Finding the nearest volume-preserving matrix for a given matrix is studied. Amatrix equation is first obtained, which is a necessary condition for the solution to the problem.Then the equation is solved by the singula...Finding the nearest volume-preserving matrix for a given matrix is studied. Amatrix equation is first obtained, which is a necessary condition for the solution to the problem.Then the equation is solved by the singular value decomposition method. Some additional results arealso provided to further characterize the solution. Using these results, a numerical algorithm isintroduced and a numerical test is given to illustrate the effectiveness of the algorithm.展开更多
We prove that, for non-uniformly hyperbolic diffeomorphisms in the sense of Young, the local central limit theorem holds, and the speed in the central limit theorem is O（1/√n）.
Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously i...Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2：=σ~2 （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A 〉 0 such that, for any integer n ≥ 1,Π（ m＊（ 1√ nS f n ）,N （0,σ~2 ） ≤A√n, where m＊（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.展开更多
Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dime...Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group’s infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.展开更多
The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacet...The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an alternative approach to quantum gravity starts with the postulate that inertial energy-momentum and gravitational energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the quantum gauge field theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The classical limit of this theory coupled to a quantized scalar field is derived for an on-shell particle where inertial energy-momentum and gravitational energy-momentum coincide. In that process the symmetry under volume-preserving diffeomorphisms disappears and a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle.展开更多
This paper is intended to study the volume-preserving procrustes problem arising from practical areas. The corresponding solution should satisfy a matrix equation which is solved by the singular value decomposition me...This paper is intended to study the volume-preserving procrustes problem arising from practical areas. The corresponding solution should satisfy a matrix equation which is solved by the singular value decomposition method. Some further results are also given to characterize the solution. Using these results, a numerical algorithm is introduced and some numerical results are provided to illustrate the effectiveness of the algorithm. Key words volume-preserving - procrustes problems - singular value decomposition MSC2000 65F30 - 65K10 Project supported by NNSFC (Grant No. 10371076), E-Institutes of Shanghai Municipal Education Commission (Grant No. N. E03004)展开更多
In this work, we propose a new variational model for multi-modal image registration and present an efficient numerical implementation. The model minimizes a new functional based on using reformulated normalized gradie...In this work, we propose a new variational model for multi-modal image registration and present an efficient numerical implementation. The model minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. A key feature of the model is its ability of guaranteeing a diffeomorphic transformation which is achieved by a control term motivated by the quasi-conformal map and Beltrami coefficient. The existence of the solution of this model is established. To solve the model numerically, we design a Gauss-Newton method to solve the resulting discrete optimization problem and prove its convergence;a multilevel technique is employed to speed up the initialization and avoid likely local minima of the underlying functional. Finally, numerical experiments demonstrate that this new model can deliver good performances for multi-modal image registration and simultaneously generate an accurate diffeomorphic transformation.展开更多
Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inn...Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space M4. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory’s Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields.展开更多
Lung image registration plays an important role in lung analysis applications,such as respiratory motion modeling.Unsupervised learning-based image registration methods that can compute the deformation without the req...Lung image registration plays an important role in lung analysis applications,such as respiratory motion modeling.Unsupervised learning-based image registration methods that can compute the deformation without the requirement of supervision attract much attention.However,it is noteworthy that they have two drawbacks:they do not handle the problem of limited data and do not guarantee diffeomorphic(topologypreserving)properties,especially when large deformation exists in lung scans.In this paper,we present an unsupervised few-shot learning-based diffeomorphic lung image registration,namely Dlung.We employ fine-tuning techniques to solve the problem of limited data and apply the scaling and squaring method to accomplish the diffeomorphic registration.Furthermore,atlas-based registration on spatio-temporal(4D)images is performed and thoroughly compared with baseline methods.Dlung achieves the highest accuracy with diffeomorphic properties.It constructs accurate and fast respiratory motion models with limited data.This research extends our knowledge of respiratory motion modeling.展开更多
Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing K...Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.展开更多
This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and i...This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and in the quantum regime a full renormalizable quantum gauge field theory. The three fundamental constraints (hamiltonian, gaussian and diffeomorphism) were formulated in 3-dimensional spatial form within LQG in Ashtekar formulation using the notion of the Kodama state with positive cosmological constant Λ. We introduce a 4-dimensional covariant version of the 3-dimensional (spatial) hamiltonian, gaussian and diffeomorphism constraints of LQG. We obtain 32 partial differential equations for the 16 variables E<sub>mn</sub> (E-tensor, inverse densitized tetrad of the metric) and 16 variables A<sub>mn</sub> (A-tensor, gravitational wave tensor). We impose the boundary condition: for large distance the E-generated metric g(E) becomes the GR-metric g (normally Schwarzschild-spacetime). The theory based on these Ashtekar-Kodama (AK) equations, and called in the following Ashtekar-Kodama (AK-) gravity has the following properties. • For Λ = 0 the AK equations become Einstein equations, A-tensor is trivial (constant), and the E-generated metric g(E) is identical with the GR-metric g. • When the AK-equations are developed into a Λ-power series, the Λ-term yields a gravitational wave equation, which has only at least quadrupole wave solutions and becomes in the limit of large distance r the (normal electromagnetic) wave equation. • AK-gravity, as opposed to GR, has no singularity at the horizon: the singularity in the metric becomes a (very high) peak. • AK-gravity has a limit scale of the gravitational quantum region 39 μm, which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. In the quantum region, the AK-gravity becomes a quantum gauge theory (AK quantum gravity) with the Lie group extended SU(2) = ε-tensor-group(four generators) as gauge group and a corresponding covariant derivative. • AK quantum gravity is fully renormalizable, we derive its Lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams.展开更多
文摘Finding the nearest volume-preserving matrix for a given matrix is studied. Amatrix equation is first obtained, which is a necessary condition for the solution to the problem.Then the equation is solved by the singular value decomposition method. Some additional results arealso provided to further characterize the solution. Using these results, a numerical algorithm isintroduced and a numerical test is given to illustrate the effectiveness of the algorithm.
基金Supported by NSF of China (10571174)the Scientific Research Foundation of Ministry of Education for Returned Overseas Chinese Scholarsthe Scientific Research Foundation of Ministry of Human and Resources and Social Security of China for Returned Overseas Scholars
文摘We prove that, for non-uniformly hyperbolic diffeomorphisms in the sense of Young, the local central limit theorem holds, and the speed in the central limit theorem is O（1/√n）.
基金supported by the National Natural Science Foundation of China(10571174)the Scientific Research Foundation of Ministry of Education for Returned Overseas Chinese ScholarsScientific Research Foundation of Ministry of Human Resources and Social Security for Returned Overseas Chinese Scholars
文摘Let T：X → X be an Axiom A diffeomorphism,m the Gibbs state for a Hlder continuous function ɡ. Assume that f：X → R^d is a Hlder continuous function with ∫_X^（fdm） = 0.If the components of f are cohomologously independent, then there exists a positive definite symmetric matrix σ~2：=σ~2 （f ） such that S^fn √ n converges in distribution with respect to m to a Gaussian random variable with expectation 0 and covariance matrix σ~2 . Moreover, there exists a real number A 〉 0 such that, for any integer n ≥ 1,Π（ m＊（ 1√ nS f n ）,N （0,σ~2 ） ≤A√n, where m＊（1√ n S^fn）denotes the distribution of 1√ n S^fn with respect to m, and Π is the Prokhorov metric.
文摘Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group’s infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed.
文摘The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an alternative approach to quantum gravity starts with the postulate that inertial energy-momentum and gravitational energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the quantum gauge field theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The classical limit of this theory coupled to a quantized scalar field is derived for an on-shell particle where inertial energy-momentum and gravitational energy-momentum coincide. In that process the symmetry under volume-preserving diffeomorphisms disappears and a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle.
文摘This paper is intended to study the volume-preserving procrustes problem arising from practical areas. The corresponding solution should satisfy a matrix equation which is solved by the singular value decomposition method. Some further results are also given to characterize the solution. Using these results, a numerical algorithm is introduced and some numerical results are provided to illustrate the effectiveness of the algorithm. Key words volume-preserving - procrustes problems - singular value decomposition MSC2000 65F30 - 65K10 Project supported by NNSFC (Grant No. 10371076), E-Institutes of Shanghai Municipal Education Commission (Grant No. N. E03004)
文摘In this work, we propose a new variational model for multi-modal image registration and present an efficient numerical implementation. The model minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. A key feature of the model is its ability of guaranteeing a diffeomorphic transformation which is achieved by a control term motivated by the quasi-conformal map and Beltrami coefficient. The existence of the solution of this model is established. To solve the model numerically, we design a Gauss-Newton method to solve the resulting discrete optimization problem and prove its convergence;a multilevel technique is employed to speed up the initialization and avoid likely local minima of the underlying functional. Finally, numerical experiments demonstrate that this new model can deliver good performances for multi-modal image registration and simultaneously generate an accurate diffeomorphic transformation.
文摘Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space M4. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory’s Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields.
基金the National Natural Science Foundation of China(No.61801413)the Natural Science Foundation of Fujian Province(Nos.2019J05123 and 2017J05110)。
文摘Lung image registration plays an important role in lung analysis applications,such as respiratory motion modeling.Unsupervised learning-based image registration methods that can compute the deformation without the requirement of supervision attract much attention.However,it is noteworthy that they have two drawbacks:they do not handle the problem of limited data and do not guarantee diffeomorphic(topologypreserving)properties,especially when large deformation exists in lung scans.In this paper,we present an unsupervised few-shot learning-based diffeomorphic lung image registration,namely Dlung.We employ fine-tuning techniques to solve the problem of limited data and apply the scaling and squaring method to accomplish the diffeomorphic registration.Furthermore,atlas-based registration on spatio-temporal(4D)images is performed and thoroughly compared with baseline methods.Dlung achieves the highest accuracy with diffeomorphic properties.It constructs accurate and fast respiratory motion models with limited data.This research extends our knowledge of respiratory motion modeling.
基金supported by NNSF of China(12101446,11401581,11971236)The second author was supported by NNSF of China(11401581)+1 种基金the third author was supported by NNSF of China(11671208,11431012)At the end,we would like to express our gratitude to Tianyuan Mathematical Center in Southwest China,Sichuan University and Southwest Jiaotong University for their support and hospitality.
文摘Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.
文摘This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and in the quantum regime a full renormalizable quantum gauge field theory. The three fundamental constraints (hamiltonian, gaussian and diffeomorphism) were formulated in 3-dimensional spatial form within LQG in Ashtekar formulation using the notion of the Kodama state with positive cosmological constant Λ. We introduce a 4-dimensional covariant version of the 3-dimensional (spatial) hamiltonian, gaussian and diffeomorphism constraints of LQG. We obtain 32 partial differential equations for the 16 variables E<sub>mn</sub> (E-tensor, inverse densitized tetrad of the metric) and 16 variables A<sub>mn</sub> (A-tensor, gravitational wave tensor). We impose the boundary condition: for large distance the E-generated metric g(E) becomes the GR-metric g (normally Schwarzschild-spacetime). The theory based on these Ashtekar-Kodama (AK) equations, and called in the following Ashtekar-Kodama (AK-) gravity has the following properties. • For Λ = 0 the AK equations become Einstein equations, A-tensor is trivial (constant), and the E-generated metric g(E) is identical with the GR-metric g. • When the AK-equations are developed into a Λ-power series, the Λ-term yields a gravitational wave equation, which has only at least quadrupole wave solutions and becomes in the limit of large distance r the (normal electromagnetic) wave equation. • AK-gravity, as opposed to GR, has no singularity at the horizon: the singularity in the metric becomes a (very high) peak. • AK-gravity has a limit scale of the gravitational quantum region 39 μm, which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. In the quantum region, the AK-gravity becomes a quantum gauge theory (AK quantum gravity) with the Lie group extended SU(2) = ε-tensor-group(four generators) as gauge group and a corresponding covariant derivative. • AK quantum gravity is fully renormalizable, we derive its Lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams.
