Objective.Objective of this work is the development and evaluation of a cortical parcellation framework based on tractography-derived brain structural connectivity.Impact Statement.The proposed framework utilizes nove...Objective.Objective of this work is the development and evaluation of a cortical parcellation framework based on tractography-derived brain structural connectivity.Impact Statement.The proposed framework utilizes novel spatial-graph representation learning methods for solving the task of cortical parcellation,an important medical image analysis and neuroscientific problem.Introduction.The concept of“connectional fingerprint”has motivated many investigations on the connectivity-based cortical parcellation,especially with the technical advancement of diffusion imaging.Previous studies on multiple brain regions have been conducted with promising results.However,performance and applicability of these models are limited by the relatively simple computational scheme and the lack of effective representation of brain imaging data.Methods.We propose the Spatial-graph Convolution Parcellation(SGCP)framework,a two-stage deep learning-based modeling for the graph representation brain imaging.In the first stage,SGCP learns an effective embedding of the input data through a self-supervised contrastive learning scheme with the backbone encoder of a spatial-graph convolution network.In the second stage,SGCP learns a supervised classifier to perform voxel-wise classification for parcellating the desired brain region.Results.SGCP is evaluated on the parcellation task for 5 brain regions in a 15-subject DWI dataset.Performance comparisons between SGCP,traditional parcellation methods,and other deep learning-based methods show that SGCP can achieve superior performance in all the cases.Conclusion.Consistent good performance of the proposed SGCP framework indicates its potential to be used as a general solution for investigating the regional/subregional composition of human brain based on one or more connectivity measurements.展开更多
In this note,we show that the solution of Kähler-Ricci flow on every Fano threefold from Family No.2.23 in the Mori-Mukai’s list develops type II singularity.In fact,we show that no Fano threefold from Family No...In this note,we show that the solution of Kähler-Ricci flow on every Fano threefold from Family No.2.23 in the Mori-Mukai’s list develops type II singularity.In fact,we show that no Fano threefold from Family No.2.23 admits Kähler-Ricci soliton and the Gromov-Hausdorff limit of the Kähler-Ricci flow must be a singular Q-Fano variety.This gives new examples of Fano manifolds of the lowest dimension on which Kähler-Ricci flow develops type II singularity.展开更多
Quantifying the shape and stiffness of extracellular vesicles(EVs)is essential for understanding their biophysical properties and roles in intercellular communication.However,achieving single-particle resolution under...Quantifying the shape and stiffness of extracellular vesicles(EVs)is essential for understanding their biophysical properties and roles in intercellular communication.However,achieving single-particle resolution under physiological conditions remains a significant challenge.Here,we introduce an approach that integrates single-molecule diffusivity mapping(SMdM)with diffusion models for spherical and discoidal shapes to quantify the geometric and mechanical properties of individual liposomes and EVs in aqueous solution.Our findings identify charged lipids and cholesterol as critical factors that enhance liposome stiffness,driving their shapes closer to spheres.Applying this method to EVs reveals that those derived from tumor cells exhibit lower stiffness compared to EVs from normal cells,consistent with the biomechanical characteristics of their parent cells.This rapid,highthroughput strategy for characterizing the shape and stiffness of single EVs in aqueous solution offers promising applications in cancer biomarker discovery and the development of EV-based therapeutics.展开更多
We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonline...We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.展开更多
Wediscuss the idea of using continuous dynamicalsystemstomodel generalhigh-dimensional nonlinear functions used in machine learning.We also discuss theconnection with deep learning.
In this paper,we study steady Ricci solitons with a linear decay of sectional curvature.In particular,we give a complete classification of 3-dimensional steady Ricci solitons and 4-dimensional K-noncollapsed steady Ri...In this paper,we study steady Ricci solitons with a linear decay of sectional curvature.In particular,we give a complete classification of 3-dimensional steady Ricci solitons and 4-dimensional K-noncollapsed steady Ricci solitons with non-negative sectional curvature under the linear curvature decay.展开更多
Distinguishing things from beings, or matters from lives, is a fundamental question. Extending E. Schr?dinger's neg-entropy and I. Prigogine's dissipative structure, we propose a chemical kinetic view that the...Distinguishing things from beings, or matters from lives, is a fundamental question. Extending E. Schr?dinger's neg-entropy and I. Prigogine's dissipative structure, we propose a chemical kinetic view that the earliest "live" process is embedded essentially in a special interaction between a pair of specific components under a particular, corresponding environmental conditions. The interaction exists as an inter-molecular-force-bond complex(IMFBC) that couples two separate chemical processes: one is the spontaneous formation of the IMFBC driven by a decrease of Gibbs free energy as a dissipative process; while the other is the disassembly of the IMFBC driven thermodynamically by free energy input from the environment. The two chemical processes coupled by the IMFBC originated independently and were considered non-living on Earth, but the IMFBC coupling of the two can be considered as the earliest form of metabolism: the first landmark on the path from things to a being. The dynamic formation and disassembly of the IMFBC, as a composite individual, follows a principle designated as "… structure for energy for structure for energy…", the cycle continues; and for short it will be referred to as "structure for energy cycle". With additional features derived from this starting point, the IMFBC-centered "live" process spontaneously evolved into more complex living organisms with the characteristics currently known.展开更多
A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in[6].For numerically solving the high order hyperbolic moment system therein,we in this paper develop a preliminary numeric...A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in[6].For numerically solving the high order hyperbolic moment system therein,we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in[8],to validate the moment system of the Wigner equation.The method developed can keep both mass and momentum conserved,and the variation of the total energy under control though it is not strictly conservative.We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion,and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method.The numerical results indicate that the high order moment system in[6]is a valid model for the Wigner equation,and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.展开更多
The sole use of single modality data often fails to capture the complex heterogeneity among patients,including the variability in resistance to anti-HER2 therapy and outcomes of combined treatment regimens,for the tre...The sole use of single modality data often fails to capture the complex heterogeneity among patients,including the variability in resistance to anti-HER2 therapy and outcomes of combined treatment regimens,for the treatment of HER2-positive gastric cancer(GC).This modality deficit has not been fully considered in many studies.Furthermore,the application of artificial intelligence in predicting the treatment response,particularly in complex diseases such as GC,is still in its infancy.Therefore,this study aimed to use a comprehensive analytic approach to accurately predict treatment responses to anti-HER2 therapy or anti-HER2 combined immunotherapy in patients with HER2-positive GC.We collected multi-modal data,comprising radiology,pathology,and clinical information from a cohort of 429 patients:310 treated with anti-HER2 therapy and 119 treated with a combination of anti-HER2 and anti-PD-1/PD-L1 inhibitors immunotherapy.We introduced a deep learning model,called the Multi-Modal model(MuMo),that integrates these data to make precise treatment response predictions.MuMo achieved an area under the curve score of 0.821 for anti-HER2 therapy and 0.914 for combined immunotherapy.Moreover,patients classified as low-risk by MuMo exhibited significantly prolonged progression-free survival and overall survival(log-rank test,P<0.05).These findings not only highlight the significance of multi-modal data analysis in enhancing treatment evaluation and personalized medicine for HER2-positive gastric cancer,but also the potential and clinical value of our model.展开更多
We described a novel single-cell RNA-seq technique called MR-seq （measure a single-cell transcriptome repeatedly）, which permits statistically assessing the technical variation and identifying the differentially exp...We described a novel single-cell RNA-seq technique called MR-seq （measure a single-cell transcriptome repeatedly）, which permits statistically assessing the technical variation and identifying the differentially expressed genes between just two single ceils by measuring each single cell twice. We demonstrated that MR-seq gave sensitivity and reproducibility similar to the standard single-cell RNA-seq and increased the positive predicate value, Application of MR-seq to early mouse embryos identified hundreds of candidate intra-embryonic heterogeneous genes among mouse 2-, 4- and 8-cell stage embryos. MR-seq should be useful for detecting differentially exnre^ed ~enes ~rnnn~ ~ ~m^ll nHmhpr nf c^ll~展开更多
We introduce an algebraicity criterion.It has the following form:Consider an analytic subvariety of some algebraic variety X over a global field K.Under certain conditions,if X contains many K-points,then X is algebra...We introduce an algebraicity criterion.It has the following form:Consider an analytic subvariety of some algebraic variety X over a global field K.Under certain conditions,if X contains many K-points,then X is algebraic over K.This gives a way to show the transcendence of points via the transcendence of analytic subvarieties.Such a situ-ation often appears when we have a dynamical system,because we can often produce infinitely many points from one point via iterates.Combining this criterion and the study of invariant subvarieties,we get some results on the transcendence in arithmetic dynamics.We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic.For this,we study the invariant subvarieties for products of endomorphisms.In particular,we partially generalize Medvedev-Scanlon’s classification of invariant subvarieties of split polynomial maps to separable endomorphisms on(P^(1))^(N) in any characteristic.We also get some high dimensional partial generalization via introducing a notion of independence.We then study dominant endomorphisms f on A^(N) over a number field of algebraic degree d≥2.We show that in most cases(e.g.when such an endomor-phism extends to an endomorphism on P^(N)),there are many analytic curves centered at infinity which are periodic.We show that for most of them,it is algebraic if and only if it contains at least one algebraic point.We also study the periodic curves.We show that for most f,all periodic curves have degree at most 2.When N=2,we get a more precise classification result.We show that under a condition which is satisfied for a general f,if f has infinitely many periodic curves,then f is homogenous up to change of origin.展开更多
We construct parabolic analogues of(global)eigenvarieties,of patched eigenvarieties and of(local)trianguline varieties,that we call,respectively,Bernstein eigenvarieties,patched Bernstein eigenvarieties,and Bernstein ...We construct parabolic analogues of(global)eigenvarieties,of patched eigenvarieties and of(local)trianguline varieties,that we call,respectively,Bernstein eigenvarieties,patched Bernstein eigenvarieties,and Bernstein paraboline varieties.We study the geometry of these rigid analytic spaces,in particular(generalising results of Breuil-Hellmann-Schraen)we show that their local geometry can be described by certain algebraic schemes related to the generalised Grothendieck-Springer resolution.We deduce several local-global compatibility results,including a classicality result(with no trianguline assumption at p),and new cases towards the locally analytic socle conjecture of Breuil in the non-trianguline case.展开更多
This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string...This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDEs）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.展开更多
In this paper,we prove a wall-crossing formula,a crucial ingredient needed to prove that the correlation function of gauged linear-model is independent of the choice of perturbations.
We prove the following result:if aℚ-Fano variety is uniformly K-stable,then it admits a Kähler–Einstein metric.This proves the uniform version of Yau–Tian–Donaldson conjecture for all(singular)Fano varieties w...We prove the following result:if aℚ-Fano variety is uniformly K-stable,then it admits a Kähler–Einstein metric.This proves the uniform version of Yau–Tian–Donaldson conjecture for all(singular)Fano varieties with discrete automorphism groups.We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments.This perturbation approach depends on the valuative criterion and non-Archimedean estimates,and is motivated by our previous paper.展开更多
文摘Objective.Objective of this work is the development and evaluation of a cortical parcellation framework based on tractography-derived brain structural connectivity.Impact Statement.The proposed framework utilizes novel spatial-graph representation learning methods for solving the task of cortical parcellation,an important medical image analysis and neuroscientific problem.Introduction.The concept of“connectional fingerprint”has motivated many investigations on the connectivity-based cortical parcellation,especially with the technical advancement of diffusion imaging.Previous studies on multiple brain regions have been conducted with promising results.However,performance and applicability of these models are limited by the relatively simple computational scheme and the lack of effective representation of brain imaging data.Methods.We propose the Spatial-graph Convolution Parcellation(SGCP)framework,a two-stage deep learning-based modeling for the graph representation brain imaging.In the first stage,SGCP learns an effective embedding of the input data through a self-supervised contrastive learning scheme with the backbone encoder of a spatial-graph convolution network.In the second stage,SGCP learns a supervised classifier to perform voxel-wise classification for parcellating the desired brain region.Results.SGCP is evaluated on the parcellation task for 5 brain regions in a 15-subject DWI dataset.Performance comparisons between SGCP,traditional parcellation methods,and other deep learning-based methods show that SGCP can achieve superior performance in all the cases.Conclusion.Consistent good performance of the proposed SGCP framework indicates its potential to be used as a general solution for investigating the regional/subregional composition of human brain based on one or more connectivity measurements.
文摘In this note,we show that the solution of Kähler-Ricci flow on every Fano threefold from Family No.2.23 in the Mori-Mukai’s list develops type II singularity.In fact,we show that no Fano threefold from Family No.2.23 admits Kähler-Ricci soliton and the Gromov-Hausdorff limit of the Kähler-Ricci flow must be a singular Q-Fano variety.This gives new examples of Fano manifolds of the lowest dimension on which Kähler-Ricci flow develops type II singularity.
基金financial supports from National Key R&D Program of China(2022YFA1305400)National Natural Science Foundation of China(22274122,22104113)+1 种基金Fundamental Research Funds for the Central Universities interdisciplinary(2042023kf1012)Innovative Talents Foundation from Renmin Hospital of Wuhan University(JCRCFZ-2022-010).
文摘Quantifying the shape and stiffness of extracellular vesicles(EVs)is essential for understanding their biophysical properties and roles in intercellular communication.However,achieving single-particle resolution under physiological conditions remains a significant challenge.Here,we introduce an approach that integrates single-molecule diffusivity mapping(SMdM)with diffusion models for spherical and discoidal shapes to quantify the geometric and mechanical properties of individual liposomes and EVs in aqueous solution.Our findings identify charged lipids and cholesterol as critical factors that enhance liposome stiffness,driving their shapes closer to spheres.Applying this method to EVs reveals that those derived from tumor cells exhibit lower stiffness compared to EVs from normal cells,consistent with the biomechanical characteristics of their parent cells.This rapid,highthroughput strategy for characterizing the shape and stiffness of single EVs in aqueous solution offers promising applications in cancer biomarker discovery and the development of EV-based therapeutics.
基金supported in part by the National Key Basic Research Program of China 2015CB856000Major Program of NNSFC under Grant 91130005,DOE Grant DE-SC0009248ONR Grant N00014-13-1-0338.
文摘We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.
基金with several collaborators,including Jiequn Han,Qianxiao Li,Jianfeng Lu and Cheng Tai.The author benefitted a great deal from discussions with them,particularly Jiequn Han.This work is supported in part by the Major Program of NNSFC under Grant91130005,ONR NO0014-13-1-0338 and DOE DE-SCo009248.
文摘Wediscuss the idea of using continuous dynamicalsystemstomodel generalhigh-dimensional nonlinear functions used in machine learning.We also discuss theconnection with deep learning.
基金supported by National Natural Science Foundation of China (Grant No. 11701030)supported by National Natural Science Foundation of China (Grant Nos. 11331001 and 11771019)
文摘In this paper,we study steady Ricci solitons with a linear decay of sectional curvature.In particular,we give a complete classification of 3-dimensional steady Ricci solitons and 4-dimensional K-noncollapsed steady Ricci solitons with non-negative sectional curvature under the linear curvature decay.
基金supported by MST (2003CB715906 to Shunong Bai)National Natural Science Foundation of China (11021463 to Qi Ouyang)
文摘Distinguishing things from beings, or matters from lives, is a fundamental question. Extending E. Schr?dinger's neg-entropy and I. Prigogine's dissipative structure, we propose a chemical kinetic view that the earliest "live" process is embedded essentially in a special interaction between a pair of specific components under a particular, corresponding environmental conditions. The interaction exists as an inter-molecular-force-bond complex(IMFBC) that couples two separate chemical processes: one is the spontaneous formation of the IMFBC driven by a decrease of Gibbs free energy as a dissipative process; while the other is the disassembly of the IMFBC driven thermodynamically by free energy input from the environment. The two chemical processes coupled by the IMFBC originated independently and were considered non-living on Earth, but the IMFBC coupling of the two can be considered as the earliest form of metabolism: the first landmark on the path from things to a being. The dynamic formation and disassembly of the IMFBC, as a composite individual, follows a principle designated as "… structure for energy for structure for energy…", the cycle continues; and for short it will be referred to as "structure for energy cycle". With additional features derived from this starting point, the IMFBC-centered "live" process spontaneously evolved into more complex living organisms with the characteristics currently known.
基金supported in part by the National Basic Research Program of China(2011CB309704)Fok Ying Tong Education and NCET in China+1 种基金T.Lu was supported in part by the NSFC(11011130029,91230107)by SRF for ROCS,SEM.
文摘A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in[6].For numerically solving the high order hyperbolic moment system therein,we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in[8],to validate the moment system of the Wigner equation.The method developed can keep both mass and momentum conserved,and the variation of the total energy under control though it is not strictly conservative.We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion,and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method.The numerical results indicate that the high order moment system in[6]is a valid model for the Wigner equation,and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.
基金supported by the National Natural Science Foundation of China(91959205 to L.S.,U22A20327 to L.S.,82203881 to Y.C.,82272627 to XT.Z.,7232018 to Y.S.,12090022 to B.D.,11831002 to B.D.,81801778 to L.Z.)Beijing Natural Science Foundation(7222021 to Y.C.,Z200015 to XT.Z.)+1 种基金Beijing Hospitals Authority Youth Programme(QML20231115 to Y.C.)Clinical Medicine Plus X-Young Scholars Project of Peking University(PKU2023LCXQ041 to Y.C.and L.Z.).
文摘The sole use of single modality data often fails to capture the complex heterogeneity among patients,including the variability in resistance to anti-HER2 therapy and outcomes of combined treatment regimens,for the treatment of HER2-positive gastric cancer(GC).This modality deficit has not been fully considered in many studies.Furthermore,the application of artificial intelligence in predicting the treatment response,particularly in complex diseases such as GC,is still in its infancy.Therefore,this study aimed to use a comprehensive analytic approach to accurately predict treatment responses to anti-HER2 therapy or anti-HER2 combined immunotherapy in patients with HER2-positive GC.We collected multi-modal data,comprising radiology,pathology,and clinical information from a cohort of 429 patients:310 treated with anti-HER2 therapy and 119 treated with a combination of anti-HER2 and anti-PD-1/PD-L1 inhibitors immunotherapy.We introduced a deep learning model,called the Multi-Modal model(MuMo),that integrates these data to make precise treatment response predictions.MuMo achieved an area under the curve score of 0.821 for anti-HER2 therapy and 0.914 for combined immunotherapy.Moreover,patients classified as low-risk by MuMo exhibited significantly prolonged progression-free survival and overall survival(log-rank test,P<0.05).These findings not only highlight the significance of multi-modal data analysis in enhancing treatment evaluation and personalized medicine for HER2-positive gastric cancer,but also the potential and clinical value of our model.
基金supported by grants from the Beijing Municipal Science and Technology Commission (D15110700240000)
文摘We described a novel single-cell RNA-seq technique called MR-seq （measure a single-cell transcriptome repeatedly）, which permits statistically assessing the technical variation and identifying the differentially expressed genes between just two single ceils by measuring each single cell twice. We demonstrated that MR-seq gave sensitivity and reproducibility similar to the standard single-cell RNA-seq and increased the positive predicate value, Application of MR-seq to early mouse embryos identified hundreds of candidate intra-embryonic heterogeneous genes among mouse 2-, 4- and 8-cell stage embryos. MR-seq should be useful for detecting differentially exnre^ed ~enes ~rnnn~ ~ ~m^ll nHmhpr nf c^ll~
文摘We introduce an algebraicity criterion.It has the following form:Consider an analytic subvariety of some algebraic variety X over a global field K.Under certain conditions,if X contains many K-points,then X is algebraic over K.This gives a way to show the transcendence of points via the transcendence of analytic subvarieties.Such a situ-ation often appears when we have a dynamical system,because we can often produce infinitely many points from one point via iterates.Combining this criterion and the study of invariant subvarieties,we get some results on the transcendence in arithmetic dynamics.We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic.For this,we study the invariant subvarieties for products of endomorphisms.In particular,we partially generalize Medvedev-Scanlon’s classification of invariant subvarieties of split polynomial maps to separable endomorphisms on(P^(1))^(N) in any characteristic.We also get some high dimensional partial generalization via introducing a notion of independence.We then study dominant endomorphisms f on A^(N) over a number field of algebraic degree d≥2.We show that in most cases(e.g.when such an endomor-phism extends to an endomorphism on P^(N)),there are many analytic curves centered at infinity which are periodic.We show that for most of them,it is algebraic if and only if it contains at least one algebraic point.We also study the periodic curves.We show that for most f,all periodic curves have degree at most 2.When N=2,we get a more precise classification result.We show that under a condition which is satisfied for a general f,if f has infinitely many periodic curves,then f is homogenous up to change of origin.
基金supported by the C.N.R.S and is a member of the A.N.R.project CLap-CLap ANR-18-CE40-0026supported by the NSFC Grant No.8200905010 and No.8200800065.
文摘We construct parabolic analogues of(global)eigenvarieties,of patched eigenvarieties and of(local)trianguline varieties,that we call,respectively,Bernstein eigenvarieties,patched Bernstein eigenvarieties,and Bernstein paraboline varieties.We study the geometry of these rigid analytic spaces,in particular(generalising results of Breuil-Hellmann-Schraen)we show that their local geometry can be described by certain algebraic schemes related to the generalised Grothendieck-Springer resolution.We deduce several local-global compatibility results,including a classicality result(with no trianguline assumption at p),and new cases towards the locally analytic socle conjecture of Breuil in the non-trianguline case.
基金supported by National Natural Science Foundation of China(Grant No.91130005)the US Army Research Office(Grant No.W911NF-11-1-0101)
文摘This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDEs）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.
文摘In this paper,we prove a wall-crossing formula,a crucial ingredient needed to prove that the correlation function of gauged linear-model is independent of the choice of perturbations.
基金supported by NSF(Grant No.DMS-1810867)research fellowship.G.Tian is partially supported by NSF(Grant No.DMS-1607091)and NSFC(Grant No.11331001)partially supported by NSFC(Grant No.11501501).
文摘We prove the following result:if aℚ-Fano variety is uniformly K-stable,then it admits a Kähler–Einstein metric.This proves the uniform version of Yau–Tian–Donaldson conjecture for all(singular)Fano varieties with discrete automorphism groups.We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments.This perturbation approach depends on the valuative criterion and non-Archimedean estimates,and is motivated by our previous paper.
