A new multi-phase active contour model is proposed for the image segmentation. It is a generalization of the C-V model with the following characteristics： （1） A key technique, called the technique of painting backg...A new multi-phase active contour model is proposed for the image segmentation. It is a generalization of the C-V model with the following characteristics： （1） A key technique, called the technique of painting background （TPBG）, is developed to remove the information of the background, which blocks the detection of weak boundaries in the object; （2） The two-phase level set is applied multiple times for getting the multi-phase segmentation model （n-1 times for the n-phase model, n〉1）; （3） A scaling-based method is introduced to improve the basic model. Experimental results show that the proposed model is effective for detecting weak boundaries.展开更多
In this paper, we present a new deformable model for shape segmentation, which makes two modifications to the original level set implementation of deformable models.The modifications are motivated by difficulties that...In this paper, we present a new deformable model for shape segmentation, which makes two modifications to the original level set implementation of deformable models.The modifications are motivated by difficulties that we have encountered in applying deformable models to segmentation of medical images.The level set algorithm has some advantages over the classical snake deformable models.However, it could develop large gaps in the boundary and holes within the objects.Such boundary gaps and holes of objects can cause inaccurate segmentation that requires manual correction.The proposed method in this paper possesses an inherent property to detect gaps and holes within the object with a single initial contour and also does not require specific initialization.The first modification is to replace the edge detector by some area constraint, and the second modification utilizes weighted length constraint to regularize the curve under evolution.The proposed method has been applied to both synthetic and real images with promising results.展开更多
In the paper, we consider Moran-type sets E;given by sequences {a;};and{n;};. we prove that E;may be decompose into the disjoint union of level sets. Moreover,we define three type of equivalence between two dimension ...In the paper, we consider Moran-type sets E;given by sequences {a;};and{n;};. we prove that E;may be decompose into the disjoint union of level sets. Moreover,we define three type of equivalence between two dimension functions associated to two Morantype sets, respectively, and we classify Moran-type sets by these equivalent relations.展开更多
Short time existence and uniqueness for the classical motion are studied by the function of the principal curvatures of a smooth surface and the Evans and Spruck's results are generalized.
This paper is to detect regions (objects) boundaries, also to isolate and extract individual components from a medical image. This can be done using an active contours to detect regions in a given image, based on tech...This paper is to detect regions (objects) boundaries, also to isolate and extract individual components from a medical image. This can be done using an active contours to detect regions in a given image, based on techniques of curve evolution, Mumford Shah functional for segmentation and level sets. The paper classified the images into different intensity regions based on Markov random field, then detected regions whose boundaries are not necessarily defined by gradient by minimizing an energy of Mumford Shah functional for segmentation which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a mean curvature flow like evolving the active contour, which will stop on the desired boundary. The stopping term does not depend on the gradient of the image, as in the classical active contour and the initial curve of level set can be anywhere in the image, and interior contours are automatically detected. The final image segmentation is one closed boundary per actual region in the image.展开更多
A data-driven model ofmultiple variable cutting(M-VCUT)level set-based substructure is proposed for the topology optimization of lattice structures.TheM-VCUTlevel setmethod is used to represent substructures,enriching...A data-driven model ofmultiple variable cutting(M-VCUT)level set-based substructure is proposed for the topology optimization of lattice structures.TheM-VCUTlevel setmethod is used to represent substructures,enriching their diversity of configuration while ensuring connectivity.To construct the data-driven model of substructure,a database is prepared by sampling the space of substructures spanned by several substructure prototypes.Then,for each substructure in this database,the stiffness matrix is condensed so that its degrees of freedomare reduced.Thereafter,the data-drivenmodel of substructures is constructed through interpolationwith compactly supported radial basis function(CS-RBF).The inputs of the data-driven model are the design variables of topology optimization,and the outputs are the condensed stiffness matrix and volume of substructures.During the optimization,this data-driven model is used,thus avoiding repeated static condensation that would requiremuch computation time.Several numerical examples are provided to verify the proposed method.展开更多
To solve the problem of false edges in a flat region of l_(1)norm total variational TV model,an edge extractor based on non-local idea is proposed in this paper.The new edge extractor can effectively suppress the infl...To solve the problem of false edges in a flat region of l_(1)norm total variational TV model,an edge extractor based on non-local idea is proposed in this paper.The new edge extractor can effectively suppress the influence of noise and extract the edge information of the image.The new edge extractor is used as the adaptive function and the weighting function of the l_(p) norm variational model to control the noise reduction ability of the model,and a new model 1 is obtained.Considering that the new model 1 only uses the gradient mode as the image feature operator,which is insufficient to express the image texture information,a new level set curvature gradient variational model 2 combined with the edge extractor is proposed.The new model 2 uses the idea of minimum curvature of the level set of clear images to obtain noise reduction images.By coupling new model 1 and new model 2 to smooth the noise and protect more textures,a new Non-local level set denoising model(NLSDM)for image noise reduction is obtained.The experimental results show that compared with the noise reduction model,the new model has significantly improved the peak signal-to-noise ratio and structural similarity,and the effect of noise reduction and edge preservation is better.展开更多
Using traditional particle tracking velocimetry based on optical flow for measuring areas with large velocity gradient changes may cause oversmoothing,resulting in significant measurement errors.To address this proble...Using traditional particle tracking velocimetry based on optical flow for measuring areas with large velocity gradient changes may cause oversmoothing,resulting in significant measurement errors.To address this problem,the traditional particle tracking velocimetry method based on an optical flow was improved.The level set segmentation algorithm was used to obtain the boundary contour of the region with large velocity gradient changes,and the non-uniform flow field was divided into regions according to the boundary contour to obtain sub-regions with uniform velocity distribution.The particle tracking velocimetry method based on optical flow was used to measure the granular flow velocity in each sub-region,thus avoiding the problem of granular flow distribution.The simulation results show that the measurement accuracy of this method is approximately 10%higher than that of traditional methods.The method was applied to a velocity measurement experiment on dense granular flow in silos,and the velocity distribution of the granular flow was obtained,verifying the practicality of the method in granular flow fields.展开更多
This paper presents an improved level set method for topology optimization of geometrically nonlinear structures accounting for the effect of thermo-mechanical couplings.It derives a new expression for element couplin...This paper presents an improved level set method for topology optimization of geometrically nonlinear structures accounting for the effect of thermo-mechanical couplings.It derives a new expression for element coupling stress resulting from the combination of mechanical and thermal loading,using geometric nonlinear finite element analysis.A topological model is then developed to minimize compliance while meeting displacement and frequency constraints to fulfill design requirements of structural members.Since the conventional Lagrange multiplier search method is unable to handle convergence instability arising from large deformation,a novel Lagrange multiplier search method is proposed.Additionally,the proposed method can be extended to multi-constrained geometrically nonlinear topology optimization,accommodating multiple physical field couplings.展开更多
Separation between water and land is vital for marine scientific research and coastal zone planning and management.The contrasting backscatter properties of land and ocean enable clear water edge line identification i...Separation between water and land is vital for marine scientific research and coastal zone planning and management.The contrasting backscatter properties of land and ocean enable clear water edge line identification in synthetic aperture radar(SAR)imagery.However,SAR images are prone to speckle noise,and the complexity of the water-land boundaries environment makes accurate water-land separation challenging.To overcome noise and complex background interference in remote sensing images,an improved level set method was employed to enhance water-land separation.In the traditional distance regularized level set method,the selection of the image correlation weight coefficient and the edge indicator function directly influences the accuracy of the final segmentation results.A novel level set segmentation algorithm incorporating an improved edge indicator function is proposed to efficiently and accurately separate the water edge lines in SAR images.The distance regularized level set evolution model is enhanced by incorporating the signed pressure force function as an adaptive parameter,which serves as an external constraint for curve evolution.A novel level set model with an adaptive edge indicator function,combining gradient and regional information,is proposed.Experimental results demonstrate that the proposed model enhances the accuracy of waterland separation in SAR images.However,further research is needed to evaluate its potential for detecting boundaries in diverse marine environments and across different types of SAR imagery.展开更多
A cylindrical chamber with a rotating bottom holds significant potential for application in cell culture bioreactors due to its ability to generate more stable swirling flows.In order to control vortex breakdown withi...A cylindrical chamber with a rotating bottom holds significant potential for application in cell culture bioreactors due to its ability to generate more stable swirling flows.In order to control vortex breakdown within the chamber,this study first establishes a computational fluid dynamics simulation coupled with the level set method.Verified by experimental results in literature,this method accurately simulates the position and shape of vortex breakdown,and also predicts the critical Reynolds numbers for the appearance and detachment of vortex breakdown bubbles from the center.Additionally,it precisely captures the gas-liquid interface and depicts the vortex breakdown phenomenon in the air above the liquid for the first time.Finally,it predicts the impact of physical property of gas-liquid systems on vortex breakdown in response to significant changes in viscosity of microbial process systems.展开更多
Structural Reliability-Based Topology Optimization(RBTO),as an efficient design methodology,serves as a crucial means to ensure the development ofmodern engineering structures towards high performance,long service lif...Structural Reliability-Based Topology Optimization(RBTO),as an efficient design methodology,serves as a crucial means to ensure the development ofmodern engineering structures towards high performance,long service life,and high reliability.However,in practical design processes,topology optimization must not only account for the static performance of structures but also consider the impacts of various responses and uncertainties under complex dynamic conditions,which traditional methods often struggle accommodate.Therefore,this study proposes an RBTO framework based on a Kriging-assisted level set function and a novel Dynamic Hybrid Particle Swarm Optimization(DHPSO)algorithm.By leveraging the Kriging model as a surrogate,the high cost associated with repeatedly running finite element analysis processes is reduced,addressing the issue of minimizing structural compliance.Meanwhile,the DHPSO algorithm enables a better balance between the population’s developmental and exploratory capabilities,significantly accelerating convergence speed and enhancing global convergence performance.Finally,the proposed method is validated through three different structural examples,demonstrating its superior performance.Observed that the computational that,compared to the traditional Solid Isotropic Material with Penalization(SIMP)method,the proposed approach reduces the upper bound of structural compliance by approximately 30%.Additionally,the optimized results exhibit clear material interfaces without grayscale elements,and the stress concentration factor is reduced by approximately 42%.Consequently,the computational results fromdifferent examples verify the effectiveness and superiority of this study across various fields,achieving the goal of providing more precise optimization results within a shorter timeframe.展开更多
We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations ut = F(▽2u,▽u,u,t) under a structural condition,and give a geometric lower bound of the...We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations ut = F(▽2u,▽u,u,t) under a structural condition,and give a geometric lower bound of the principal curvature of the spatial level surfaces.展开更多
For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and th...For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.展开更多
We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature...We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.展开更多
In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued...In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.展开更多
For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／...For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／f-1（0）） automatically sat- isfies f ∈M a（Ω）, if it is Caj-1smooth in the z/variable, α ∈ Zn＋ up to the boundary. For a submanifold U C Cn, denote by ma（U）, the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal （off the singularities） obey the boundary maximum modulus principle. We prove, that for a C3-smooth hypersurface, Ω, a member of ma （Ω）, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.展开更多
For the p-harmonic function with strictly convex level sets,we find an auxiliary function which comes from the combination of the norm of gradient of the p-harmonic function and the Gaussian curvature of the level set...For the p-harmonic function with strictly convex level sets,we find an auxiliary function which comes from the combination of the norm of gradient of the p-harmonic function and the Gaussian curvature of the level sets of p-harmonic function.We prove that this curvature function is concave with respect to the height of the p-harmonic function.This auxiliary function is an affine function of the height when the p-harmonic function is the p-Green function on ball.展开更多
文摘A new multi-phase active contour model is proposed for the image segmentation. It is a generalization of the C-V model with the following characteristics： （1） A key technique, called the technique of painting background （TPBG）, is developed to remove the information of the background, which blocks the detection of weak boundaries in the object; （2） The two-phase level set is applied multiple times for getting the multi-phase segmentation model （n-1 times for the n-phase model, n〉1）; （3） A scaling-based method is introduced to improve the basic model. Experimental results show that the proposed model is effective for detecting weak boundaries.
基金Supported by the National Natural Science Foundation of China (No.60472071, 60532080, 60602062)the Natural Science Foundation of Beijing (No.4051002)
文摘In this paper, we present a new deformable model for shape segmentation, which makes two modifications to the original level set implementation of deformable models.The modifications are motivated by difficulties that we have encountered in applying deformable models to segmentation of medical images.The level set algorithm has some advantages over the classical snake deformable models.However, it could develop large gaps in the boundary and holes within the objects.Such boundary gaps and holes of objects can cause inaccurate segmentation that requires manual correction.The proposed method in this paper possesses an inherent property to detect gaps and holes within the object with a single initial contour and also does not require specific initialization.The first modification is to replace the edge detector by some area constraint, and the second modification utilizes weighted length constraint to regularize the curve under evolution.The proposed method has been applied to both synthetic and real images with promising results.
基金supported by NSFC (11201152)supported by NSFC(11371148)+4 种基金STCSM(13dz2260400)FDPHEC(20120076120001)Fundamental Research Funds for the central Universities,scut(2012zz0073)Fundamental Research Funds for the Central Universities SCUT(D2154240)Guangdong Natural Science Foundation(2014A030313230)
文摘In the paper, we consider Moran-type sets E;given by sequences {a;};and{n;};. we prove that E;may be decompose into the disjoint union of level sets. Moreover,we define three type of equivalence between two dimension functions associated to two Morantype sets, respectively, and we classify Moran-type sets by these equivalent relations.
文摘Short time existence and uniqueness for the classical motion are studied by the function of the principal curvatures of a smooth surface and the Evans and Spruck's results are generalized.
文摘This paper is to detect regions (objects) boundaries, also to isolate and extract individual components from a medical image. This can be done using an active contours to detect regions in a given image, based on techniques of curve evolution, Mumford Shah functional for segmentation and level sets. The paper classified the images into different intensity regions based on Markov random field, then detected regions whose boundaries are not necessarily defined by gradient by minimizing an energy of Mumford Shah functional for segmentation which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a mean curvature flow like evolving the active contour, which will stop on the desired boundary. The stopping term does not depend on the gradient of the image, as in the classical active contour and the initial curve of level set can be anywhere in the image, and interior contours are automatically detected. The final image segmentation is one closed boundary per actual region in the image.
基金supported by the National Natural Science Foundation of China(Grant No.12272144).
文摘A data-driven model ofmultiple variable cutting(M-VCUT)level set-based substructure is proposed for the topology optimization of lattice structures.TheM-VCUTlevel setmethod is used to represent substructures,enriching their diversity of configuration while ensuring connectivity.To construct the data-driven model of substructure,a database is prepared by sampling the space of substructures spanned by several substructure prototypes.Then,for each substructure in this database,the stiffness matrix is condensed so that its degrees of freedomare reduced.Thereafter,the data-drivenmodel of substructures is constructed through interpolationwith compactly supported radial basis function(CS-RBF).The inputs of the data-driven model are the design variables of topology optimization,and the outputs are the condensed stiffness matrix and volume of substructures.During the optimization,this data-driven model is used,thus avoiding repeated static condensation that would requiremuch computation time.Several numerical examples are provided to verify the proposed method.
基金funded by National Nature Science Foundation of China,grant number 61302188.
文摘To solve the problem of false edges in a flat region of l_(1)norm total variational TV model,an edge extractor based on non-local idea is proposed in this paper.The new edge extractor can effectively suppress the influence of noise and extract the edge information of the image.The new edge extractor is used as the adaptive function and the weighting function of the l_(p) norm variational model to control the noise reduction ability of the model,and a new model 1 is obtained.Considering that the new model 1 only uses the gradient mode as the image feature operator,which is insufficient to express the image texture information,a new level set curvature gradient variational model 2 combined with the edge extractor is proposed.The new model 2 uses the idea of minimum curvature of the level set of clear images to obtain noise reduction images.By coupling new model 1 and new model 2 to smooth the noise and protect more textures,a new Non-local level set denoising model(NLSDM)for image noise reduction is obtained.The experimental results show that compared with the noise reduction model,the new model has significantly improved the peak signal-to-noise ratio and structural similarity,and the effect of noise reduction and edge preservation is better.
文摘Using traditional particle tracking velocimetry based on optical flow for measuring areas with large velocity gradient changes may cause oversmoothing,resulting in significant measurement errors.To address this problem,the traditional particle tracking velocimetry method based on an optical flow was improved.The level set segmentation algorithm was used to obtain the boundary contour of the region with large velocity gradient changes,and the non-uniform flow field was divided into regions according to the boundary contour to obtain sub-regions with uniform velocity distribution.The particle tracking velocimetry method based on optical flow was used to measure the granular flow velocity in each sub-region,thus avoiding the problem of granular flow distribution.The simulation results show that the measurement accuracy of this method is approximately 10%higher than that of traditional methods.The method was applied to a velocity measurement experiment on dense granular flow in silos,and the velocity distribution of the granular flow was obtained,verifying the practicality of the method in granular flow fields.
基金supported by grants from the National Natural Science Foundation of China (51478130)the Guangzhou Municipal Education Bureau’s Scientific Research Project, China (2024312217)+1 种基金the China Scholarship Council (201808440070)the 111 Project of China (D21021).
文摘This paper presents an improved level set method for topology optimization of geometrically nonlinear structures accounting for the effect of thermo-mechanical couplings.It derives a new expression for element coupling stress resulting from the combination of mechanical and thermal loading,using geometric nonlinear finite element analysis.A topological model is then developed to minimize compliance while meeting displacement and frequency constraints to fulfill design requirements of structural members.Since the conventional Lagrange multiplier search method is unable to handle convergence instability arising from large deformation,a novel Lagrange multiplier search method is proposed.Additionally,the proposed method can be extended to multi-constrained geometrically nonlinear topology optimization,accommodating multiple physical field couplings.
基金The National Natural Science Foundation of China under contract Nos 61701416 and 61901195Natural Science Foundation of Jiangsu Province of China under contract No.BK20211341China Postdoctoral Science Foundation under contract No.2022M712687.
文摘Separation between water and land is vital for marine scientific research and coastal zone planning and management.The contrasting backscatter properties of land and ocean enable clear water edge line identification in synthetic aperture radar(SAR)imagery.However,SAR images are prone to speckle noise,and the complexity of the water-land boundaries environment makes accurate water-land separation challenging.To overcome noise and complex background interference in remote sensing images,an improved level set method was employed to enhance water-land separation.In the traditional distance regularized level set method,the selection of the image correlation weight coefficient and the edge indicator function directly influences the accuracy of the final segmentation results.A novel level set segmentation algorithm incorporating an improved edge indicator function is proposed to efficiently and accurately separate the water edge lines in SAR images.The distance regularized level set evolution model is enhanced by incorporating the signed pressure force function as an adaptive parameter,which serves as an external constraint for curve evolution.A novel level set model with an adaptive edge indicator function,combining gradient and regional information,is proposed.Experimental results demonstrate that the proposed model enhances the accuracy of waterland separation in SAR images.However,further research is needed to evaluate its potential for detecting boundaries in diverse marine environments and across different types of SAR imagery.
基金National Natural Science Foundation of China(22178228,22178326)
文摘A cylindrical chamber with a rotating bottom holds significant potential for application in cell culture bioreactors due to its ability to generate more stable swirling flows.In order to control vortex breakdown within the chamber,this study first establishes a computational fluid dynamics simulation coupled with the level set method.Verified by experimental results in literature,this method accurately simulates the position and shape of vortex breakdown,and also predicts the critical Reynolds numbers for the appearance and detachment of vortex breakdown bubbles from the center.Additionally,it precisely captures the gas-liquid interface and depicts the vortex breakdown phenomenon in the air above the liquid for the first time.Finally,it predicts the impact of physical property of gas-liquid systems on vortex breakdown in response to significant changes in viscosity of microbial process systems.
基金fundings supported by Sichuan Science and Technology Program(2025YFHZ0065).
文摘Structural Reliability-Based Topology Optimization(RBTO),as an efficient design methodology,serves as a crucial means to ensure the development ofmodern engineering structures towards high performance,long service life,and high reliability.However,in practical design processes,topology optimization must not only account for the static performance of structures but also consider the impacts of various responses and uncertainties under complex dynamic conditions,which traditional methods often struggle accommodate.Therefore,this study proposes an RBTO framework based on a Kriging-assisted level set function and a novel Dynamic Hybrid Particle Swarm Optimization(DHPSO)algorithm.By leveraging the Kriging model as a surrogate,the high cost associated with repeatedly running finite element analysis processes is reduced,addressing the issue of minimizing structural compliance.Meanwhile,the DHPSO algorithm enables a better balance between the population’s developmental and exploratory capabilities,significantly accelerating convergence speed and enhancing global convergence performance.Finally,the proposed method is validated through three different structural examples,demonstrating its superior performance.Observed that the computational that,compared to the traditional Solid Isotropic Material with Penalization(SIMP)method,the proposed approach reduces the upper bound of structural compliance by approximately 30%.Additionally,the optimized results exhibit clear material interfaces without grayscale elements,and the stress concentration factor is reduced by approximately 42%.Consequently,the computational results fromdifferent examples verify the effectiveness and superiority of this study across various fields,achieving the goal of providing more precise optimization results within a shorter timeframe.
基金supported by National Natural Science Foundation of China (Grant No. 10871187)supported by the Science Research Program from the Education Department of Heilongjiang Province (Grant No. 11551137)
文摘We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations ut = F(▽2u,▽u,u,t) under a structural condition,and give a geometric lower bound of the principal curvature of the spatial level surfaces.
基金supported by the Chinese Universities Scientific Fund(No.WK0010000028)supported by the National Science Fund for Distinguished Young Scholars of China and Wu Wen-Tsun Key Laboratory of Mathematics+1 种基金partially supported by the National Natural Science Foundation of China(Nos.11101110,11326144)the Foundation of Harbin Normal University(No.KGB201224)
文摘For the Monge-Amp`ere equation det D2u = 1, the authors find new auxiliary curvature functions which attain their respective maxima on the boundary. Moreover, the upper bounded estimates for the Gauss curvature and the mean curvature of the level sets for the solution to this equation are obtained.
文摘We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.
基金Supported partly by the NNSF of China(Nos.10371092,10171015 and No.10271027)
文摘In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.
文摘For an open set V C Cn, denote by Mα （V） the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded ＂harmonically fat＂ domain Ω C Cn, a function f ∈M a（Ω／f-1（0）） automatically sat- isfies f ∈M a（Ω）, if it is Caj-1smooth in the z/variable, α ∈ Zn＋ up to the boundary. For a submanifold U C Cn, denote by ma（U）, the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal （off the singularities） obey the boundary maximum modulus principle. We prove, that for a C3-smooth hypersurface, Ω, a member of ma （Ω）, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form. The result can be partially generalized to C4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.
基金Research of the first author was supported by NSFC and Wu Wen-Tsun Key Laboratory of Mathematics.We finished this paper in the winter of 2009 as a part of the thesis of the second author in University of Science and Technology of China.
文摘For the p-harmonic function with strictly convex level sets,we find an auxiliary function which comes from the combination of the norm of gradient of the p-harmonic function and the Gaussian curvature of the level sets of p-harmonic function.We prove that this curvature function is concave with respect to the height of the p-harmonic function.This auxiliary function is an affine function of the height when the p-harmonic function is the p-Green function on ball.
