To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constr...To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constraints in event space is presented, secondly, the Noether's theorem and the Noether's inverse theorem for the nonholonomic systems of non_Chetaev's type with unilateral constraints in event space are studied and obtained, which is based upon the invariance of the differential variational principle under the infinitesimal transformations of group, finally, an example is given to illustrate the application of the result.展开更多
The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether's theorem and to reestablish rather complete conservation laws and balance equations for co...The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether's theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.展开更多
The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff＇s equations and the corresponding ...The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff＇s equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether＇s theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether＇s theorems are established. Finally, an example is given to illustrate the application of the results.展开更多
This paper focuses on the Noether symmetries and the conserved quantities for both holonomic and nonholonomic systems based on a new non-conservative dynamical model introduced by E1-Nabulsi. First, the E1-Nabulsi dyn...This paper focuses on the Noether symmetries and the conserved quantities for both holonomic and nonholonomic systems based on a new non-conservative dynamical model introduced by E1-Nabulsi. First, the E1-Nabulsi dynamical model which is based on a fractional integral extended by periodic laws is introduced, and E1-Nabulsi-Hamilton＇s canoni- cal equations for non-conservative Hamilton system with holonomic or nonholonomic constraints are established. Second, the definitions and criteria of E1-Nabulsi-Noether symmetrical transformations and quasi-symmetrical transformations are presented in terms of the invariance of E1-Nabulsi-Hamilton action under the infinitesimal transformations of the group. Fi- nally, Noether＇s theorems for the non-conservative Hamilton system under the E1-Nabulsi dynamical system are established, which reveal the relationship between the Noether symmetry and the conserved quantity of the system.展开更多
The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations wit...The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.展开更多
Let C be a smooth projective curve over C, and K be the canonical line bundle. A well-known Theorem of Noether says that if C is not a hyperelliptic curve, then the map H^o(C,K)(?)H^o(C,K)→H^o(C, 2K) is surjective,wh...Let C be a smooth projective curve over C, and K be the canonical line bundle. A well-known Theorem of Noether says that if C is not a hyperelliptic curve, then the map H^o(C,K)(?)H^o(C,K)→H^o(C, 2K) is surjective,which means that K is normally generated.展开更多
Noether’s theorem[1],relating the symmetry of a physical system to its constant of motion,is a fundamental physical law. It has been shown that the expectation value of a time-independent operator is a constant of mo...Noether’s theorem[1],relating the symmetry of a physical system to its constant of motion,is a fundamental physical law. It has been shown that the expectation value of a time-independent operator is a constant of motion if it commutes with the Hermitian Hamiltonian. However,applying Noether’s theorem to open systems and obtaining the corresponding conserved quantities are still tricky.展开更多
Noether’s theorem is one of the fundamental laws in physics,relating the symmetry of a physical system to its constant of motion and conservation law.On the other hand,there exist a variety of non-Hermitian parity-ti...Noether’s theorem is one of the fundamental laws in physics,relating the symmetry of a physical system to its constant of motion and conservation law.On the other hand,there exist a variety of non-Hermitian parity-time(PT)-symmetric systems,which exhibit novel quantum properties and have attracted increasing interest.In this work,we extend Noether’s theorem to a class of significant PT-symmetry systems for which the eigenvalues of the PT-symmetry Hamiltonian HPTchange from purely real numbers to purely imaginary numbers,and introduce a generalized expectation value of an operator based on biorthogonal quantum mechanics.We find that the generalized expectation value of a time-independent operator is a constant of motion when the operator presents a standard symmetry in the PT-symmetry unbroken regime,or a chiral symmetry in the PT-symmetry broken regime.In addition,we experimentally investigate the extended Noether’s theorem in PT-symmetry single-qubit and two-qubit systems using an optical setup.Our experiment demonstrates the existence of the constant of motion and reveals how this constant of motion can be used to judge whether the PT-symmetry of a system is broken.Furthermore,a novel phenomenon of masking quantum information is first observed in a PT-symmetry two-qubit system.This study not only contributes to full understanding of the relation between symmetry and conservation law in PT-symmetry physics,but also has potential applications in quantum information theory and quantum communication protocols.展开更多
Fractional calculus is widely used to deal with nonconservative dynamics because of its memorability and non-local properties.In this paper,the Herglotz principle with generalized operators is discussed,and the Herglo...Fractional calculus is widely used to deal with nonconservative dynamics because of its memorability and non-local properties.In this paper,the Herglotz principle with generalized operators is discussed,and the Herglotz type equations for nonholonomic systems are established.Then,the Noether symmetries are studied,and the conserved quantities are obtained.The results are extended to nonholonomic canonical systems,and the Herglotz type canonical equations and the Noether theorems are obtained.Two examples are provided to demonstrate the validity of the methods and results.展开更多
In this paper,the Paley-Wiener theorem is extended to the analytic function spaces with general weights.We first generalize the theorem to weighted Hardy spaces Hp(0<p<∞)on tube domains by constructing a sequen...In this paper,the Paley-Wiener theorem is extended to the analytic function spaces with general weights.We first generalize the theorem to weighted Hardy spaces Hp(0<p<∞)on tube domains by constructing a sequence of L^(1)functions converging to the given function and verifying their representation in the form of Fourier transform to establish the desired result of the given function.Applying this main result,we further generalize the Paley-Wiener theorem for band-limited functions to the analytic function spaces L^(p)(0<p<∞)with general weights.展开更多
By using the synmeby and the qusi-symmetry of the infinitesimal transformation of the transformation group G1 and by imposing restrictions of constraints on the transformation, the Noether's theory of constrained ...By using the synmeby and the qusi-symmetry of the infinitesimal transformation of the transformation group G1 and by imposing restrictions of constraints on the transformation, the Noether's theory of constrained Birkhoffian system has been established. The theory includes the generalized Noether's theorem obtaining the first integrals from the known symmetry and quasi-symmetry and its inverse obtaining the corresponding symmetry and quasi-symmetry from the known first integrals for the systerm.展开更多
Quantum software development utilizes quantum phenomena such as superposition and entanglement to address problems that are challenging for classical systems.However,it must also adhere to critical quantum constraints...Quantum software development utilizes quantum phenomena such as superposition and entanglement to address problems that are challenging for classical systems.However,it must also adhere to critical quantum constraints,notably the no-cloning theorem,which prohibits the exact duplication of unknown quantum states and has profound implications for cryptography,secure communication,and error correction.While existing quantum circuit representations implicitly honor such constraints,they lack formal mechanisms for early-stage verification in software design.Addressing this constraint at the design phase is essential to ensure the correctness and reliability of quantum software.This paper presents a formal metamodeling framework using UML-style notation and and Object Constraint Language(OCL)to systematically capture and enforce the no-cloning theorem within quantum software models.The proposed metamodel formalizes key quantum concepts—such as entanglement and teleportation—and encodes enforceable invariants that reflect core quantum mechanical laws.The framework’s effectiveness is validated by analyzing two critical edge cases—conditional copying with CNOT gates and quantum teleportation—through instance model evaluations.These cases demonstrate that the metamodel can capture nuanced scenarios that are often mistaken as violations of the no-cloning theorem but are proven compliant under formal analysis.Thus,these serve as constructive validations that demonstrate the metamodel’s expressiveness and correctness in representing operations that may appear to challenge the no-cloning theorem but,upon rigorous analysis,are shown to comply with it.The approach supports early detection of conceptual design errors,promoting correctness prior to implementation.The framework’s extensibility is also demonstrated by modeling projective measurement,further reinforcing its applicability to broader quantum software engineering tasks.By integrating the rigor of metamodeling with fundamental quantum mechanical principles,this work provides a structured,model-driven approach that enables traditional software engineers to address quantum computing challenges.It offers practical insights into embedding quantum correctness at the modeling level and advances the development of reliable,error-resilient quantum software systems.展开更多
We propose the scaling rule of Morse oscillator,based on this rule and by virtue of the Her-mann-Feymann theorem,we respectively obtain the distribution of potential and kinetic ener-gy of the Morse Hamiltonian.Also,w...We propose the scaling rule of Morse oscillator,based on this rule and by virtue of the Her-mann-Feymann theorem,we respectively obtain the distribution of potential and kinetic ener-gy of the Morse Hamiltonian.Also,we derive the exact upper limit of physical energy level.Further,we derive some recursive relations for energy matrix elements of the potential and other similar operators in the context of Morse oscillator theory.展开更多
The Steiner-Lehmus equal bisectors theorem originated in the mid 19th century.Despite its age,it would have been accessible to Euclid and his contemporaries.The theorem remains evergreen,with new proofs continuing to ...The Steiner-Lehmus equal bisectors theorem originated in the mid 19th century.Despite its age,it would have been accessible to Euclid and his contemporaries.The theorem remains evergreen,with new proofs continuing to appear steadily.The theorem has fostered discussion about the nature of proof itself,direct and indirect.Here we continue the momentum by providing a trigonometric proof,relatively short,based on an analytic estimate that leverages algebraic trigonometric identities.Many proofs of the theorem exist in the literature.Some of these contain key ideas that already appeared in C.L.Lehmus’1850 proofs,not always with citation.In the aim of increasing awareness of and making more accessible Lehmus’proofs,we provide an annotated translation.We conclude with remarks on different proofs and relations among them.展开更多
In this paper,we study Liouville theorem for the 3D stationary Q-tensor system of liquid crystal in Lorentz and Morrey spaces.Under some additional hypotheses,stated in terms of Lorentz and Morrey spaces,using energy ...In this paper,we study Liouville theorem for the 3D stationary Q-tensor system of liquid crystal in Lorentz and Morrey spaces.Under some additional hypotheses,stated in terms of Lorentz and Morrey spaces,using energy estimation,we obtain that the trivial solution u=Q=0 is the unique solution.Our theorems correspond to improvements of some recent results and contain some known results as particular cases.展开更多
In this paper,we study the basic p-harmonic forms on the complete foliated Riemannian manifolds.By using the method in[1],we show that if the basic mean curvature form is bounded and co-closed,and the transversal curv...In this paper,we study the basic p-harmonic forms on the complete foliated Riemannian manifolds.By using the method in[1],we show that if the basic mean curvature form is bounded and co-closed,and the transversal curvature operator is nonnegative and positive at least one point,then we obtain a vanishing theorem for L^(p)-integrably p-harmonic r-forms.展开更多
In this paper,we obtain a vector bundle valued mixed hard Lefschetz theorem.The argument is mainly based on the works of Tien-Cuong Dinh and Viet-Anh Nguyen.
In this paper,we compute sub-Riemannian limits of some important curvature variants associated with the connection with torsion for four dimensional twisted BCV spaces and derive a Gauss-Bonnet theorem for four dimens...In this paper,we compute sub-Riemannian limits of some important curvature variants associated with the connection with torsion for four dimensional twisted BCV spaces and derive a Gauss-Bonnet theorem for four dimensional twisted BCV spaces.展开更多
this paper,we study Liouville theorem for 3D steady Q-tensor system of liquid crystal in mixed Lorentz spaces.We obtain u=0,Q=0 on the conditions that μ∈L^(p,∞x_(1)L^(q,∞x_(2)L^(r,∞x_(3)(R^(4)∩H^(1)(R^(3),Q∈H^(...this paper,we study Liouville theorem for 3D steady Q-tensor system of liquid crystal in mixed Lorentz spaces.We obtain u=0,Q=0 on the conditions that μ∈L^(p,∞x_(1)L^(q,∞x_(2)L^(r,∞x_(3)(R^(4)∩H^(1)(R^(3),Q∈H^(2)(R^(3),p,q,r∈(3,∞],and 1/p+1/q+1/r≥2/3, which extends some known results.展开更多
文摘To study the Noether's theorem of nonholonomic systems of non_Chetaev's type with unilateral constraints in event space, firstly, the principle of D'Alembert_Lagrange for the systems with unilateral constraints in event space is presented, secondly, the Noether's theorem and the Noether's inverse theorem for the nonholonomic systems of non_Chetaev's type with unilateral constraints in event space are studied and obtained, which is based upon the invariance of the differential variational principle under the infinitesimal transformations of group, finally, an example is given to illustrate the application of the result.
文摘The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether's theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10972151 and 11272227the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(Grant No.CXZZ11 0949)
文摘The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff＇s equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether＇s theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether＇s theorems are established. Finally, an example is given to illustrate the application of the results.
基金supported by the National Natural Science Foundation of China(Grant Nos.10972151 and 11272227)the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(Grant No.CXLX11_0961)
文摘This paper focuses on the Noether symmetries and the conserved quantities for both holonomic and nonholonomic systems based on a new non-conservative dynamical model introduced by E1-Nabulsi. First, the E1-Nabulsi dynamical model which is based on a fractional integral extended by periodic laws is introduced, and E1-Nabulsi-Hamilton＇s canoni- cal equations for non-conservative Hamilton system with holonomic or nonholonomic constraints are established. Second, the definitions and criteria of E1-Nabulsi-Noether symmetrical transformations and quasi-symmetrical transformations are presented in terms of the invariance of E1-Nabulsi-Hamilton action under the infinitesimal transformations of the group. Fi- nally, Noether＇s theorems for the non-conservative Hamilton system under the E1-Nabulsi dynamical system are established, which reveal the relationship between the Noether symmetry and the conserved quantity of the system.
基金National Natural Science Foundations of China(Nos.11572212,11272227,10972151)the Innovation Program for Scientific Research of Nanjing University of Science and Technology,Chinathe Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(No.KYLX15_0405)
文摘The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.
文摘Let C be a smooth projective curve over C, and K be the canonical line bundle. A well-known Theorem of Noether says that if C is not a hyperelliptic curve, then the map H^o(C,K)(?)H^o(C,K)→H^o(C, 2K) is surjective,which means that K is normally generated.
文摘Noether’s theorem[1],relating the symmetry of a physical system to its constant of motion,is a fundamental physical law. It has been shown that the expectation value of a time-independent operator is a constant of motion if it commutes with the Hermitian Hamiltonian. However,applying Noether’s theorem to open systems and obtaining the corresponding conserved quantities are still tricky.
基金supported by the National Natural Science Foundation of China(Grant Nos.12264040,12204311,11804228,11865013,and U21A20436)the Jiangxi Natural Science Foundation(Grant Nos.20212BAB211018,20192ACBL20051)+8 种基金the Project of Jiangxi Province Higher Educational Science and Technology Program(Grant Nos.GJJ190891,and GJJ211735)the Key-Area Research and Development Program of Guangdong Province(Grant No.2018B03-0326001)supported in part by the Nippon Telegraph and Telephone(NTT)Corporation Researchthe Japan Science and Technology(JST)Agency[via the Quantum Leap Flagship Program(Q-LEAP)Moonshot R&D Grant Number JPMJMS2061]the Japan Society for the Promotion of Science(JSPS)[via the Grants-in-Aid for Scientific Research(KAKENHI)Grant No.JP20H00134]the Army Research Office(ARO)(Grant No.W911NF-18-1-0358)the Asian Office of Aerospace Research and Development(AOARD)(Grant No.FA2386-20-1-4069)the Foundational Questions Institute Fund(FQXi)(Grant No.FQXi-IAF19-06)。
文摘Noether’s theorem is one of the fundamental laws in physics,relating the symmetry of a physical system to its constant of motion and conservation law.On the other hand,there exist a variety of non-Hermitian parity-time(PT)-symmetric systems,which exhibit novel quantum properties and have attracted increasing interest.In this work,we extend Noether’s theorem to a class of significant PT-symmetry systems for which the eigenvalues of the PT-symmetry Hamiltonian HPTchange from purely real numbers to purely imaginary numbers,and introduce a generalized expectation value of an operator based on biorthogonal quantum mechanics.We find that the generalized expectation value of a time-independent operator is a constant of motion when the operator presents a standard symmetry in the PT-symmetry unbroken regime,or a chiral symmetry in the PT-symmetry broken regime.In addition,we experimentally investigate the extended Noether’s theorem in PT-symmetry single-qubit and two-qubit systems using an optical setup.Our experiment demonstrates the existence of the constant of motion and reveals how this constant of motion can be used to judge whether the PT-symmetry of a system is broken.Furthermore,a novel phenomenon of masking quantum information is first observed in a PT-symmetry two-qubit system.This study not only contributes to full understanding of the relation between symmetry and conservation law in PT-symmetry physics,but also has potential applications in quantum information theory and quantum communication protocols.
基金supported by the National Natural Science Foundation of China(Grant No.12272248)the Postgraduate Research and Practice Innovation Program of Jiangsu Province of China(Grant No.KYCX23_3296).
文摘Fractional calculus is widely used to deal with nonconservative dynamics because of its memorability and non-local properties.In this paper,the Herglotz principle with generalized operators is discussed,and the Herglotz type equations for nonholonomic systems are established.Then,the Noether symmetries are studied,and the conserved quantities are obtained.The results are extended to nonholonomic canonical systems,and the Herglotz type canonical equations and the Noether theorems are obtained.Two examples are provided to demonstrate the validity of the methods and results.
基金Supported by the National Natural Science Foundation of China(12301101)the Guangdong Basic and Applied Basic Research Foundation(2022A1515110019 and 2020A1515110585)。
文摘In this paper,the Paley-Wiener theorem is extended to the analytic function spaces with general weights.We first generalize the theorem to weighted Hardy spaces Hp(0<p<∞)on tube domains by constructing a sequence of L^(1)functions converging to the given function and verifying their representation in the form of Fourier transform to establish the desired result of the given function.Applying this main result,we further generalize the Paley-Wiener theorem for band-limited functions to the analytic function spaces L^(p)(0<p<∞)with general weights.
文摘By using the synmeby and the qusi-symmetry of the infinitesimal transformation of the transformation group G1 and by imposing restrictions of constraints on the transformation, the Noether's theory of constrained Birkhoffian system has been established. The theory includes the generalized Noether's theorem obtaining the first integrals from the known symmetry and quasi-symmetry and its inverse obtaining the corresponding symmetry and quasi-symmetry from the known first integrals for the systerm.
文摘Quantum software development utilizes quantum phenomena such as superposition and entanglement to address problems that are challenging for classical systems.However,it must also adhere to critical quantum constraints,notably the no-cloning theorem,which prohibits the exact duplication of unknown quantum states and has profound implications for cryptography,secure communication,and error correction.While existing quantum circuit representations implicitly honor such constraints,they lack formal mechanisms for early-stage verification in software design.Addressing this constraint at the design phase is essential to ensure the correctness and reliability of quantum software.This paper presents a formal metamodeling framework using UML-style notation and and Object Constraint Language(OCL)to systematically capture and enforce the no-cloning theorem within quantum software models.The proposed metamodel formalizes key quantum concepts—such as entanglement and teleportation—and encodes enforceable invariants that reflect core quantum mechanical laws.The framework’s effectiveness is validated by analyzing two critical edge cases—conditional copying with CNOT gates and quantum teleportation—through instance model evaluations.These cases demonstrate that the metamodel can capture nuanced scenarios that are often mistaken as violations of the no-cloning theorem but are proven compliant under formal analysis.Thus,these serve as constructive validations that demonstrate the metamodel’s expressiveness and correctness in representing operations that may appear to challenge the no-cloning theorem but,upon rigorous analysis,are shown to comply with it.The approach supports early detection of conceptual design errors,promoting correctness prior to implementation.The framework’s extensibility is also demonstrated by modeling projective measurement,further reinforcing its applicability to broader quantum software engineering tasks.By integrating the rigor of metamodeling with fundamental quantum mechanical principles,this work provides a structured,model-driven approach that enables traditional software engineers to address quantum computing challenges.It offers practical insights into embedding quantum correctness at the modeling level and advances the development of reliable,error-resilient quantum software systems.
基金supported by the National Natural Science Foundation of China(No.10874174)。
文摘We propose the scaling rule of Morse oscillator,based on this rule and by virtue of the Her-mann-Feymann theorem,we respectively obtain the distribution of potential and kinetic ener-gy of the Morse Hamiltonian.Also,we derive the exact upper limit of physical energy level.Further,we derive some recursive relations for energy matrix elements of the potential and other similar operators in the context of Morse oscillator theory.
文摘The Steiner-Lehmus equal bisectors theorem originated in the mid 19th century.Despite its age,it would have been accessible to Euclid and his contemporaries.The theorem remains evergreen,with new proofs continuing to appear steadily.The theorem has fostered discussion about the nature of proof itself,direct and indirect.Here we continue the momentum by providing a trigonometric proof,relatively short,based on an analytic estimate that leverages algebraic trigonometric identities.Many proofs of the theorem exist in the literature.Some of these contain key ideas that already appeared in C.L.Lehmus’1850 proofs,not always with citation.In the aim of increasing awareness of and making more accessible Lehmus’proofs,we provide an annotated translation.We conclude with remarks on different proofs and relations among them.
基金Supported by National Natural Science Foundation of China(11871305,11901346).
文摘In this paper,we study Liouville theorem for the 3D stationary Q-tensor system of liquid crystal in Lorentz and Morrey spaces.Under some additional hypotheses,stated in terms of Lorentz and Morrey spaces,using energy estimation,we obtain that the trivial solution u=Q=0 is the unique solution.Our theorems correspond to improvements of some recent results and contain some known results as particular cases.
基金supported by Guangzhou Science and Technology Program(202102021174)Guangdong Basic and Applied Basic Research Foundation(2023A1515012121).The second author was supported by the Natural Science Foundation of Jiangsu Province(BK20230900)+1 种基金the Fundamental Research Funds for the Central Universities(30924010838)Both authors are partially supported by NSF in China(12141104).
文摘In this paper,we study the basic p-harmonic forms on the complete foliated Riemannian manifolds.By using the method in[1],we show that if the basic mean curvature form is bounded and co-closed,and the transversal curvature operator is nonnegative and positive at least one point,then we obtain a vanishing theorem for L^(p)-integrably p-harmonic r-forms.
基金supported by the National key R and D Program of China 2020YFA0713100the NSFC(12141104,12371062 and 12431004).
文摘In this paper,we obtain a vector bundle valued mixed hard Lefschetz theorem.The argument is mainly based on the works of Tien-Cuong Dinh and Viet-Anh Nguyen.
基金Supported by National Natural Science Foundation of China(Grant No.11771070).
文摘In this paper,we compute sub-Riemannian limits of some important curvature variants associated with the connection with torsion for four dimensional twisted BCV spaces and derive a Gauss-Bonnet theorem for four dimensional twisted BCV spaces.
基金Supported by the National Natural Science Foundation of China(11871305)。
文摘this paper,we study Liouville theorem for 3D steady Q-tensor system of liquid crystal in mixed Lorentz spaces.We obtain u=0,Q=0 on the conditions that μ∈L^(p,∞x_(1)L^(q,∞x_(2)L^(r,∞x_(3)(R^(4)∩H^(1)(R^(3),Q∈H^(2)(R^(3),p,q,r∈(3,∞],and 1/p+1/q+1/r≥2/3, which extends some known results.
