The invertibility of combinations of two orthogonal projectors aP ＋ bQ - cPQ - dQP is researched by using the CSdecomposition of matrices and properties of orthogonal projectors. The Moore-enrose inverse of the combi...The invertibility of combinations of two orthogonal projectors aP ＋ bQ - cPQ - dQP is researched by using the CSdecomposition of matrices and properties of orthogonal projectors. The Moore-enrose inverse of the combinations is presented under some special conditions.展开更多
In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group invers...In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group inverses of them are derived. The work of this paper extends some previous results.展开更多
This paper establishes some new equalities and inequalities for the null and column spaces of combinations of two projectors P and Q. Some new necessary and sufficient conditions for P ± Q to be invertible are gi...This paper establishes some new equalities and inequalities for the null and column spaces of combinations of two projectors P and Q. Some new necessary and sufficient conditions for P ± Q to be invertible are given by the structure of null and column space of some combinations of P and Q. In addition, the inclusion relation of Af（PQ ＋ QP） and N（PQ - QP） is discussed and necessary and sufficient conditions for them to be equal are also studied.展开更多
We give in this note some expansion formulas for the orthogonal projectors onto the range of the row block matrix [ A, B ], and use the expansion formulas to examine relations among the orthogonal projectors onto the ...We give in this note some expansion formulas for the orthogonal projectors onto the range of the row block matrix [ A, B ], and use the expansion formulas to examine relations among the orthogonal projectors onto the ranges of A, B and [A, B]. In particular, we present some identifying conditions for a pair of orthogonal projectors of the same size to commute.展开更多
Necessary and sufficient conditions for equalities between a 2 y′(I-P Xx)y and minimum norm quadratic unbiased estimator of variance under the general linear model, where a 2 is a known positive number, are...Necessary and sufficient conditions for equalities between a 2 y′(I-P Xx)y and minimum norm quadratic unbiased estimator of variance under the general linear model, where a 2 is a known positive number, are derived. Further, when the Gauss? Markov estimators and the ordinary least squares estimator are identical, a relative simply equivalent condition is obtained. At last, this condition is applied to an interesting example.展开更多
Based on the constructivism, using the College English（new edition） CD-ROMs for classroom teaching as main media, this paper explores the feasibility of the classroom teaching mode of appropriate combination of trad...Based on the constructivism, using the College English（new edition） CD-ROMs for classroom teaching as main media, this paper explores the feasibility of the classroom teaching mode of appropriate combination of traditional methods with the multimedia overhead projector. The new teaching mode proves to be successful in that it has greatly improved the students＇ ability to use English in an all-round way and has obvious advantages over the only traditional method.展开更多
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the uni...In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ?by using the rotation group [3] [4]. It will be proved that the group acts on elements of ?in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ?in terms of matrix operations using the operator and the Hadamard Product;this construction is consistent with the group operation defined in the first article.展开更多
By using the Riemann-Hilbert method and the Corona theorem, Wiener-Hopf factorization for a class of matrix functions is studied. Under appropriate assumption, a sufficient and neces- sary condition for the existence ...By using the Riemann-Hilbert method and the Corona theorem, Wiener-Hopf factorization for a class of matrix functions is studied. Under appropriate assumption, a sufficient and neces- sary condition for the existence of the matrix function admitting canonical factorization is obtained and the solution to a class of non-linear Riemann-Hilbert problems is also given. Furthermore, by means of non-standard Corona theorem partial estimation of the general factorization can be obtained.展开更多
In the present study,the influence of dynamic process parameters such as water pressure,traverse speed and abrasive mass flow rate on depth of penetration and surface topography in high strength AZ91 magnesium alloy w...In the present study,the influence of dynamic process parameters such as water pressure,traverse speed and abrasive mass flow rate on depth of penetration and surface topography in high strength AZ91 magnesium alloy were investigated using Abrasive Water Jet(AWJ)cutting technology.Process parameters were varied at 3 levels and influences of each parameter on penetration ability were identified using analysis of variance(ANOVA).Contribution of water pressure and traverse speed on jet penetration found higher compared to abrasive mass flow rate.Profile projector was used to measure depth of penetration.Microstructural features and topography of cut surfaces were examined using Scanning Electron Microscopy(SEM).Micro cutting and ploughing were observed on the top and bottom portion of the cut which were similar to that of modes of deformation in other ductile materials like aluminium and steel.Surface roughness of cut surfaces was measured using Taylor Hobson surface roughness tester.Surface roughness found higher at higher traverse speeds and lower at lower traverse speeds.This study also highlights the suitability of AWJ cutting technology for cutting magnesium and its alloys.展开更多
This paper concerns the optimization problem for multilayered ultrasonic transducer with active porous piezoelectric layer. The dependences of the effective moduli for porous piezoelectric material on porosity have be...This paper concerns the optimization problem for multilayered ultrasonic transducer with active porous piezoelectric layer. The dependences of the effective moduli for porous piezoelectric material on porosity have been previously obtained and allowed to decrease the number of design variables. The multiobjective optimization problem based on the Pareto-frontier calculation has been solved using the live-link of finite-element (FE) package Comsol Multiphysics with MATLAB.展开更多
The co-rotational finite element formulation is an attractive technique extending the capabilities of an existing high performing linear element to geometrically nonlinear analysis.This paper presents a modified co-ro...The co-rotational finite element formulation is an attractive technique extending the capabilities of an existing high performing linear element to geometrically nonlinear analysis.This paper presents a modified co-rotational framework,unified for beam,shell,and brick elements.A unified zero-spin criterion is proposed to specify the local element frame,whose origin is always located at the centroid.Utilizing this criterion,a spin matrix is introduced,and the local frame is invariant to the element nodal ordering.Additionally,the projector matrix is redefined in a more intuitive way,which is the derivative of local co-rotational element frame with respect to the global one.Furthermore,the nodal rotation is obtained with pseudo vector and instantaneous rotation,under a high-order accurate transformation.The resulting formulations are achieved in unified expression and thus a series of linear elements can be embedded into the framework.Several examples are presented to demonstrate the efficiency and accuracy of the proposed framework for large displacement analysis.展开更多
基金Supported by the Hubei Normal University Research Grant (2008D54)
文摘The invertibility of combinations of two orthogonal projectors aP ＋ bQ - cPQ - dQP is researched by using the CSdecomposition of matrices and properties of orthogonal projectors. The Moore-enrose inverse of the combinations is presented under some special conditions.
基金Supported by the National Natural Science Foundation of China(11271105)the Key Research Project of Educational Department of Hubei Province(D20122202)Youth Research Project of Educational Department of Hubei Province(B20122203)
文摘In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group inverses of them are derived. The work of this paper extends some previous results.
基金Supported by the Tianyuan of National Natural Science Foundation of China(Grant No.70871050)the Youth Project of Education Department of Hubei Province(Grant No.B20122203)
文摘This paper establishes some new equalities and inequalities for the null and column spaces of combinations of two projectors P and Q. Some new necessary and sufficient conditions for P ± Q to be invertible are given by the structure of null and column space of some combinations of P and Q. In addition, the inclusion relation of Af（PQ ＋ QP） and N（PQ - QP） is discussed and necessary and sufficient conditions for them to be equal are also studied.
基金Supported by National Natural Science Foundation of China(Grant No.11271384)
文摘We give in this note some expansion formulas for the orthogonal projectors onto the range of the row block matrix [ A, B ], and use the expansion formulas to examine relations among the orthogonal projectors onto the ranges of A, B and [A, B]. In particular, we present some identifying conditions for a pair of orthogonal projectors of the same size to commute.
文摘Necessary and sufficient conditions for equalities between a 2 y′(I-P Xx)y and minimum norm quadratic unbiased estimator of variance under the general linear model, where a 2 is a known positive number, are derived. Further, when the Gauss? Markov estimators and the ordinary least squares estimator are identical, a relative simply equivalent condition is obtained. At last, this condition is applied to an interesting example.
文摘Based on the constructivism, using the College English（new edition） CD-ROMs for classroom teaching as main media, this paper explores the feasibility of the classroom teaching mode of appropriate combination of traditional methods with the multimedia overhead projector. The new teaching mode proves to be successful in that it has greatly improved the students＇ ability to use English in an all-round way and has obvious advantages over the only traditional method.
文摘In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on ?by using the rotation group [3] [4]. It will be proved that the group acts on elements of ?in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation ?in terms of matrix operations using the operator and the Hadamard Product;this construction is consistent with the group operation defined in the first article.
基金Supported by the National Natural Science Foundation of China(10471107)the Specialized Research Fund for the Doctoral Program of Higher Education of China (20060486001)
文摘By using the Riemann-Hilbert method and the Corona theorem, Wiener-Hopf factorization for a class of matrix functions is studied. Under appropriate assumption, a sufficient and neces- sary condition for the existence of the matrix function admitting canonical factorization is obtained and the solution to a class of non-linear Riemann-Hilbert problems is also given. Furthermore, by means of non-standard Corona theorem partial estimation of the general factorization can be obtained.
基金The authors would like to express their sincere thanks and appreciation to Center of Excellence(CoE)in advanced materials research and TEQIP,BMS College of Engineering,Ben-galuru,Karnataka,India,for providing financial assistance in conducting this research work.
文摘In the present study,the influence of dynamic process parameters such as water pressure,traverse speed and abrasive mass flow rate on depth of penetration and surface topography in high strength AZ91 magnesium alloy were investigated using Abrasive Water Jet(AWJ)cutting technology.Process parameters were varied at 3 levels and influences of each parameter on penetration ability were identified using analysis of variance(ANOVA).Contribution of water pressure and traverse speed on jet penetration found higher compared to abrasive mass flow rate.Profile projector was used to measure depth of penetration.Microstructural features and topography of cut surfaces were examined using Scanning Electron Microscopy(SEM).Micro cutting and ploughing were observed on the top and bottom portion of the cut which were similar to that of modes of deformation in other ductile materials like aluminium and steel.Surface roughness of cut surfaces was measured using Taylor Hobson surface roughness tester.Surface roughness found higher at higher traverse speeds and lower at lower traverse speeds.This study also highlights the suitability of AWJ cutting technology for cutting magnesium and its alloys.
文摘This paper concerns the optimization problem for multilayered ultrasonic transducer with active porous piezoelectric layer. The dependences of the effective moduli for porous piezoelectric material on porosity have been previously obtained and allowed to decrease the number of design variables. The multiobjective optimization problem based on the Pareto-frontier calculation has been solved using the live-link of finite-element (FE) package Comsol Multiphysics with MATLAB.
基金the National Natural Science Foundation of China(Grant Nos.11972297 and 11972300)the Fundamental Research Funds for the Central Universities of China(Grant No.G2019KY05203).
文摘The co-rotational finite element formulation is an attractive technique extending the capabilities of an existing high performing linear element to geometrically nonlinear analysis.This paper presents a modified co-rotational framework,unified for beam,shell,and brick elements.A unified zero-spin criterion is proposed to specify the local element frame,whose origin is always located at the centroid.Utilizing this criterion,a spin matrix is introduced,and the local frame is invariant to the element nodal ordering.Additionally,the projector matrix is redefined in a more intuitive way,which is the derivative of local co-rotational element frame with respect to the global one.Furthermore,the nodal rotation is obtained with pseudo vector and instantaneous rotation,under a high-order accurate transformation.The resulting formulations are achieved in unified expression and thus a series of linear elements can be embedded into the framework.Several examples are presented to demonstrate the efficiency and accuracy of the proposed framework for large displacement analysis.
