This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By i...This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.展开更多
In this paper we analyze the optimal control problem for a class of afflne nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which def...In this paper we analyze the optimal control problem for a class of afflne nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.展开更多
Using the direct method,we investigate the generalized Hyers-Ulam stability of the following quadratic functional inequality■inβ-homogeneous complex Banach spaces.
We consider the Hyers-Ulam stability problem of the generalized quadratic functional equationuoA＋voB-2woP1 - 2ko P2 =0, which is a distributional version of the classical generalized quadratic functional equation f（...We consider the Hyers-Ulam stability problem of the generalized quadratic functional equationuoA＋voB-2woP1 - 2ko P2 =0, which is a distributional version of the classical generalized quadratic functional equation f（x＋y）＋g（x - y） - 2h（x） - 2k（y）=0展开更多
In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n,...In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[展开更多
The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the...The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.展开更多
A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.
Quadratic Discrimination Function （QDF） is commonly used in speech emotion recognition, which proceeds on the premise that the input data is normal distribution. In this paper, we propose a transformation to normali...Quadratic Discrimination Function （QDF） is commonly used in speech emotion recognition, which proceeds on the premise that the input data is normal distribution. In this paper, we propose a transformation to normalize the emotional features, emotion recognition. Features based on prosody then derivate a Modified QDF （MQDF） to speech and voice quality are extracted and Principal Component Analysis Neural Network （PCANN） is used to reduce dimension of the feature vectors. The results show that voice quality features are effective supplement for recognition, and the method in this paper could improve the recognition ratio effectively.展开更多
This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via statedependent switching rules.Several attempts have been made to solve this problem via different types ...This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via statedependent switching rules.Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid.These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type.In some cases,the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function.In this paper,we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently.The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size.Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size,i.e.,a purely numerical method and a numerical-analytical one.In the former,both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities.The latter is established on a principle that the attractiveness of a set implies its invariance.Thus,the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension.Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.展开更多
A quadratic yield function which can describe the anisotropic behaviors of sheet metals with tension/compression symmetry and asymmetry is proposed.Five mechanical properties are adopted to determine the coefficients ...A quadratic yield function which can describe the anisotropic behaviors of sheet metals with tension/compression symmetry and asymmetry is proposed.Five mechanical properties are adopted to determine the coefficients of each part of the yield function.For particular cases,the proposed yield function can be simplified to Mises or Hill’s quadratic yield function.The anisotropic mechanical properties are expressed by defining an angle between the current normalized principal stress space and the reference direction with the assumption of orthotropic anisotropy.The accuracy of the proposed yield function in describing the anisotropy under tension and compression is demonstrated.展开更多
This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attaine...This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions （pwLDIs） model of synchronisation error dynamics, then we design a switching （piecewise-linear） feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.展开更多
In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ...In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ＋ f（t, x） we give the instability of the zero solution when f is sufficiently small.展开更多
This study investigates the efficacy of the Mathematics Independent Learning Activity Practice and Play Unite Scheme(MILAPlus)as an instructional strategy to improve the proficiency levels of Grade 9 students in quadr...This study investigates the efficacy of the Mathematics Independent Learning Activity Practice and Play Unite Scheme(MILAPlus)as an instructional strategy to improve the proficiency levels of Grade 9 students in quadratic equations and functions through a study carried out at Quezon National High School.The research involved 116 Grade 9 students and utilized a quantitative approach,incorporating both pre-assessment and post-assessment measures.The research utilizes a quasi-experimental design,examining the academic performance of students before and after the introduction of MILAPlus.The pre-assessment establishes a baseline,and the subsequent post-assessment measures the impact of the instructional strategy.Statistical analyses,including t-tests,assess the significance of differences in mean scores and mean percentage scores,providing quantitative insights into the effectiveness of MILAPlus.Findings from the study revealed a statistically significant improvement in both mean scores and mean percentage scores after the utilization of MILAPlus,indicating enhanced proficiency in quadratic equations and functions.The Mean Proficiency Scores(MPS)also showed a substantial increase,demonstrating a marked improvement in overall proficiency levels among Grade 9 students.In light of the results,recommendations were given including the continued utilization of MILAPlus as an instructional strategy and aligning its development with prescribed learning competencies.Emphasizing the consistent adherence to policies and guidelines for MILAPlus implementation is suggested for sustaining positive effects on students’long-term performance in mathematics.This research contributes valuable insights into the practical application and effectiveness of MILAPlus within the context of Grade 9 mathematics education at Quezon National High School.展开更多
Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search d...Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search directions are obtained using the quadratic hypothesis of objective functions.Based on derivatives,quadratic function assumptions,and directional derivatives,the computational formulas of numerical first-order partial derivatives,second-order partial derivatives,and numerical second-order mixed partial derivatives were constructed.Based on the coordinate transformation relation,a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction.A numerical algorithm was proposed,taking the second order approximation direction as an example.A large stepsize numerical algorithm based on coordinate transformation was proposed.Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function.The numerical second order approximation direction with the numerical mixed partial derivatives showed good results.Its calculated amount is 0.2843%of that of without second-order mixed partial derivative.In the process of rotating the local coordinate system 360°,because the objective function is more complex than the quadratic function,if the numerical direction derivative is used instead of the analytic partial derivative,the optimization direction varies with a range of 103.05°.Because theoretical error is in the numerical negative gradient direction,the calculation with the coordinate transformation is 94.71%less than the calculation without coordinate transformation.If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation,the sawtooth phenomenon occurs.When each numerical mixed partial derivative takes more than one point,the optimization results cannot be improved.The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained,but does not require derivability and does not take into account truncation error and rounding error.Thus,the application scopes of many optimization methods are extended.展开更多
Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary m...Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.展开更多
Differential inequalities generated in an extended Lyapunov framework are employed in the stability and instability analyses of a class of switched continuous-time second-and higher order linear systems with an arbitr...Differential inequalities generated in an extended Lyapunov framework are employed in the stability and instability analyses of a class of switched continuous-time second-and higher order linear systems with an arbitrary number of switching matrices.The exponential stability and instability(ESI)conditions so obtained involve the supremum and infimum of ratios of certain quadratic forms of the matrices,leading to global time-averages of their activity intervals.Further,motivated by linear switching system examples of(i)instability with stable matrices and(ii)stability with unstable matrices(found in the literature primarily for second-order systems),the proposed framework is generalized to establish ESI conditions that include both the activity intervals of the matrices and their switching rates,the latter being governed by a certain logarithmic measure of the normalized magnitudes of discontinuities caused by switching.In effect,(the new,globally averaged)dwell-time is flexibly traded,apparently for the first time,but under specific conditions(related,in part,to the eigenvalues of the matrices),for switching discontinuity-based conditions.Two further novel aspects of the proposed approach are:(i)For second-order matrices,switching lines in phase space can be chosen for periodic switching to stabilize or destabilize the system,and even generate oscillations,depending on the eigenvalues of the system matrices.But for third-(and higher)order matrices,such an analytically tractable(and controlled)periodical switching entails solution of an explicit non-convex multi-parameter optimization problem for which a stochastic optimization algorithm from the literature can be invoked.(ii)Lower and upper bounds on the solutions of the system equations can be quantified to reflect the stability/instability/oscillatory property of the system.Illustrative examples,which demonstrate the novelty of the derived stability and instability conditions,are presented in part 2 which is advisedly to be read along with this part 1 for a coherent merging of theory with practice.展开更多
In this second part of the paper,bearing the same title as above,but with the last hyphenated phrase replaced by part 1(Theory),the exponential stability and instability(ESI)Theorems 1–4 of part 1 are illustrated by ...In this second part of the paper,bearing the same title as above,but with the last hyphenated phrase replaced by part 1(Theory),the exponential stability and instability(ESI)Theorems 1–4 of part 1 are illustrated by applying them to second-andby,say,third-order linear switched systems with different eigenvalue structures to demonstrate the versatility,novelty and superiority(over many of the results found in the literature,especially for second-order switched lined systems)of the new theoretical results.The computational procedure that is employed with reference to the third-order systems is generic,in the sense that it is applicable to higher(i.e.,greater than third-)order linear switched systems.A pseudo-code for a computer implementation of the stability/instability conditions is also presented.With the principal aim of facilitating an independent reading of this part 2 of the paper,some crucial mathematical notations,definitions and results of part 1 have been repeated,thereby making the contents as self-contained as possible.展开更多
In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector f...In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.展开更多
In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to deri...In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.展开更多
基金the National Natural Science Foundation of China(62273058,U22A2045)the Key Science and Technology Projects of Jilin Province(20200401075GX)the Youth Science and Technology Innovation and Entrepreneurship Outstanding Talents Project of Jilin Province(20230508043RC)。
文摘This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.
文摘In this paper we analyze the optimal control problem for a class of afflne nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.
基金Supported by the National Natural Science Foundation of China(Grant No.11401190)Humanities and Social Sciences of Ministry of Education Planning Fund(Grant No.17YJA790098)
文摘Using the direct method,we investigate the generalized Hyers-Ulam stability of the following quadratic functional inequality■inβ-homogeneous complex Banach spaces.
基金Supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (Grant No. KRF-2007-521-C00016)
文摘We consider the Hyers-Ulam stability problem of the generalized quadratic functional equationuoA＋voB-2woP1 - 2ko P2 =0, which is a distributional version of the classical generalized quadratic functional equation f（x＋y）＋g（x - y） - 2h（x） - 2k（y）=0
文摘In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[
文摘The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.
基金Supported by National Natural Science Foundation of China （Grant No. 10131010）
文摘A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.
基金the Ministry of Education Fund (No: 20050286001)Ministry of Education "New Century Tal-ents Support Plan" (No:NCET-04-0483)Doctoral Foundation of Ministry of Education (No:20050286001).
文摘Quadratic Discrimination Function （QDF） is commonly used in speech emotion recognition, which proceeds on the premise that the input data is normal distribution. In this paper, we propose a transformation to normalize the emotional features, emotion recognition. Features based on prosody then derivate a Modified QDF （MQDF） to speech and voice quality are extracted and Principal Component Analysis Neural Network （PCANN） is used to reduce dimension of the feature vectors. The results show that voice quality features are effective supplement for recognition, and the method in this paper could improve the recognition ratio effectively.
文摘This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via statedependent switching rules.Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid.These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type.In some cases,the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function.In this paper,we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently.The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size.Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size,i.e.,a purely numerical method and a numerical-analytical one.In the former,both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities.The latter is established on a principle that the attractiveness of a set implies its invariance.Thus,the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension.Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.
基金supported by the National Natural Science Foundation of China (Grant Nos.51475003 and 51205004)Beijing Natural Science Foundation (Grant No.3152010)+1 种基金open project of "State Key Laboratory of Solidification Processing" of Northwestern Polytechnical University (No.SKLSP201635)Beijing Education Committee Science and Technology Program (Grant No.KM201510009004)
文摘A quadratic yield function which can describe the anisotropic behaviors of sheet metals with tension/compression symmetry and asymmetry is proposed.Five mechanical properties are adopted to determine the coefficients of each part of the yield function.For particular cases,the proposed yield function can be simplified to Mises or Hill’s quadratic yield function.The anisotropic mechanical properties are expressed by defining an angle between the current normalized principal stress space and the reference direction with the assumption of orthotropic anisotropy.The accuracy of the proposed yield function in describing the anisotropy under tension and compression is demonstrated.
基金Project partially supported by the grant from the Research Grants Council of the Hong Kong Special Administrative Region,China (Grant No. 101005)the National Natural Science Foundation of China (Grant No. 60904004)the Key Youth Science and Technology Foundation of University of Electronic Science and Technology of China (Grant No. L08010201JX0720)
文摘This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions （pwLDIs） model of synchronisation error dynamics, then we design a switching （piecewise-linear） feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.
文摘In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ＋ f（t, x） we give the instability of the zero solution when f is sufficiently small.
文摘This study investigates the efficacy of the Mathematics Independent Learning Activity Practice and Play Unite Scheme(MILAPlus)as an instructional strategy to improve the proficiency levels of Grade 9 students in quadratic equations and functions through a study carried out at Quezon National High School.The research involved 116 Grade 9 students and utilized a quantitative approach,incorporating both pre-assessment and post-assessment measures.The research utilizes a quasi-experimental design,examining the academic performance of students before and after the introduction of MILAPlus.The pre-assessment establishes a baseline,and the subsequent post-assessment measures the impact of the instructional strategy.Statistical analyses,including t-tests,assess the significance of differences in mean scores and mean percentage scores,providing quantitative insights into the effectiveness of MILAPlus.Findings from the study revealed a statistically significant improvement in both mean scores and mean percentage scores after the utilization of MILAPlus,indicating enhanced proficiency in quadratic equations and functions.The Mean Proficiency Scores(MPS)also showed a substantial increase,demonstrating a marked improvement in overall proficiency levels among Grade 9 students.In light of the results,recommendations were given including the continued utilization of MILAPlus as an instructional strategy and aligning its development with prescribed learning competencies.Emphasizing the consistent adherence to policies and guidelines for MILAPlus implementation is suggested for sustaining positive effects on students’long-term performance in mathematics.This research contributes valuable insights into the practical application and effectiveness of MILAPlus within the context of Grade 9 mathematics education at Quezon National High School.
基金supported in part by the Teaching Reform Research Foundation of Shengli College in China University of Petroleum(East China)(JG201725)the Natural Science Foundation Shandong Province of China(ZR2018PEE009)the Project of Science and Technology of Shandong Universities in China(J17KA044,J17KB061)。
文摘Many effective optimization algorithms require partial derivatives of objective functions,while some optimization problems'objective functions have no derivatives.According to former research studies,some search directions are obtained using the quadratic hypothesis of objective functions.Based on derivatives,quadratic function assumptions,and directional derivatives,the computational formulas of numerical first-order partial derivatives,second-order partial derivatives,and numerical second-order mixed partial derivatives were constructed.Based on the coordinate transformation relation,a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction.A numerical algorithm was proposed,taking the second order approximation direction as an example.A large stepsize numerical algorithm based on coordinate transformation was proposed.Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function.The numerical second order approximation direction with the numerical mixed partial derivatives showed good results.Its calculated amount is 0.2843%of that of without second-order mixed partial derivative.In the process of rotating the local coordinate system 360°,because the objective function is more complex than the quadratic function,if the numerical direction derivative is used instead of the analytic partial derivative,the optimization direction varies with a range of 103.05°.Because theoretical error is in the numerical negative gradient direction,the calculation with the coordinate transformation is 94.71%less than the calculation without coordinate transformation.If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation,the sawtooth phenomenon occurs.When each numerical mixed partial derivative takes more than one point,the optimization results cannot be improved.The numerical direction based on the quadratic hypothesis only requires the objective function to be obtained,but does not require derivability and does not take into account truncation error and rounding error.Thus,the application scopes of many optimization methods are extended.
基金The authors would like to thank Prof. Y.D. Zhang for selfless helps and valuable discussions.
文摘Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.
文摘Differential inequalities generated in an extended Lyapunov framework are employed in the stability and instability analyses of a class of switched continuous-time second-and higher order linear systems with an arbitrary number of switching matrices.The exponential stability and instability(ESI)conditions so obtained involve the supremum and infimum of ratios of certain quadratic forms of the matrices,leading to global time-averages of their activity intervals.Further,motivated by linear switching system examples of(i)instability with stable matrices and(ii)stability with unstable matrices(found in the literature primarily for second-order systems),the proposed framework is generalized to establish ESI conditions that include both the activity intervals of the matrices and their switching rates,the latter being governed by a certain logarithmic measure of the normalized magnitudes of discontinuities caused by switching.In effect,(the new,globally averaged)dwell-time is flexibly traded,apparently for the first time,but under specific conditions(related,in part,to the eigenvalues of the matrices),for switching discontinuity-based conditions.Two further novel aspects of the proposed approach are:(i)For second-order matrices,switching lines in phase space can be chosen for periodic switching to stabilize or destabilize the system,and even generate oscillations,depending on the eigenvalues of the system matrices.But for third-(and higher)order matrices,such an analytically tractable(and controlled)periodical switching entails solution of an explicit non-convex multi-parameter optimization problem for which a stochastic optimization algorithm from the literature can be invoked.(ii)Lower and upper bounds on the solutions of the system equations can be quantified to reflect the stability/instability/oscillatory property of the system.Illustrative examples,which demonstrate the novelty of the derived stability and instability conditions,are presented in part 2 which is advisedly to be read along with this part 1 for a coherent merging of theory with practice.
文摘In this second part of the paper,bearing the same title as above,but with the last hyphenated phrase replaced by part 1(Theory),the exponential stability and instability(ESI)Theorems 1–4 of part 1 are illustrated by applying them to second-andby,say,third-order linear switched systems with different eigenvalue structures to demonstrate the versatility,novelty and superiority(over many of the results found in the literature,especially for second-order switched lined systems)of the new theoretical results.The computational procedure that is employed with reference to the third-order systems is generic,in the sense that it is applicable to higher(i.e.,greater than third-)order linear switched systems.A pseudo-code for a computer implementation of the stability/instability conditions is also presented.With the principal aim of facilitating an independent reading of this part 2 of the paper,some crucial mathematical notations,definitions and results of part 1 have been repeated,thereby making the contents as self-contained as possible.
文摘In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.
基金supported by the National Natural Science Foundation of China(Nos.12101572,12371256)2023 Shanxi Province Graduate Innovation Project(No.2023KY614)the 19th Graduate Science and Technology Project of North University of China(No.20231943)。
文摘In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.
