Structural optimization of lead compounds is a crucial step in drug discovery.One optimization strategy is to modify the molecular structure of a scaffold to improve both its biological activities and absorption,distr...Structural optimization of lead compounds is a crucial step in drug discovery.One optimization strategy is to modify the molecular structure of a scaffold to improve both its biological activities and absorption,distribution,metabolism,excretion,and toxicity(ADMET)properties.One of the deep molecular generative model approaches preserves the scaffold while generating drug-like molecules,thereby accelerating the molecular optimization process.Deep molecular diffusion generative models simulate a gradual process that creates novel,chemically feasible molecules from noise.However,the existing models lack direct interatomic constraint features and struggle with capturing long-range dependencies in macromolecules,leading to challenges in modifying the scaffold-based molecular structures,and creates limitations in the stability and diversity of the generated molecules.To address these challenges,we propose a deep molecular diffusion generative model,the three-dimensional(3D)equivariant diffusion-driven molecular generation(3D-EDiffMG)model.The dual strong and weak atomic interaction force-based long-range dependency capturing equivariant encoder(dual-SWLEE)is introduced to encode both the bonding and non-bonding information based on strong and weak atomic interactions.Addi-tionally,a gate multilayer perceptron(gMLP)block with tiny attention is incorporated to explicitly model complex long-sequence feature interactions and long-range dependencies.The experimental results show that 3D-EDiffMG effectively generates unique,novel,stable,and diverse drug-like molecules,highlighting its potential for lead optimization and accelerating drug discovery.展开更多
This article is concerned with finite element implementations of the three- dimensional geometrically exact rod. The special attention is paid to identifying the con- dition that ensures the frame invariance of the re...This article is concerned with finite element implementations of the three- dimensional geometrically exact rod. The special attention is paid to identifying the con- dition that ensures the frame invariance of the resulting discrete approximations. From the perspective of symmetry, this requirement is equivalent to the commutativity of the employed interpolation operator I with the action of the special Euclidean group SE（3）, or I is SE（3）-equivariant. This geometric criterion helps to clarify several subtle issues about the interpolation of finite rotation. It leads us to reexamine the finite element for- mulation first proposed by Simo in his work on energy-momentum conserving algorithms. That formulation is often mistakenly regarded as non-objective. However, we show that the obtained approximation is invariant under the superposed rigid body motions, and as a corollary, the objectivity of the continuum model is preserved. The key of this proof comes from the observation that since the numerical quadrature is used to compute the integrals, by storing the rotation field and its derivative at the Gauss points, the equiv- ariant conditions can be relaxed only at these points. Several numerical examples are presented to confirm the theoretical results and demonstrate the performance of this al- gorithm.展开更多
In this paper versal unfolding theorem of multiparameter equivariant bifurcation problem with parameter symmetry is given. The necessary and sufficient condition that unfolding of multiparameter equivariant bifurcatio...In this paper versal unfolding theorem of multiparameter equivariant bifurcation problem with parameter symmetry is given. The necessary and sufficient condition that unfolding of multiparameter equivariant bifurcation problem with parameter symmetry factors through another is given. The corresponding results in [1]-[6] are generalized.展开更多
For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry,the versal unfolding theorem with respect to left-right equivalence is obtained by using th...For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry,the versal unfolding theorem with respect to left-right equivalence is obtained by using the related methods and techniques in the singularity theory of smooth map-germs.The corresponding results in[4,9]can be considered as its special cases.A relationship between the versal unfolding w.r.t.left-right equivalence and the versal deformation w.r.t.contact equivalence is established.展开更多
In this paper we investigate the estimator for the rth power of the scale parameter in a class of exponential family under symmetric entropy loss L（θ, δ） = v（θ/δ ＋ δ/θ - 2）. An exact form of the minimum ris...In this paper we investigate the estimator for the rth power of the scale parameter in a class of exponential family under symmetric entropy loss L（θ, δ） = v（θ/δ ＋ δ/θ - 2）. An exact form of the minimum risk equivariant estimator under symmetric entropy loss is given, and the minimaxity of the minimum risk equivariant estimator is proved. The results with regard to admissibility and inadmissibility of a class of linear estimators of the form cT（X） ＋ d are given, where T（X） Gamma（v, θ）.展开更多
Based on the left_right equivalent relation of smooth map_germs in singularity theory, the unfoldings of multiparameter equivariant bifurcation problems with respect to left_right equivalence are discussed. The state ...Based on the left_right equivalent relation of smooth map_germs in singularity theory, the unfoldings of multiparameter equivariant bifurcation problems with respect to left_right equivalence are discussed. The state variables of such an equivariant bifurcation problem were divided into two groups, in which the first can vary independently, while the others depend on the first in the varying process. By applying related methods and techniques in the unfolding theory of smooth map_germs, the necessary and sufficient condition for an unfolding of a multiparameter equivariant bifurcation problem with two groups of state variables to be versal is obtained.展开更多
Associated with an immersion φ : S^3→ ■, we can define a canonical bundle endomorphism F : TS^3→ TS^3 by the pull back of the K?hler form of ■. In this article,related to F we study equivariant minimal immersions...Associated with an immersion φ : S^3→ ■, we can define a canonical bundle endomorphism F : TS^3→ TS^3 by the pull back of the K?hler form of ■. In this article,related to F we study equivariant minimal immersions from S^3 into ■ under the additional condition(?_XF)X = 0 for all X ∈ ker(F). As main result, we give a complete classification of such kinds of immersions. Moreover, we also construct a typical example of equivariant non-minimal immersion φ : S^3→ ■ satisfying(?_XF)X = 0 for all X ∈ ker(F), which is neither Lagrangian nor of CR type.展开更多
In this paper,we compute the first two equivariant heat kernel coeffcients of the Bochner Laplacian on differential forms.The first two equivariant heat kernel coeffcients of the Bochner Laplacian with torsion are als...In this paper,we compute the first two equivariant heat kernel coeffcients of the Bochner Laplacian on differential forms.The first two equivariant heat kernel coeffcients of the Bochner Laplacian with torsion are also given.We also study the equivariant heat kernel coeffcients of nonminimal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula.展开更多
Based on the contact equivalent relation of smooth map-germs in singularity theory, the stability of equivariant bifurcation problems with two types of state variables and their unfoldings in the presence of parameter...Based on the contact equivalent relation of smooth map-germs in singularity theory, the stability of equivariant bifurcation problems with two types of state variables and their unfoldings in the presence of parameter symmetry is discussed. Some basic results are obtained. Transversality condition is used to characterize the stability of equavariant bifurcation problems.展开更多
Von Rosen (1989) proposed the MLE of parameters in multivariate linear normal model MLNM(sumfromn= lto ∞AiBiCi). This paper discusses some properties of Rosen's MLE for general distributions which includs invaria...Von Rosen (1989) proposed the MLE of parameters in multivariate linear normal model MLNM(sumfromn= lto ∞AiBiCi). This paper discusses some properties of Rosen's MLE for general distributions which includs invariant, equivariant, strong consistency and asymptotic normality. Furthermore, we can construct the consistent confidence region for the parameter of experctation in MLNM(sumfromn=1to∞, AiBiCi) and obtain asymptotic distribu- tion and consistent confidence region of the linear discrimination function for canonical correlation by Kahtri (1988).展开更多
In addressing the challenge of motion artifacts in Positron Emission Tomography (PET) lung scans, our studyintroduces the Triple Equivariant Motion Transformer (TEMT), an innovative, unsupervised, deep-learningbasedfr...In addressing the challenge of motion artifacts in Positron Emission Tomography (PET) lung scans, our studyintroduces the Triple Equivariant Motion Transformer (TEMT), an innovative, unsupervised, deep-learningbasedframework for efficient respiratory motion correction in PET imaging. Unlike traditional techniques,which segment PET data into bins throughout a respiratory cycle and often face issues such as inefficiency andoveremphasis on certain artifacts, TEMT employs Convolutional Neural Networks (CNNs) for effective featureextraction and motion decomposition.TEMT’s unique approach involves transforming motion sequences into Liegroup domains to highlight fundamental motion patterns, coupled with employing competitive weighting forprecise target deformation field generation. Our empirical evaluations confirm TEMT’s superior performancein handling diverse PET lung datasets compared to existing image registration networks. Experimental resultsdemonstrate that TEMT achieved Dice indices of 91.40%, 85.41%, 79.78%, and 72.16% on simulated geometricphantom data, lung voxel phantom data, cardiopulmonary voxel phantom data, and clinical data, respectively. Tofacilitate further research and practical application, the TEMT framework, along with its implementation detailsand part of the simulation data, is made publicly accessible at https://github.com/yehaowei/temt.展开更多
The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R3n Jordan blocks....The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R3n Jordan blocks. We use an algorithm based on the notion of transvectants from classical invariant theory known as boosting to equivariants in determining the normal form when the Stanley decomposition for the ring of invariants is known.展开更多
Several equivalent formulations are given for equivariant coarse embedding into Hilbert space.Using these equivalent definitions,it is proved that for a metric space X and a Hilbert space H with proper and isometric g...Several equivalent formulations are given for equivariant coarse embedding into Hilbert space.Using these equivalent definitions,it is proved that for a metric space X and a Hilbert space H with proper and isometric group actions on both of them,if X is coarsely embeddable into H and the group is amenable,then the coarse embedding can be modified to be equivariant by using the invariant mean property of the amenable group.展开更多
In this paper,a statistical prediction problem under ordered location and scale parameters are considered.Double-shrinkage predictors are given which use all the available data and improve on single-shrinkage predicto...In this paper,a statistical prediction problem under ordered location and scale parameters are considered.Double-shrinkage predictors are given which use all the available data and improve on single-shrinkage predictors,and hence the best equivariant predictors.展开更多
The minimum risk equivariant estimator of a quantile of the common marginal distribution in a multivariate Lomax distribution with unknown location and scale parameters under Linex loss function is considered.
1. Introduction. Throughout this note, G is a finite group, M is a compact connected smooth on-dimensional manifold with or without boundary M, and G acts smoothly on M. We follow the standard notations ([B], [tD]). T...1. Introduction. Throughout this note, G is a finite group, M is a compact connected smooth on-dimensional manifold with or without boundary M, and G acts smoothly on M. We follow the standard notations ([B], [tD]). The isotropy subgroup of a point展开更多
Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal subgroup of G . For principal fibre bundle (E,p, E/G;G) and (E/H,p′,E/G;G/H), the relation between aut G(E) ...Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal subgroup of G . For principal fibre bundle (E,p, E/G;G) and (E/H,p′,E/G;G/H), the relation between aut G(E) (resp. aut * G(E) ) and aut G/H (E/H) (resp.aut * G/H (E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group F G(E) (resp. E G(E)) while the group F G/H (E/H) is known.展开更多
Geometric graphs are a special kind of graph with geometric features,which are vital to model many scientific problems.Unlike generic graphs,geometric graphs often exhibit physical symmetries of translations,rotations...Geometric graphs are a special kind of graph with geometric features,which are vital to model many scientific problems.Unlike generic graphs,geometric graphs often exhibit physical symmetries of translations,rotations,and reflections,making them ineffectively processed by current Graph Neural Networks(GNNs).To address this issue,researchers proposed a variety of geometric GNNs equipped with invariant/equivariant properties to better characterize the geometry and topology of geometric graphs.Given the current progress in this field,it is imperative to conduct a comprehensive survey of data structures,models,and applications related to geometric GNNs.In this paper,based on the necessary but concise mathematical preliminaries,we formalize geometric graph as the data structure,on top of which we provide a unified view of existing models from the geometric message passing perspective.Additionally,we summarize the applications as well as the related datasets to facilitate later research for methodology development and experimental evaluation.We also discuss the challenges and future potential directions of geometric GNNs at the end of this survey.展开更多
Constructing Lie superalgebras from Lie algebras is an important subject for studying Lie superalgebras.In this paper,all the superizations are presented for finite-dimensional trivial central extensions of a class of...Constructing Lie superalgebras from Lie algebras is an important subject for studying Lie superalgebras.In this paper,all the superizations are presented for finite-dimensional trivial central extensions of a class of 3-dimensional complex or real Lie algebras.In particular,up to isomorphism,superizations are determined for 1-dimensional trivial central extensions of these Lie algebras by considering orbits of an action of G(g0)on Symad(g0),where Symad(g0)consists of go-invariant symmetric bilinear mappings and G(g0)={(σ,τ)∈Aut(g0)×GL(g0)|[σ(x),τ(y)]=τ([x,y]),x,y∈g0}.展开更多
基金supported by the National Key R&D Program of China(Grant No.:2023YFF1205102)the National Natural Science Foundation of China(Grant Nos.:82273856,22077143,and 21977127)the Science Foundation of Guangzhou,China(No.:2Grant024A04J2172).
文摘Structural optimization of lead compounds is a crucial step in drug discovery.One optimization strategy is to modify the molecular structure of a scaffold to improve both its biological activities and absorption,distribution,metabolism,excretion,and toxicity(ADMET)properties.One of the deep molecular generative model approaches preserves the scaffold while generating drug-like molecules,thereby accelerating the molecular optimization process.Deep molecular diffusion generative models simulate a gradual process that creates novel,chemically feasible molecules from noise.However,the existing models lack direct interatomic constraint features and struggle with capturing long-range dependencies in macromolecules,leading to challenges in modifying the scaffold-based molecular structures,and creates limitations in the stability and diversity of the generated molecules.To address these challenges,we propose a deep molecular diffusion generative model,the three-dimensional(3D)equivariant diffusion-driven molecular generation(3D-EDiffMG)model.The dual strong and weak atomic interaction force-based long-range dependency capturing equivariant encoder(dual-SWLEE)is introduced to encode both the bonding and non-bonding information based on strong and weak atomic interactions.Addi-tionally,a gate multilayer perceptron(gMLP)block with tiny attention is incorporated to explicitly model complex long-sequence feature interactions and long-range dependencies.The experimental results show that 3D-EDiffMG effectively generates unique,novel,stable,and diverse drug-like molecules,highlighting its potential for lead optimization and accelerating drug discovery.
文摘This article is concerned with finite element implementations of the three- dimensional geometrically exact rod. The special attention is paid to identifying the con- dition that ensures the frame invariance of the resulting discrete approximations. From the perspective of symmetry, this requirement is equivalent to the commutativity of the employed interpolation operator I with the action of the special Euclidean group SE（3）, or I is SE（3）-equivariant. This geometric criterion helps to clarify several subtle issues about the interpolation of finite rotation. It leads us to reexamine the finite element for- mulation first proposed by Simo in his work on energy-momentum conserving algorithms. That formulation is often mistakenly regarded as non-objective. However, we show that the obtained approximation is invariant under the superposed rigid body motions, and as a corollary, the objectivity of the continuum model is preserved. The key of this proof comes from the observation that since the numerical quadrature is used to compute the integrals, by storing the rotation field and its derivative at the Gauss points, the equiv- ariant conditions can be relaxed only at these points. Several numerical examples are presented to confirm the theoretical results and demonstrate the performance of this al- gorithm.
基金This work is supported by NNSF of China(10271023) and Hunan Provincial Natural Science Foundation of China(04JJ3072).
文摘In this paper versal unfolding theorem of multiparameter equivariant bifurcation problem with parameter symmetry is given. The necessary and sufficient condition that unfolding of multiparameter equivariant bifurcation problem with parameter symmetry factors through another is given. The corresponding results in [1]-[6] are generalized.
文摘For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry,the versal unfolding theorem with respect to left-right equivalence is obtained by using the related methods and techniques in the singularity theory of smooth map-germs.The corresponding results in[4,9]can be considered as its special cases.A relationship between the versal unfolding w.r.t.left-right equivalence and the versal deformation w.r.t.contact equivalence is established.
基金The SRFDPHE(20070183023)the NSF(10571073,J0630104)of China
文摘In this paper we investigate the estimator for the rth power of the scale parameter in a class of exponential family under symmetric entropy loss L（θ, δ） = v（θ/δ ＋ δ/θ - 2）. An exact form of the minimum risk equivariant estimator under symmetric entropy loss is given, and the minimaxity of the minimum risk equivariant estimator is proved. The results with regard to admissibility and inadmissibility of a class of linear estimators of the form cT（X） ＋ d are given, where T（X） Gamma（v, θ）.
文摘Based on the left_right equivalent relation of smooth map_germs in singularity theory, the unfoldings of multiparameter equivariant bifurcation problems with respect to left_right equivalence are discussed. The state variables of such an equivariant bifurcation problem were divided into two groups, in which the first can vary independently, while the others depend on the first in the varying process. By applying related methods and techniques in the unfolding theory of smooth map_germs, the necessary and sufficient condition for an unfolding of a multiparameter equivariant bifurcation problem with two groups of state variables to be versal is obtained.
文摘Associated with an immersion φ : S^3→ ■, we can define a canonical bundle endomorphism F : TS^3→ TS^3 by the pull back of the K?hler form of ■. In this article,related to F we study equivariant minimal immersions from S^3 into ■ under the additional condition(?_XF)X = 0 for all X ∈ ker(F). As main result, we give a complete classification of such kinds of immersions. Moreover, we also construct a typical example of equivariant non-minimal immersion φ : S^3→ ■ satisfying(?_XF)X = 0 for all X ∈ ker(F), which is neither Lagrangian nor of CR type.
基金supported by NSFC(10801027)Fok Ying Tong Education Foundation(121003)
文摘In this paper,we compute the first two equivariant heat kernel coeffcients of the Bochner Laplacian on differential forms.The first two equivariant heat kernel coeffcients of the Bochner Laplacian with torsion are also given.We also study the equivariant heat kernel coeffcients of nonminimal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula.
基金Project supported by the National Natural Science Foundation of China(No.10671002)the Natural Science Foundation of Hunan Province of China(No.04JJ3072)the Science Foundation of the Education Department of Hunan Province of China(No.04C383)
文摘Based on the contact equivalent relation of smooth map-germs in singularity theory, the stability of equivariant bifurcation problems with two types of state variables and their unfoldings in the presence of parameter symmetry is discussed. Some basic results are obtained. Transversality condition is used to characterize the stability of equavariant bifurcation problems.
文摘Von Rosen (1989) proposed the MLE of parameters in multivariate linear normal model MLNM(sumfromn= lto ∞AiBiCi). This paper discusses some properties of Rosen's MLE for general distributions which includs invariant, equivariant, strong consistency and asymptotic normality. Furthermore, we can construct the consistent confidence region for the parameter of experctation in MLNM(sumfromn=1to∞, AiBiCi) and obtain asymptotic distribu- tion and consistent confidence region of the linear discrimination function for canonical correlation by Kahtri (1988).
基金the National Natural Science Foundation of China(No.82160347)Yunnan Provincial Science and Technology Department(No.202102AE090031)Yunnan Key Laboratory of Smart City in Cyberspace Security(No.202105AG070010).
文摘In addressing the challenge of motion artifacts in Positron Emission Tomography (PET) lung scans, our studyintroduces the Triple Equivariant Motion Transformer (TEMT), an innovative, unsupervised, deep-learningbasedframework for efficient respiratory motion correction in PET imaging. Unlike traditional techniques,which segment PET data into bins throughout a respiratory cycle and often face issues such as inefficiency andoveremphasis on certain artifacts, TEMT employs Convolutional Neural Networks (CNNs) for effective featureextraction and motion decomposition.TEMT’s unique approach involves transforming motion sequences into Liegroup domains to highlight fundamental motion patterns, coupled with employing competitive weighting forprecise target deformation field generation. Our empirical evaluations confirm TEMT’s superior performancein handling diverse PET lung datasets compared to existing image registration networks. Experimental resultsdemonstrate that TEMT achieved Dice indices of 91.40%, 85.41%, 79.78%, and 72.16% on simulated geometricphantom data, lung voxel phantom data, cardiopulmonary voxel phantom data, and clinical data, respectively. Tofacilitate further research and practical application, the TEMT framework, along with its implementation detailsand part of the simulation data, is made publicly accessible at https://github.com/yehaowei/temt.
文摘The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R3n Jordan blocks. We use an algorithm based on the notion of transvectants from classical invariant theory known as boosting to equivariants in determining the normal form when the Stanley decomposition for the ring of invariants is known.
基金Supported by National Natural Science Foundation of China(11871342)。
文摘Several equivalent formulations are given for equivariant coarse embedding into Hilbert space.Using these equivalent definitions,it is proved that for a metric space X and a Hilbert space H with proper and isometric group actions on both of them,if X is coarsely embeddable into H and the group is amenable,then the coarse embedding can be modified to be equivariant by using the invariant mean property of the amenable group.
文摘In this paper,a statistical prediction problem under ordered location and scale parameters are considered.Double-shrinkage predictors are given which use all the available data and improve on single-shrinkage predictors,and hence the best equivariant predictors.
文摘The minimum risk equivariant estimator of a quantile of the common marginal distribution in a multivariate Lomax distribution with unknown location and scale parameters under Linex loss function is considered.
文摘1. Introduction. Throughout this note, G is a finite group, M is a compact connected smooth on-dimensional manifold with or without boundary M, and G acts smoothly on M. We follow the standard notations ([B], [tD]). The isotropy subgroup of a point
文摘Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal subgroup of G . For principal fibre bundle (E,p, E/G;G) and (E/H,p′,E/G;G/H), the relation between aut G(E) (resp. aut * G(E) ) and aut G/H (E/H) (resp.aut * G/H (E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group F G(E) (resp. E G(E)) while the group F G/H (E/H) is known.
基金supported by the following projects:The National Natural Science Foundation of China(Grant Nos.62376276 and 62172422)Beijing Nova Program(Grant No.20230484278)+1 种基金the Fundamental Research Funds for the Central Universities,and the Research Funds of Renmin University of China(Grant No.23XNKJ19)Tencent AI Lab Rhino-Bird Focused Research Program.
文摘Geometric graphs are a special kind of graph with geometric features,which are vital to model many scientific problems.Unlike generic graphs,geometric graphs often exhibit physical symmetries of translations,rotations,and reflections,making them ineffectively processed by current Graph Neural Networks(GNNs).To address this issue,researchers proposed a variety of geometric GNNs equipped with invariant/equivariant properties to better characterize the geometry and topology of geometric graphs.Given the current progress in this field,it is imperative to conduct a comprehensive survey of data structures,models,and applications related to geometric GNNs.In this paper,based on the necessary but concise mathematical preliminaries,we formalize geometric graph as the data structure,on top of which we provide a unified view of existing models from the geometric message passing perspective.Additionally,we summarize the applications as well as the related datasets to facilitate later research for methodology development and experimental evaluation.We also discuss the challenges and future potential directions of geometric GNNs at the end of this survey.
基金S.Q.Zhao was supported by the NSF of Hainan Province(No.121MS0784)Postgraduate Innovation and Scientific Research Topic of the School of Mathematical Statistics of Hainan Normal University(No.styc202201)+1 种基金W.D.Liu was supported by the NSF of China(No.12061029)the NSF of Hainan Province(No.120RC587).
文摘Constructing Lie superalgebras from Lie algebras is an important subject for studying Lie superalgebras.In this paper,all the superizations are presented for finite-dimensional trivial central extensions of a class of 3-dimensional complex or real Lie algebras.In particular,up to isomorphism,superizations are determined for 1-dimensional trivial central extensions of these Lie algebras by considering orbits of an action of G(g0)on Symad(g0),where Symad(g0)consists of go-invariant symmetric bilinear mappings and G(g0)={(σ,τ)∈Aut(g0)×GL(g0)|[σ(x),τ(y)]=τ([x,y]),x,y∈g0}.
