposition algorithms. Many studies work on accelerating these algorithms by FPGA or ASIC in a case by case style. In this paper, we propose a unified framework for the matrix decomposition algorithms, combining three QR decomposition algorithms and LU algorithm with pivoting into a unified linear array structure. The QR and LU decomposition algorithms exhibit the same two-level loop structure and the same data dependency. Utilizing the similarities in loop structure and data dependency of matrix decomposition, we unify a fine-grained algorithm for all four matrix decomposition algorithms. Furthermore, we present a unified co-processor structure with a scalable linear array of processing elements （PEs）, in which four types of PEs are same in the structure of memory channels and PE connections, but the only difference exists in the internal structure of data path. Our unified co-processor, which is IEEE 32-bit floating-point precision, is implemented and mapped onto a Xilinx Virtex5 FPGA chip. Experimental results show that our co-processors can achieve speedup of 2.3 to 14.9 factors compared to a Pentium Dual CPU with double SSE threads.展开更多
To fully display the modeling mechanism of the novelfractional order grey model （FGM （q,1））, this paper decomposesthe data matrix of the model into the mean generation matrix, theaccumulative generation matrix and...To fully display the modeling mechanism of the novelfractional order grey model （FGM （q,1））, this paper decomposesthe data matrix of the model into the mean generation matrix, theaccumulative generation matrix and the raw data matrix, whichare consistent with the fractional order accumulative grey model（FAGM （1,1））. Following this, this paper decomposes the accumulativedata difference matrix into the accumulative generationmatrix, the q-order reductive accumulative matrix and the rawdata matrix, and then combines the least square method, findingthat the differential order affects the model parameters only byaffecting the formation of differential sequences. This paper thensummarizes matrix decomposition of some special sequences,such as the sequence generated by the strengthening and weakeningoperators, the jumping sequence, and the non-equidistancesequence. Finally, this paper expresses the influences of the rawdata transformation, the accumulation sequence transformation,and the differential matrix transformation on the model parametersas matrices, and takes the non-equidistance sequence as an exampleto show the modeling mechanism.展开更多
Advances in quantum computation threaten to break public key eryptosystems that are based on the difficulty of fac- torization or the difficulty of discrete logariths, although , no quantum algorithms have been found ...Advances in quantum computation threaten to break public key eryptosystems that are based on the difficulty of fac- torization or the difficulty of discrete logariths, although , no quantum algorithms have been found to be able to solve certain mathematical problems on non-commutative algebraic structures up to now. The proposed new quasi-inverse based cryptography scheme is vulnerable to a linear algebra attack based on the probable occurrence of weak keys in the generation process. In this paper, we illustrate that two of the quasi-inverse based cryptography are vulnerable to a structural attack and that it only requires polynomial time to obtain the equivalent keys for some given public keys. In addition, we conduct a detailed analysis on attack methods and provide some improved suggestions on these two schemes.展开更多
Under certain load pattern, the geometrically indeterminate pin-jointed mechanisms will present certain shapes to keep static equalization. This paper proposes a matrix-based method to determine the mobility and equil...Under certain load pattern, the geometrically indeterminate pin-jointed mechanisms will present certain shapes to keep static equalization. This paper proposes a matrix-based method to determine the mobility and equilibrium stability of mechanisms according to the effects of the external loads. The first and second variations of the potential energy function of mechanisms under conservative force field are analyzed. Based on the singular value decomposition (SVD) method, a new crite- rion for the mobility and equilibrium stability of mechanisms can be concluded by analyzing the equilibrium matrix. The mobility and stability of mechanisms can be classified by unified matrix formulae. A number of examples are given to demonstrate the proposed criterion. In the end, criteria are summarized in a table.展开更多
With the popularity of deep learning tools in image decomposition and natural language processing,how to support and store a large number of parameters required by deep learning algorithms has become an urgent problem...With the popularity of deep learning tools in image decomposition and natural language processing,how to support and store a large number of parameters required by deep learning algorithms has become an urgent problem to be solved.These parameters are huge and can be as many as millions.At present,a feasible direction is to use the sparse representation technique to compress the parameter matrix to achieve the purpose of reducing parameters and reducing the storage pressure.These methods include matrix decomposition and tensor decomposition.To let vector take advance of the compressing performance of matrix decomposition and tensor decomposition,we use reshaping and unfolding to let vector be the input and output of Tensor-Factorized Neural Networks.We analyze how reshaping can get the best compress ratio.According to the relationship between the shape of tensor and the number of parameters,we get a lower bound of the number of parameters.We take some data sets to verify the lower bound.展开更多
The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersi...The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.展开更多
In this paper, the synthesis and implementation of three-qubit SWAP gate is discussed. The three-qubit SWAP gate can be decomposed into product of 2 two-qubit SWAP gates, and it can be realized by 6 CNOT gates. Resear...In this paper, the synthesis and implementation of three-qubit SWAP gate is discussed. The three-qubit SWAP gate can be decomposed into product of 2 two-qubit SWAP gates, and it can be realized by 6 CNOT gates. Research illustrated that although the result is very simple, the current methods of matrix decomposition for multi-qubit gate can not get that. Then the implementation of three-qubit SWAP gate in the three spin system with Ising interaction is investigated and the sequence of control pulse and drift process to implement the gate is given. It needs 23 control pulses and 12 drift processes. Since the interaction can not be switched on and off at will, the realization of three-qubit SWAP gate in specific quantum system also can not simply come down to 2 two-qubit SWAP gates.展开更多
We have recently proposed an optical method for assessing heart structure that uses polarized light measurement of birefringence as an indicator of tissue anisotropy.The highly aligned nature of healthy cardiac muscle...We have recently proposed an optical method for assessing heart structure that uses polarized light measurement of birefringence as an indicator of tissue anisotropy.The highly aligned nature of healthy cardiac muscle tissue has a detectable effect on the polarization of light,resulting in a measurable phase shift(“retardance”).When this organized tissue structure is perturbed,for example after cardiac infarction(heart attack),scar tissue containing disorganized collagen is formed,causing a decrease in the measured retardance values.However,these are dependent not only on tissue anisotropy,but also on the angle between the tissue’s optical anisotropy direction and the beam interrogating the sample.To remove this experimental ambiguity,we present a method that interrogates the sample at two different incident beam angles,thus yielding enough information to uniquely determine the true magnitude and orientation of the tissue optical anisotropy.We use an infarcted porcine heart model to compare these polarimetryderived anisotropy metrics with those obtained with diffusion tensor magnetic resonance imaging(DT-MRI).The latter yields the anisotropy and the direction of tissue water diffusivity,providing an independent measure of tissue anisotropy.The optical and MR results are thus directly compared in a common ex vivo biological model of interest,yielding reasonable agreement but also highlighting some technique-specific differences.展开更多
This paper is concerned with the event-triggered control of positive semi-Markovian jump systems without/with input saturation.The considered systems are subject to a stochastic semi-Markovian process whose sojourn ti...This paper is concerned with the event-triggered control of positive semi-Markovian jump systems without/with input saturation.The considered systems are subject to a stochastic semi-Markovian process whose sojourn time is dependent on a non-exponential distribution.First,an event-triggering condition is introduced in a linear form for the systems.A class of event-triggered feedback controllers is proposed using matrix decomposition technique.By using a stochastic co-positive Lyapunov function,the systems’positivity and stability are guaranteed.Then,the obtained results are developed for the systems with input saturation.A cone set is chosen as the attraction domain and the corresponding attraction domain gain matrix is designed in terms of standard linear programming approach.Finally,two numerical examples are provided to verify the validity and effectiveness of the presented theoretical findings.展开更多
It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert time-domain problems into frequency-domain problems. Are there similar tools for a matrix...It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert time-domain problems into frequency-domain problems. Are there similar tools for a matrix？ By pairing a matrix to a piecewise function,a Haar-like wavelet is used to set up a similar tool for matrix analyzing, resulting in new methods for matrix approximation and orthogonal decomposition. By using our method, one can approximate a matrix by matrices with different orders. Our method also results in a new matrix orthogonal decomposition, reproducing Haar transformation for matrices with orders of powers of two. The computational complexity of the new orthogonal decomposition is linear. That is, for an m × n matrix, the computational complexity is O（mn）. In addition,when the method is applied to k-means clustering, one can obtain that k-means clustering can be equivalently converted to the problem of finding a best approximation solution of a function. In fact, the results in this paper could be applied to any matrix related problems.In addition, one can also employ other wavelet transformations and Fourier transformation to obtain similar results.展开更多
The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equ...The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters＇ values. It makes the equation＇s solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.展开更多
A noise-reduction method with sliding called the local f-x Cadzow noise-reduction method, windows in the frequency-space （f-x） domain, is presented in this paper. This method is based on the assumption that the sign...A noise-reduction method with sliding called the local f-x Cadzow noise-reduction method, windows in the frequency-space （f-x） domain, is presented in this paper. This method is based on the assumption that the signal in each window is linearly predictable in the spatial direction while the random noise is not. For each Toeplitz matrix constructed by constant frequency slice, a singular value decomposition （SVD） is applied to separate signal from noise. To avoid edge artifacts caused by zero percent overlap between windows and to remove more noise, an appropriate overlap is adopted. Besides flat and dipping events, this method can enhance curved and conflicting events. However, it is not suitable for seismic data that contains big spikes or null traces. It is also compared with the SVD, f-x deconvolution, and Cadzow method without windows. The comparison results show that the local Cadzow method performs well in removing random noise and preserving signal. In addition, a real data example proves that it is a potential noise-reduction technique for seismic data obtained in areas of complex formations.展开更多
In the teaching and researching of linear regression analysis, it is interesting and enlightening to explore how the dependent variable vector can be inner-transformed into regression coefficient estimator vector from...In the teaching and researching of linear regression analysis, it is interesting and enlightening to explore how the dependent variable vector can be inner-transformed into regression coefficient estimator vector from a visible geometrical view. As an example, the roadmap of such inner transformation is presented based on a simple multiple linear regression model in this work. By applying the matrix algorithms like singular value decomposition (SVD) and Moore-Penrose generalized matrix inverse, the dependent variable vector lands into the right space of the independent variable matrix and is metamorphosed into regression coefficient estimator vector through the three-step of inner transformation. This work explores the geometrical relationship between the dependent variable vector and regression coefficient estimator vector as well as presents a new approach for vector rotating.展开更多
Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics...Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.展开更多
The implementation of product development process management (PDPM) is an effective means of developing products with higher quality in shorter lead time. It is argued in this paper that product, data, person and acti...The implementation of product development process management (PDPM) is an effective means of developing products with higher quality in shorter lead time. It is argued in this paper that product, data, person and activity are basic factors in PDPM With detailed analysis of these basic factors and their relations in product developmed process, all product development activities are considered as tasks and the management of product development process is regarded as the management of task execution A task decomposition based product development model is proposed with methods of constructing task relation matrix from layer model and constraint model resulted from task decomposition. An algorithm for constructing directed task graph is given and is used in the management of tasks. Finally, the usage and limitation of the proposed PDPM model is given with further work proposed.展开更多
Calculation of eigen-solutions plays an important role in the small signal stability analysis of power systems.In this paper,a novel approach based on matrix perturbation theory is proposed for the calculation of eige...Calculation of eigen-solutions plays an important role in the small signal stability analysis of power systems.In this paper,a novel approach based on matrix perturbation theory is proposed for the calculation of eigen-solutions in a perturbed system.Rigorous theoretical analysis is conducted on the solution of distinct,multiple,and close eigen-solutions,respectively,under perturbations of parameters.The computational flowchart of the unified solution of eigen-solutions is then proposed,aimed toward obtaining eigen-solutions of a perturbed system directly with algebraic formulas without solving an eigenvalue problem repeatedly.Finally,the effectiveness of the matrix perturbation based approach for eigen-solutions’calculation in power systems is verified by numerical examples on a two-area four-machine system.展开更多
Knowledge graph embedding, which maps the entities and relations into low-dimensional vector spaces, has demonstrated its effectiveness in many tasks such as link prediction and relation extraction. Typical methods in...Knowledge graph embedding, which maps the entities and relations into low-dimensional vector spaces, has demonstrated its effectiveness in many tasks such as link prediction and relation extraction. Typical methods include TransE, TransH, and TransR. All these methods map different relations into the vector space separately and the intrinsic correlations of these relations are ignored. It is obvious that there exist some correlations among relations because different relations may connect to a common entity. For example, the triples （Steve Jobs, PlaceOfBrith, California） and （Apple Inc., Location, California） share the same entity California as their tail entity. We analyze the embedded relation matrices learned by TransE/TransH/TransR, and find that the correlations of relations do exist and they are showed as low-rank structure over the embedded relation matrix. It is natural to ask whether we can leverage these correlations to learn better embeddings for the entities and relations in a knowledge graph. In this paper, we propose to learn the embedded relation matrix by decomposing it as a product of two low-dimensional matrices, for characterizing the low-rank structure. The proposed method, called TransCoRe （Translation-Based Method via Modeling the Correlations of Relations）, learns the embeddings of entities and relations with translation-based framework. Experimental results based on the benchmark datasets of WordNet and Freebase demonstrate that our method outperforms the typical baselines on link prediction and triple classification tasks.展开更多
A local radial basis function method(LRBF)is applied for the solution of boundary value problems in annular domains governed by the Poisson equation,the inhomogeneous biharmonic equation and the inhomogeneous Cauchy-N...A local radial basis function method(LRBF)is applied for the solution of boundary value problems in annular domains governed by the Poisson equation,the inhomogeneous biharmonic equation and the inhomogeneous Cauchy-Navier equations of elasticity.By appropriately choosing the collocation points we obtain linear systems in which the coefficient matrices possess block sparse circulant structures and which can be solved efficiently using matrix decomposition algorithms(MDAs)and fast Fourier transforms(FFTs).The MDAs used are appropriately modified to take into account the sparsity of the arrays involved in the discretization.The leave-one-out cross validation(LOOCV)algorithm is employed to obtain a suitable value for the shape parameter in the radial basis functions(RBFs)used.The selection of the nearest centres for each local influence domain is carried out using a modification of the kdtree algorithm.In several numerical experiments,it is demonstrated that the proposed algorithm is both accurate and capable of solving large scale problems.展开更多
Approximations based on random Fourier features have recently emerged as an efficient and elegant method for designing large-scale machine learning tasks.Unlike approaches using the Nystr?m method,which randomly sampl...Approximations based on random Fourier features have recently emerged as an efficient and elegant method for designing large-scale machine learning tasks.Unlike approaches using the Nystr?m method,which randomly samples the training examples,we make use of random Fourier features,whose basis functions(i.e.,cosine and sine)are sampled from a distribution independent from the training sample set,to cluster preference data which appears extensively in recommender systems.Firstly,we propose a two-stage preference clustering framework.In this framework,we make use of random Fourier features to map the preference matrix into the feature matrix,soon afterwards,utilize the traditional k-means approach to cluster preference data in the transformed feature space.Compared with traditional preference clustering,our method solves the problem of insufficient memory and greatly improves the efficiency of the operation.Experiments on movie data sets containing 100000 ratings,show that the proposed method is more effective in clustering accuracy than the Nystr?m and k-means,while also achieving better performance than these clustering approaches.展开更多
文摘As the development of smart grid and energy internet, this leads to a significantincrease in the amount of data transmitted in real time. Due to the mismatch withcommunication networks that were not designed to carry high-speed and real time data,data losses and data quality degradation may happen constantly. For this problem,according to the strong spatial and temporal correlation of electricity data which isgenerated by human’s actions and feelings, we build a low-rank electricity data matrixwhere the row is time and the column is user. Inspired by matrix decomposition, we dividethe low-rank electricity data matrix into the multiply of two small matrices and use theknown data to approximate the low-rank electricity data matrix and recover the missedelectrical data. Based on the real electricity data, we analyze the low-rankness of theelectricity data matrix and perform the Matrix Decomposition-based method on the realdata. The experimental results verify the efficiency and efficiency of the proposed scheme.
基金Supported by the National Natural Science Foundation of China under Grant Nos.60633050 and 60833004,60903057the National High-Technology Research and Development 863 Program of China under Grant No.2009AA01Z101
文摘QR and LU decompositions are the most important matrix decomposition algorithms. Many studies work on accelerating these algorithms by FPGA or ASIC in a case by case style. In this paper, we propose a unified framework for the matrix decomposition algorithms, combining three QR decomposition algorithms and LU algorithm with pivoting into a unified linear array structure. The QR and LU decomposition algorithms exhibit the same two-level loop structure and the same data dependency. Utilizing the similarities in loop structure and data dependency of matrix decomposition, we unify a fine-grained algorithm for all four matrix decomposition algorithms. Furthermore, we present a unified co-processor structure with a scalable linear array of processing elements （PEs）, in which four types of PEs are same in the structure of memory channels and PE connections, but the only difference exists in the internal structure of data path. Our unified co-processor, which is IEEE 32-bit floating-point precision, is implemented and mapped onto a Xilinx Virtex5 FPGA chip. Experimental results show that our co-processors can achieve speedup of 2.3 to 14.9 factors compared to a Pentium Dual CPU with double SSE threads.
基金supported by the National Natural Science Foundation of China(5147915151279149+2 种基金71540027)the China Postdoctoral Science Foundation Special Foundation Project(2013T607552012M521487)
文摘To fully display the modeling mechanism of the novelfractional order grey model （FGM （q,1））, this paper decomposesthe data matrix of the model into the mean generation matrix, theaccumulative generation matrix and the raw data matrix, whichare consistent with the fractional order accumulative grey model（FAGM （1,1））. Following this, this paper decomposes the accumulativedata difference matrix into the accumulative generationmatrix, the q-order reductive accumulative matrix and the rawdata matrix, and then combines the least square method, findingthat the differential order affects the model parameters only byaffecting the formation of differential sequences. This paper thensummarizes matrix decomposition of some special sequences,such as the sequence generated by the strengthening and weakeningoperators, the jumping sequence, and the non-equidistancesequence. Finally, this paper expresses the influences of the rawdata transformation, the accumulation sequence transformation,and the differential matrix transformation on the model parametersas matrices, and takes the non-equidistance sequence as an exampleto show the modeling mechanism.
基金Supported by the National Natural Science Foundation of China(61303212,61170080,61202386)the State Key Program of National Natural Science of China(61332019,U1135004)+2 种基金the Major Research Plan of the National Natural Science Foundation of China(91018008)Major State Basic Research Development Program of China(973 Program)(2014CB340600)the Natural Science Foundation of Hubei Province(2011CDB453,2014CFB440)
文摘Advances in quantum computation threaten to break public key eryptosystems that are based on the difficulty of fac- torization or the difficulty of discrete logariths, although , no quantum algorithms have been found to be able to solve certain mathematical problems on non-commutative algebraic structures up to now. The proposed new quasi-inverse based cryptography scheme is vulnerable to a linear algebra attack based on the probable occurrence of weak keys in the generation process. In this paper, we illustrate that two of the quasi-inverse based cryptography are vulnerable to a structural attack and that it only requires polynomial time to obtain the equivalent keys for some given public keys. In addition, we conduct a detailed analysis on attack methods and provide some improved suggestions on these two schemes.
基金Project supported by the National Natural Science Foundation of China (Nos. 50378083 and 50638050)the Research Foundation for the Doctoral Program of Higher Education of China (No. 20050335097)
文摘Under certain load pattern, the geometrically indeterminate pin-jointed mechanisms will present certain shapes to keep static equalization. This paper proposes a matrix-based method to determine the mobility and equilibrium stability of mechanisms according to the effects of the external loads. The first and second variations of the potential energy function of mechanisms under conservative force field are analyzed. Based on the singular value decomposition (SVD) method, a new crite- rion for the mobility and equilibrium stability of mechanisms can be concluded by analyzing the equilibrium matrix. The mobility and stability of mechanisms can be classified by unified matrix formulae. A number of examples are given to demonstrate the proposed criterion. In the end, criteria are summarized in a table.
基金This work was supported by National Natural Science Foundation of China(Nos.61802030,61572184)the Science and Technology Projects of Hunan Province(No.2016JC2075)the International Cooperative Project for“Double First-Class”,CSUST(No.2018IC24).
文摘With the popularity of deep learning tools in image decomposition and natural language processing,how to support and store a large number of parameters required by deep learning algorithms has become an urgent problem to be solved.These parameters are huge and can be as many as millions.At present,a feasible direction is to use the sparse representation technique to compress the parameter matrix to achieve the purpose of reducing parameters and reducing the storage pressure.These methods include matrix decomposition and tensor decomposition.To let vector take advance of the compressing performance of matrix decomposition and tensor decomposition,we use reshaping and unfolding to let vector be the input and output of Tensor-Factorized Neural Networks.We analyze how reshaping can get the best compress ratio.According to the relationship between the shape of tensor and the number of parameters,we get a lower bound of the number of parameters.We take some data sets to verify the lower bound.
文摘The aim of this paper is to investigate the numerical solution of the hypersingular integral equation reduced by the harmonic equation. First, we transform the hypersingular integral equation into 2π-periodic hypersingular integral equation with the map x=cot（θ/2）. Second, we initiate the study of the multiscale Galerkin method for the 2π-periodic hypersingular integral equation. The trigonometric wavelets are used as trial functions. Consequently, the 2j＋1 × 2j＋1 stiffness matrix Kj can be partitioned j×j block matrices. Furthermore, these block matrices are zeros except main diagonal block matrices. These main diagonal block matrices are symmetrical and circulant matrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform and the inverse fast Fourier transform instead of the inverse matrix. Finally, we provide several numerical examples to demonstrate our method has good accuracy even though the exact solutions are multi-peak and almost singular.
基金Supported by the Natural Science Foundation of Jiangsu Education Bureau under Grant No.09KJB140010the Project Prepared for National Natural Science Foundation of Xuzhou Normal University under Grant No.08XLY03
文摘In this paper, the synthesis and implementation of three-qubit SWAP gate is discussed. The three-qubit SWAP gate can be decomposed into product of 2 two-qubit SWAP gates, and it can be realized by 6 CNOT gates. Research illustrated that although the result is very simple, the current methods of matrix decomposition for multi-qubit gate can not get that. Then the implementation of three-qubit SWAP gate in the three spin system with Ising interaction is investigated and the sequence of control pulse and drift process to implement the gate is given. It needs 23 control pulses and 12 drift processes. Since the interaction can not be switched on and off at will, the realization of three-qubit SWAP gate in specific quantum system also can not simply come down to 2 two-qubit SWAP gates.
基金Support from the Natural Sciences and Engineering Research Council of Canada,the Canadian Institutes of Health Research and the Canadian Foundation for Innovation,is gratefully acknowledged
文摘We have recently proposed an optical method for assessing heart structure that uses polarized light measurement of birefringence as an indicator of tissue anisotropy.The highly aligned nature of healthy cardiac muscle tissue has a detectable effect on the polarization of light,resulting in a measurable phase shift(“retardance”).When this organized tissue structure is perturbed,for example after cardiac infarction(heart attack),scar tissue containing disorganized collagen is formed,causing a decrease in the measured retardance values.However,these are dependent not only on tissue anisotropy,but also on the angle between the tissue’s optical anisotropy direction and the beam interrogating the sample.To remove this experimental ambiguity,we present a method that interrogates the sample at two different incident beam angles,thus yielding enough information to uniquely determine the true magnitude and orientation of the tissue optical anisotropy.We use an infarcted porcine heart model to compare these polarimetryderived anisotropy metrics with those obtained with diffusion tensor magnetic resonance imaging(DT-MRI).The latter yields the anisotropy and the direction of tissue water diffusivity,providing an independent measure of tissue anisotropy.The optical and MR results are thus directly compared in a common ex vivo biological model of interest,yielding reasonable agreement but also highlighting some technique-specific differences.
基金the National Natural Science Foundation of China(Nos.62073111 and 61803134)the Fundamental Research Funds for the Provincial Universities of Zhejiang(No.GK209907299001-007)+2 种基金the Natural Science Foundation of Zhejiang Province,China(Nos.LY20F030008 and LY20F030011)the Open Research Project of Zhejiang Lab(No.2021MC0AB04)the Foundation of Zhejiang Provincial Education Department of China(No.Y202044263)。
文摘This paper is concerned with the event-triggered control of positive semi-Markovian jump systems without/with input saturation.The considered systems are subject to a stochastic semi-Markovian process whose sojourn time is dependent on a non-exponential distribution.First,an event-triggering condition is introduced in a linear form for the systems.A class of event-triggered feedback controllers is proposed using matrix decomposition technique.By using a stochastic co-positive Lyapunov function,the systems’positivity and stability are guaranteed.Then,the obtained results are developed for the systems with input saturation.A cone set is chosen as the attraction domain and the corresponding attraction domain gain matrix is designed in terms of standard linear programming approach.Finally,two numerical examples are provided to verify the validity and effectiveness of the presented theoretical findings.
文摘It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert time-domain problems into frequency-domain problems. Are there similar tools for a matrix？ By pairing a matrix to a piecewise function,a Haar-like wavelet is used to set up a similar tool for matrix analyzing, resulting in new methods for matrix approximation and orthogonal decomposition. By using our method, one can approximate a matrix by matrices with different orders. Our method also results in a new matrix orthogonal decomposition, reproducing Haar transformation for matrices with orders of powers of two. The computational complexity of the new orthogonal decomposition is linear. That is, for an m × n matrix, the computational complexity is O（mn）. In addition,when the method is applied to k-means clustering, one can obtain that k-means clustering can be equivalently converted to the problem of finding a best approximation solution of a function. In fact, the results in this paper could be applied to any matrix related problems.In addition, one can also employ other wavelet transformations and Fourier transformation to obtain similar results.
基金Supported by National Natural Science Foundation of China(Grants 61100129)Open Program of Key Laboratory of Intelligent Information Processing,Institute of Computing Technology,Chinese Academy of Sciences(IIP2014-7)
文摘The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters＇ values. It makes the equation＇s solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.
基金support from the National Key Basic Research Development Program(Grant No.2007CB209600)National Major Science and Technology Program(Grant No.2008ZX05010-002)
文摘A noise-reduction method with sliding called the local f-x Cadzow noise-reduction method, windows in the frequency-space （f-x） domain, is presented in this paper. This method is based on the assumption that the signal in each window is linearly predictable in the spatial direction while the random noise is not. For each Toeplitz matrix constructed by constant frequency slice, a singular value decomposition （SVD） is applied to separate signal from noise. To avoid edge artifacts caused by zero percent overlap between windows and to remove more noise, an appropriate overlap is adopted. Besides flat and dipping events, this method can enhance curved and conflicting events. However, it is not suitable for seismic data that contains big spikes or null traces. It is also compared with the SVD, f-x deconvolution, and Cadzow method without windows. The comparison results show that the local Cadzow method performs well in removing random noise and preserving signal. In addition, a real data example proves that it is a potential noise-reduction technique for seismic data obtained in areas of complex formations.
文摘In the teaching and researching of linear regression analysis, it is interesting and enlightening to explore how the dependent variable vector can be inner-transformed into regression coefficient estimator vector from a visible geometrical view. As an example, the roadmap of such inner transformation is presented based on a simple multiple linear regression model in this work. By applying the matrix algorithms like singular value decomposition (SVD) and Moore-Penrose generalized matrix inverse, the dependent variable vector lands into the right space of the independent variable matrix and is metamorphosed into regression coefficient estimator vector through the three-step of inner transformation. This work explores the geometrical relationship between the dependent variable vector and regression coefficient estimator vector as well as presents a new approach for vector rotating.
基金supported by Open Fund of Engineering Laboratory of Spatial Information Technology of Highway Geological Disaster Early Warning in Hunan Province(Changsha University of Science&Technology,kfj150602)Hunan Province Science and Technology Program Funded Projects,China(2015NK3035)+1 种基金the Land and Resources Department Scientific Research Project of Hunan Province,China(2013-27)the Education Department Scientific Research Project of Hunan Province,China(13C1011)
文摘Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.
文摘The implementation of product development process management (PDPM) is an effective means of developing products with higher quality in shorter lead time. It is argued in this paper that product, data, person and activity are basic factors in PDPM With detailed analysis of these basic factors and their relations in product developmed process, all product development activities are considered as tasks and the management of product development process is regarded as the management of task execution A task decomposition based product development model is proposed with methods of constructing task relation matrix from layer model and constraint model resulted from task decomposition. An algorithm for constructing directed task graph is given and is used in the management of tasks. Finally, the usage and limitation of the proposed PDPM model is given with further work proposed.
基金supported in part by the National Science Foundation of United States(NSF)(Grant No.0844707)in part by the International S&T Cooperation Program of China(ISTCP)(Grant No.2013DFA60930)
文摘Calculation of eigen-solutions plays an important role in the small signal stability analysis of power systems.In this paper,a novel approach based on matrix perturbation theory is proposed for the calculation of eigen-solutions in a perturbed system.Rigorous theoretical analysis is conducted on the solution of distinct,multiple,and close eigen-solutions,respectively,under perturbations of parameters.The computational flowchart of the unified solution of eigen-solutions is then proposed,aimed toward obtaining eigen-solutions of a perturbed system directly with algebraic formulas without solving an eigenvalue problem repeatedly.Finally,the effectiveness of the matrix perturbation based approach for eigen-solutions’calculation in power systems is verified by numerical examples on a two-area four-machine system.
基金This work was supported by the National Basic Research 973 Program of China under Grant No. 2014CB340405, the National Key Research and Development Program of China under Grant No. 2016YFB1000902, and the National Natural Science Foundation of China under Grant Nos. 61402442, 61272177, 61173008, 61232010, 61303244, 61572469, 91646120 and 61572473.
文摘Knowledge graph embedding, which maps the entities and relations into low-dimensional vector spaces, has demonstrated its effectiveness in many tasks such as link prediction and relation extraction. Typical methods include TransE, TransH, and TransR. All these methods map different relations into the vector space separately and the intrinsic correlations of these relations are ignored. It is obvious that there exist some correlations among relations because different relations may connect to a common entity. For example, the triples （Steve Jobs, PlaceOfBrith, California） and （Apple Inc., Location, California） share the same entity California as their tail entity. We analyze the embedded relation matrices learned by TransE/TransH/TransR, and find that the correlations of relations do exist and they are showed as low-rank structure over the embedded relation matrix. It is natural to ask whether we can leverage these correlations to learn better embeddings for the entities and relations in a knowledge graph. In this paper, we propose to learn the embedded relation matrix by decomposing it as a product of two low-dimensional matrices, for characterizing the low-rank structure. The proposed method, called TransCoRe （Translation-Based Method via Modeling the Correlations of Relations）, learns the embeddings of entities and relations with translation-based framework. Experimental results based on the benchmark datasets of WordNet and Freebase demonstrate that our method outperforms the typical baselines on link prediction and triple classification tasks.
基金acknowledges HPC at The University of Southern Mississippi supported by the National Science Foundation under the Major Research Instrumentation(MRI)program via Grant#ACI 1626217.
文摘A local radial basis function method(LRBF)is applied for the solution of boundary value problems in annular domains governed by the Poisson equation,the inhomogeneous biharmonic equation and the inhomogeneous Cauchy-Navier equations of elasticity.By appropriately choosing the collocation points we obtain linear systems in which the coefficient matrices possess block sparse circulant structures and which can be solved efficiently using matrix decomposition algorithms(MDAs)and fast Fourier transforms(FFTs).The MDAs used are appropriately modified to take into account the sparsity of the arrays involved in the discretization.The leave-one-out cross validation(LOOCV)algorithm is employed to obtain a suitable value for the shape parameter in the radial basis functions(RBFs)used.The selection of the nearest centres for each local influence domain is carried out using a modification of the kdtree algorithm.In several numerical experiments,it is demonstrated that the proposed algorithm is both accurate and capable of solving large scale problems.
基金supported by the National Natural Science Foundation of China(Nos.61872260 and 61592419)the Natural Science Foundation of Shanxi Province(No.201703D421013).
文摘Approximations based on random Fourier features have recently emerged as an efficient and elegant method for designing large-scale machine learning tasks.Unlike approaches using the Nystr?m method,which randomly samples the training examples,we make use of random Fourier features,whose basis functions(i.e.,cosine and sine)are sampled from a distribution independent from the training sample set,to cluster preference data which appears extensively in recommender systems.Firstly,we propose a two-stage preference clustering framework.In this framework,we make use of random Fourier features to map the preference matrix into the feature matrix,soon afterwards,utilize the traditional k-means approach to cluster preference data in the transformed feature space.Compared with traditional preference clustering,our method solves the problem of insufficient memory and greatly improves the efficiency of the operation.Experiments on movie data sets containing 100000 ratings,show that the proposed method is more effective in clustering accuracy than the Nystr?m and k-means,while also achieving better performance than these clustering approaches.
