The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses ...The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses were independent of the choice of the generalized inverse. Based on this result, two sufficient and necessary conditions for the lower semi-continuity of generalized inverses as the set-valued mappings are given.展开更多
Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and...Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.展开更多
If an operator is not invertible, we are interested if there is a subspace such that the reduction of the operator to that subspace is invertible. In this paper we give a spectral approach to generalized inverses cons...If an operator is not invertible, we are interested if there is a subspace such that the reduction of the operator to that subspace is invertible. In this paper we give a spectral approach to generalized inverses considering the subspace determined by the range of the spectral projection associated with an operator and a spectral set containing the point 0. We compare the cases, 0 is a simple pole of the resolvent function, 0 is a pole of order n of the resolvent function, 0 is an isolated point of the spectrum, and 0 is contained in a circularly isolated spectral set.展开更多
Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear ...Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear operator on X,and a bound on its spectral radius is also obtained.This generalizes the classic Banach lemma.We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.展开更多
The generalized inversion of S-wave amplitude spectra from the free-field strong motion recordings of the China National Strong Motion Observation Network System (NSMONS) are used to evaluate the site effects in the...The generalized inversion of S-wave amplitude spectra from the free-field strong motion recordings of the China National Strong Motion Observation Network System (NSMONS) are used to evaluate the site effects in the Wenchuan area. In this regard, a total of 602 recordings from 96 aftershocks of the Wenchuan earthquake with magnitudes of M3.7-M6.5 were selected as a dataset. These recordings were obtained from 28 stations at a hypocenter distance ranging from 30 km to 150 km. The inversion results have been verified as reliable by comparing the site response at station 62WUD using the Generalized Inversion Technique (GIT) and the Standard Spectral Ratio method (SSR). For all 28 stations, the site predominant frequency F and the average site amplification in different frequency bands of 1.0-5.0 Hz, 5.0-10.0 Hz and 1.0-10.0 Hz have been calculated based on the inversion results. Compared with the results from the horizontal-to-vertical spectral ratio (HVSR) method, it shows that the HVSR method can reasonably estimate the site predominant frequency but underestimates the site amplification. The linear fitting between the average site amplification for each frequency band and the V20 (the average uppermost-20 m shear wave velocity) shows good correlation. A distance measurement called the asperity distance DAspt is proposed to reasonably characterize the source-to-site distance for large earthquakes. Finally, the inversed site response is used to identify the soil nonlinearity in the main shock and aftershocks of Wenchuan earthquake. In ten of the 28 stations analyzed in the main shock, the soil behaved nonlinearly, where the ground motion level is apparently beyond a threshold ofPGA 〉 300 cm/s^2 or PGV 〉 20 cm/s, and only one station coded 51SFB has evidence of soil nonlinear behavior in the aftershocks.展开更多
In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is "the generalized Neumann lemma"...In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is "the generalized Neumann lemma" which is quite different from the method in [12] where "the generalized Banach lemma" was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.展开更多
Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expr...Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator. This is theoretically important for studying the stabilization and asymptotic stability of the second order generalized distributed parameter system.展开更多
Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergenc...Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergence of the power series with function_valued coefficients and to estimate characteristic value of the integral equations. Famous Wynn identities of the Pad approximants is also established by means of the connection of two algorithms.展开更多
This paper deals with the Bayesian estimation of Shannon entropy for the generalized inverse exponential distribution.Assuming that the observed samples are taken from the upper record ranked set sampling(URRSS)and up...This paper deals with the Bayesian estimation of Shannon entropy for the generalized inverse exponential distribution.Assuming that the observed samples are taken from the upper record ranked set sampling(URRSS)and upper record values(URV)schemes.Formulas of Bayesian estimators are derived depending on a gamma prior distribution considering the squared error,linear exponential and precautionary loss functions,in addition,we obtain Bayesian credible intervals.The random-walk Metropolis-Hastings algorithm is handled to generate Markov chain Monte Carlo samples from the posterior distribution.Then,the behavior of the estimates is examined at various record values.The output of the study shows that the entropy Bayesian estimates under URRSS are more convenient than the other estimates under URV in the majority of the situations.Also,the entropy Bayesian estimates perform well as the number of records increases.The obtained results validate the usefulness and efficiency of the URV method.Real data is analyzed for more clarifying purposes which validate the theoretical results.展开更多
For an integer n ≥2, we say that an operator A is an n-idempotent if A^n = A; A is a generalized n-idempotent if A^n = A^*; A is a hyper-generalized n. idempotent if A^n = A^+. The set of all n-idempotents, all gen...For an integer n ≥2, we say that an operator A is an n-idempotent if A^n = A; A is a generalized n-idempotent if A^n = A^*; A is a hyper-generalized n. idempotent if A^n = A^+. The set of all n-idempotents, all generalized n-idempotents and all hyper-generalized n-idempotents are denoted by In(H), gIn(H) and HgIn(H), respectively. In this note, we obtain a chain of proper inclusions gIn(H) belong to HgIn(H) belong to In+2(H).展开更多
A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discusse...A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discussed and constructed. An example was given to illustrate its application.展开更多
After choosing weight functions suitably, we define a Banach spaceH ω μ (L) and discuss the generalized inverse of singular integral operators on an open arc. Using the generalized inverse, we obtain the solutions f...After choosing weight functions suitably, we define a Banach spaceH ω μ (L) and discuss the generalized inverse of singular integral operators on an open arc. Using the generalized inverse, we obtain the solutions for the following singular integral equation展开更多
In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit n...In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.展开更多
Let X=Rn +×R denote the underlying manifold of polyradial functions on the Heisenberg group H n. We construct a generalized translation on X=Rn +×R, and establish the Plancherel formula on L2(X,dμ). Usin...Let X=Rn +×R denote the underlying manifold of polyradial functions on the Heisenberg group H n. We construct a generalized translation on X=Rn +×R, and establish the Plancherel formula on L2(X,dμ). Using the Gelfand transform we give the condition of generalized wavelets on L2(X,dμ). Moreover, we show the reconstruction formulas for wavelet packet trnasforms and an inversion formula of the Radon transform on X.展开更多
This article continues to study the research suggestions in depth made by M.Z.Nashed and G.F.Votruba in the journal"Bull.Amer.Math.Soc."in 1974.Concerned with the pricing of non-reachable"contingent cla...This article continues to study the research suggestions in depth made by M.Z.Nashed and G.F.Votruba in the journal"Bull.Amer.Math.Soc."in 1974.Concerned with the pricing of non-reachable"contingent claims"in an incomplete financial market,when constructing a specific bounded linear operator A:l_(1)^(n)→l_(2) from a non-reflexive Banach space l_(1)^(n) to a Hilbert space l_(2),the problem of non-reachable"contingent claims"pricing is reduced to researching the(single-valued)selection of the(set-valued)metric generalized inverse A■ of the operator A.In this paper,by using the Banach space structure theory and the generalized inverse method of operators,we obtain a bounded linear single-valued selection A^(σ)=A+of A■.展开更多
A method that attempts to recover signal using generalized inverse theory is presented to obtain a good approximation of the signal in reconstruction space from its generalized samples. The proposed approaches differ ...A method that attempts to recover signal using generalized inverse theory is presented to obtain a good approximation of the signal in reconstruction space from its generalized samples. The proposed approaches differ with the assumptions on reconstruction space. If the reconstruction space satisfies one-to-one relationship between the samples and the reconstruction model, then we propose a method, which achieves consistent signal reconstruction. At the same time, when the number of samples is more than the number of reconstruction functions, the minimal-norm reconstruction signal can be obtained. Finally, it is demonstrated that the minimal-norm reconstruction can outperform consistent signal reconstruction in both theory and simulations for the problem.展开更多
In this paper,we prove that if X is an almost convex and 2-strictly convex space,linear operator T:X→Y is bounded,N(T)is an approximative compact Chebyshev subspace of X and R(T)is a 3-Chebyshev hyperplane,then there...In this paper,we prove that if X is an almost convex and 2-strictly convex space,linear operator T:X→Y is bounded,N(T)is an approximative compact Chebyshev subspace of X and R(T)is a 3-Chebyshev hyperplane,then there exists a homogeneous selection T^(σ)of T^(■)such that continuous points of T^(σ)and T^(■)are dense on Y.展开更多
This paper presents the design and implementation of Adaptive Generalized Dynamic Inversion(AGDI)to track the position of a Linear Flexible Joint Cart(LFJC)system along with vibration suppression of the flexible joint...This paper presents the design and implementation of Adaptive Generalized Dynamic Inversion(AGDI)to track the position of a Linear Flexible Joint Cart(LFJC)system along with vibration suppression of the flexible joint.The proposed AGDI control law will be comprised of two control elements.The baseline(continuous)control law is based on principle of conventional GDI approach and is established by prescribing the constraint dynamics of controlled state variables that reflect the control objectives.The control law is realized by inverting the prescribed dynamics using dynamically scaledMoore-Penrose generalized inversion.To boost the robust attributes against system nonlinearities,parametric uncertainties and external perturbations,a discontinuous control law will be augmented which is based on the concept of sliding mode principle.In discontinuous control law,the sliding mode gain is made adaptive in order to achieve improved tracking performance and chattering reduction.The closed-loop stability of resultant control law is established by introducing a positive define Lyapunov candidate function such that semi-global asymptotic attitude tracking of LFJC system is guaranteed.Rigorous computer simulations followed by experimental investigation will be performed on Quanser’s LFJC system to authenticate the feasibility of proposed control approach for its application to real world problems.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos. 10571150 and 10271053)
文摘The perturbation problem of generalized inverse is studied. And some new stability characteristics of generalized inverses were presented. It was also proved that the stability characteristics of generalized inverses were independent of the choice of the generalized inverse. Based on this result, two sufficient and necessary conditions for the lower semi-continuity of generalized inverses as the set-valued mappings are given.
基金This work is Supported by NSF of Heilongjiang Province
文摘Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.
文摘If an operator is not invertible, we are interested if there is a subspace such that the reduction of the operator to that subspace is invertible. In this paper we give a spectral approach to generalized inverses considering the subspace determined by the range of the spectral projection associated with an operator and a spectral set containing the point 0. We compare the cases, 0 is a simple pole of the resolvent function, 0 is a pole of order n of the resolvent function, 0 is an isolated point of the spectrum, and 0 is contained in a circularly isolated spectral set.
基金Supported by the National Natural Science Foundation of China(12001142).
文摘Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear operator on X,and a bound on its spectral radius is also obtained.This generalizes the classic Banach lemma.We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.
基金Nonprofit Industry Research Project of CEA under Grant No. 201208014National Natural Science Fund No. 51278473Environmental Protection Research Fund for Public Interest No. 201209040
文摘The generalized inversion of S-wave amplitude spectra from the free-field strong motion recordings of the China National Strong Motion Observation Network System (NSMONS) are used to evaluate the site effects in the Wenchuan area. In this regard, a total of 602 recordings from 96 aftershocks of the Wenchuan earthquake with magnitudes of M3.7-M6.5 were selected as a dataset. These recordings were obtained from 28 stations at a hypocenter distance ranging from 30 km to 150 km. The inversion results have been verified as reliable by comparing the site response at station 62WUD using the Generalized Inversion Technique (GIT) and the Standard Spectral Ratio method (SSR). For all 28 stations, the site predominant frequency F and the average site amplification in different frequency bands of 1.0-5.0 Hz, 5.0-10.0 Hz and 1.0-10.0 Hz have been calculated based on the inversion results. Compared with the results from the horizontal-to-vertical spectral ratio (HVSR) method, it shows that the HVSR method can reasonably estimate the site predominant frequency but underestimates the site amplification. The linear fitting between the average site amplification for each frequency band and the V20 (the average uppermost-20 m shear wave velocity) shows good correlation. A distance measurement called the asperity distance DAspt is proposed to reasonably characterize the source-to-site distance for large earthquakes. Finally, the inversed site response is used to identify the soil nonlinearity in the main shock and aftershocks of Wenchuan earthquake. In ten of the 28 stations analyzed in the main shock, the soil behaved nonlinearly, where the ground motion level is apparently beyond a threshold ofPGA 〉 300 cm/s^2 or PGV 〉 20 cm/s, and only one station coded 51SFB has evidence of soil nonlinear behavior in the aftershocks.
基金Supported by the Nature Science Foundation of China(11471091 and 11401143)
文摘In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is "the generalized Neumann lemma" which is quite different from the method in [12] where "the generalized Banach lemma" was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.
文摘Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator. This is theoretically important for studying the stabilization and asymptotic stability of the second order generalized distributed parameter system.
文摘Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergence of the power series with function_valued coefficients and to estimate characteristic value of the integral equations. Famous Wynn identities of the Pad approximants is also established by means of the connection of two algorithms.
基金A.R.A.Alanzi would like to thank the Deanship of Scientific Research at Majmaah University for financial support and encouragement.
文摘This paper deals with the Bayesian estimation of Shannon entropy for the generalized inverse exponential distribution.Assuming that the observed samples are taken from the upper record ranked set sampling(URRSS)and upper record values(URV)schemes.Formulas of Bayesian estimators are derived depending on a gamma prior distribution considering the squared error,linear exponential and precautionary loss functions,in addition,we obtain Bayesian credible intervals.The random-walk Metropolis-Hastings algorithm is handled to generate Markov chain Monte Carlo samples from the posterior distribution.Then,the behavior of the estimates is examined at various record values.The output of the study shows that the entropy Bayesian estimates under URRSS are more convenient than the other estimates under URV in the majority of the situations.Also,the entropy Bayesian estimates perform well as the number of records increases.The obtained results validate the usefulness and efficiency of the URV method.Real data is analyzed for more clarifying purposes which validate the theoretical results.
文摘For an integer n ≥2, we say that an operator A is an n-idempotent if A^n = A; A is a generalized n-idempotent if A^n = A^*; A is a hyper-generalized n. idempotent if A^n = A^+. The set of all n-idempotents, all generalized n-idempotents and all hyper-generalized n-idempotents are denoted by In(H), gIn(H) and HgIn(H), respectively. In this note, we obtain a chain of proper inclusions gIn(H) belong to HgIn(H) belong to In+2(H).
文摘A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discussed and constructed. An example was given to illustrate its application.
基金Supported by the National Natural Science Foundation of China( No.2 0 1980 6 33)
文摘After choosing weight functions suitably, we define a Banach spaceH ω μ (L) and discuss the generalized inverse of singular integral operators on an open arc. Using the generalized inverse, we obtain the solutions for the following singular integral equation
基金supported by the National Natural Science Foundation of China(Grants 11472161,11102102,and 91130017)the Independent Innovation Foundation of Shandong University(Grant 2013ZRYQ002)the Natural Science Foundation of Shandong Province(Grant ZR2014AQ015)
文摘In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.
基金Supported by the Foundation of the National Natural Science of China( No.1 0 0 71 0 39) and the Foundation of Edu-cation Commission of Jiangsu Province
文摘Let X=Rn +×R denote the underlying manifold of polyradial functions on the Heisenberg group H n. We construct a generalized translation on X=Rn +×R, and establish the Plancherel formula on L2(X,dμ). Using the Gelfand transform we give the condition of generalized wavelets on L2(X,dμ). Moreover, we show the reconstruction formulas for wavelet packet trnasforms and an inversion formula of the Radon transform on X.
基金supported by the National Science Foundation (12001142)Harbin Normal University doctoral initiation Fund (XKB201812)supported by the Science Foundation Grant of Heilongjiang Province (LH2019A017)
文摘This article continues to study the research suggestions in depth made by M.Z.Nashed and G.F.Votruba in the journal"Bull.Amer.Math.Soc."in 1974.Concerned with the pricing of non-reachable"contingent claims"in an incomplete financial market,when constructing a specific bounded linear operator A:l_(1)^(n)→l_(2) from a non-reflexive Banach space l_(1)^(n) to a Hilbert space l_(2),the problem of non-reachable"contingent claims"pricing is reduced to researching the(single-valued)selection of the(set-valued)metric generalized inverse A■ of the operator A.In this paper,by using the Banach space structure theory and the generalized inverse method of operators,we obtain a bounded linear single-valued selection A^(σ)=A+of A■.
文摘A method that attempts to recover signal using generalized inverse theory is presented to obtain a good approximation of the signal in reconstruction space from its generalized samples. The proposed approaches differ with the assumptions on reconstruction space. If the reconstruction space satisfies one-to-one relationship between the samples and the reconstruction model, then we propose a method, which achieves consistent signal reconstruction. At the same time, when the number of samples is more than the number of reconstruction functions, the minimal-norm reconstruction signal can be obtained. Finally, it is demonstrated that the minimal-norm reconstruction can outperform consistent signal reconstruction in both theory and simulations for the problem.
基金supported by the“China Natural Science Fund under grant 11871181”the“China Natural Science Fund under grant 11561053”。
文摘In this paper,we prove that if X is an almost convex and 2-strictly convex space,linear operator T:X→Y is bounded,N(T)is an approximative compact Chebyshev subspace of X and R(T)is a 3-Chebyshev hyperplane,then there exists a homogeneous selection T^(σ)of T^(■)such that continuous points of T^(σ)and T^(■)are dense on Y.
基金This research work was funded by Institutional Fund Projects under Grant No.(IFPHI-106-135-2020).
文摘This paper presents the design and implementation of Adaptive Generalized Dynamic Inversion(AGDI)to track the position of a Linear Flexible Joint Cart(LFJC)system along with vibration suppression of the flexible joint.The proposed AGDI control law will be comprised of two control elements.The baseline(continuous)control law is based on principle of conventional GDI approach and is established by prescribing the constraint dynamics of controlled state variables that reflect the control objectives.The control law is realized by inverting the prescribed dynamics using dynamically scaledMoore-Penrose generalized inversion.To boost the robust attributes against system nonlinearities,parametric uncertainties and external perturbations,a discontinuous control law will be augmented which is based on the concept of sliding mode principle.In discontinuous control law,the sliding mode gain is made adaptive in order to achieve improved tracking performance and chattering reduction.The closed-loop stability of resultant control law is established by introducing a positive define Lyapunov candidate function such that semi-global asymptotic attitude tracking of LFJC system is guaranteed.Rigorous computer simulations followed by experimental investigation will be performed on Quanser’s LFJC system to authenticate the feasibility of proposed control approach for its application to real world problems.