Achieving higher true positive rate when decreasing false positive rate is always a great challenge to the imbalance learning community.This work combines penalized empirical likelihood method,lower bound algorithm an...Achieving higher true positive rate when decreasing false positive rate is always a great challenge to the imbalance learning community.This work combines penalized empirical likelihood method,lower bound algorithm and Nyströmmethod and applies these techniques along with kernel method to density ratio model.The resulting classifier,density ratio classifier(DRC),is a combination of kernelization,regularization,efficient implementation and threshold moving,all of which are critical to enable DRC to be an effective and powerful method for solving difficult imbalance problems.Compared with other methods,DRC is competitive in that it is widely applicable and it is simple and easy to use without additional imbalance handling skills.In addition,the convergence rate of the estimate of log density ratio is discussed as well.And the results of numerical analysis also show that DRC outperforms other methods in AUC and G-mean score.展开更多
Let pod_(3)(n)denote the number of 3-regular partitions with distinct odd parts(and even parts are unrestricted)of a non-negative integer n.In this paper,we present infinite families of Ramanujan-type congruences modu...Let pod_(3)(n)denote the number of 3-regular partitions with distinct odd parts(and even parts are unrestricted)of a non-negative integer n.In this paper,we present infinite families of Ramanujan-type congruences modulo 2 and 3 for pod_(3)(n).展开更多
In this article,we address two issues related to the perturbation method introduced by Zhang and Lu(J Comput Phys 194:773-794,2004),and applied to solving linear stochastic parabolic PDE.Those issues are the construct...In this article,we address two issues related to the perturbation method introduced by Zhang and Lu(J Comput Phys 194:773-794,2004),and applied to solving linear stochastic parabolic PDE.Those issues are the construction of the perturbation series,and its convergence.展开更多
Let f be a formation function and G an A-group.It is said that A acts f-hypercentrally on G if A acts f-centrally on every A-composition factor of G.In this paper,groups are investigated by f-hypercentral actions.In p...Let f be a formation function and G an A-group.It is said that A acts f-hypercentrally on G if A acts f-centrally on every A-composition factor of G.In this paper,groups are investigated by f-hypercentral actions.In particular,some well-known results,including a theorem of Huppert and a theorem of Hall-Higman,are generalized.展开更多
In this paper,we introduce a new averaging rule,the nonlinear weighted averaging rule.As an application,this averaging rule is used to replace the midpoint averaging in the de Casteljau evaluation algorithm and with t...In this paper,we introduce a new averaging rule,the nonlinear weighted averaging rule.As an application,this averaging rule is used to replace the midpoint averaging in the de Casteljau evaluation algorithm and with this scheme we can also generate transcendental functions which cannot be generated by the classical de Casteljau algorithm.We also investigate the properties of the curves of the functions generated by blossoming,where the results show that these curves and the classical Bézier curves have some similar properties,including variation diminishing property and endpoint interpolation.However,the curves obtained by blossoming using nonlinear weighted averaging rules induced by certain functions violate some properties like convex hull property.展开更多
Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,...Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,receive a single arrow from one of its facets,or conversely,send a single arrow into a cell of which it is a facet.By following arrows,one can then construct a natural Floer-type boundary operator.Here,we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows(but not both),again in the absence of closed orbits.Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.展开更多
In the present paper,we study the structure of cyclic DNA codes of even length over the ring F_(2)+uF_(2)+u^(2)F_(2)where u^(3)=0.We investigate two presentations of cyclic codes of even length over F_(2)+uF_(2)+u^(2)...In the present paper,we study the structure of cyclic DNA codes of even length over the ring F_(2)+uF_(2)+u^(2)F_(2)where u^(3)=0.We investigate two presentations of cyclic codes of even length over F_(2)+uF_(2)+u^(2)F_(2)satisfying the reverse constraint and the reverse-complement constraint.展开更多
Let R be a prime ring of characteristic different from 2,Q_(r) be its right MartindalequotientringandC beitsextendedcentroid,G beanonzero X-generalized skew derivation of R,and S be the set of the evaluations of a mul...Let R be a prime ring of characteristic different from 2,Q_(r) be its right MartindalequotientringandC beitsextendedcentroid,G beanonzero X-generalized skew derivation of R,and S be the set of the evaluations of a multilinear polynomial f(x_(1),...,x_(n))over C with n non-commuting variables.Let u,v∈R be such that uG(x)x+G(x)xv=0 for all x∈S.Then one of the following statements holds:(a)v∈C and there exist a,b,c∈Q_(r) such that G(x)=ax+bxc for any x∈R with(u+v)a=(u+v)b=0;(b)f(x_(1),...,x_(n))2 is central-valued on R and there exists a∈Q r such that G(x)=ax for all x∈R with ua+av=0.展开更多
Recently,it was found that during the process of certain refinement of hier-archical T-meshes,some basis functions of PHT-splines decay severely,which is not expected in solving numerical PDEs and in least square data...Recently,it was found that during the process of certain refinement of hier-archical T-meshes,some basis functions of PHT-splines decay severely,which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned.In this paper,we present a method to modify the basis functions of PHT-splines in the case that the sup-ports of the original truncated basis functions are rectangular domains to overcome the decay problem.The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity,local support,linear independency.Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions.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.
Consider n nodes{X_(i)}_(1≤i≤n) independently and identically distributed(i.i.d.)across N cities located within the unit square S.Each city is modelled as an r_(n)×r_(n)square,and MSTC_(n)denotes the weighted l...Consider n nodes{X_(i)}_(1≤i≤n) independently and identically distributed(i.i.d.)across N cities located within the unit square S.Each city is modelled as an r_(n)×r_(n)square,and MSTC_(n)denotes the weighted length of the minimum spanning tree containing all the n nodes,where the edge length between nodes X_(i)and X_(j)is weighted by a factor that depends on the individual locations of X_(i)and X_(j).We use approximation methods to obtain variance estimates for MSTC_(n)and prove that if the cities are well connected in a certain sense,then MSTC_(n)appropriately centred and scaled converges to zero in probability.Using the above proof techniques we also study MST_(n),the length of the minimum weighted spanning tree for nodes distributed throughout the unit square S with location-dependent edge weights.In this case,the variance of MST_(n)grows at most as a power of the logarithm of n and we use a subsequence argument to get almost sure convergence of MST_(n),appropriately centred and scaled.展开更多
The aim of this paper is to investigate different radicals(Wedderburn radical,lower nil radical,Levitzky radical,upper nil radical,the set of all nilpotent elements,the sum of all nil left ideals)of the noncommutative...The aim of this paper is to investigate different radicals(Wedderburn radical,lower nil radical,Levitzky radical,upper nil radical,the set of all nilpotent elements,the sum of all nil left ideals)of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions.We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions.Finally,we establish the transfer of several ring-theoretical properties(reduced,symmetric,reversible,2-primal)from the coefficients ring of a skew PBW extension to the extension itself.展开更多
Which Levy processes satisfy Hunt’s hypothesis(H)is a long-standing open problem in probabilistic potential theory.The study of this problem for one-dimensional Levy processes suggests us to consider(H)from the point...Which Levy processes satisfy Hunt’s hypothesis(H)is a long-standing open problem in probabilistic potential theory.The study of this problem for one-dimensional Levy processes suggests us to consider(H)from the point of view of the sum of Levy processes.In this paper,we present theorems and examples on the validity of(H)for the sum of two independent Levy processes.We also give a novel condition on the Levy measure which implies(H)for a large class of one-dimensional Levy processes.展开更多
This paper presents exponential-type ratio and product estimators for a finite population mean in double sampling using information on several auxiliary variates.The proposed estimators can be viewed as a generalizati...This paper presents exponential-type ratio and product estimators for a finite population mean in double sampling using information on several auxiliary variates.The proposed estimators can be viewed as a generalization over the estimators suggested by Singh and Vishwakarma(Austrian J Stat 36(3):217–225,2007).The expressions for biases and mean square errors(MSEs)of the proposed estimators have been derived to the first degree of approximation.In addition,the expressions for minimum attainable MSEs are also investigated using the criterion for optimality of the weights.An empirical study is carried out in the support of the present study.Both theoretical andempirical findings are encouraging and support thesoundness that the proposed procedures for mean estimation perform better than the usual unbiased estimators and other well-known estimators under some realistic conditions.展开更多
We investigated the false-negative,true-negative,false-positive,and true-positive predictive values from a general group testing procedure for a heterogeneous population.We show that its false(true)-negative predictiv...We investigated the false-negative,true-negative,false-positive,and true-positive predictive values from a general group testing procedure for a heterogeneous population.We show that its false(true)-negative predictive value of a specimen is larger(smaller),and the false(true)-positive predictive value is smaller(larger)than that from individual testing procedure,where the former is in aversion.Then we propose a nested group testing procedure,and show that it can keep the sterling characteristics and also improve the false-negative predictive values for a specimen,not larger than that from individual testing.These characteristics are studied from both theoretical and numerical points of view.The nested group testing procedure is better than individual testing on both false-positive and false-negative predictive values,while retains the efficiency as a basic characteristic of a group testing procedure.Applications to Dorfman’s,Halving and Sterrett procedures are discussed.Results from extensive simulation studies and an application to malaria infection in microscopy-negative Malawian women exemplify the findings.展开更多
We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound.As...We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound.As an application we show that the Coleman–Oort conjecture holds for Shimura curves associated with partial corestriction upon a suitable choice of parameters,which generalizes a construction due to Mumford.展开更多
A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator...A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator subgroup of the normal closures in the link group of themeridians excluding themeridian of the last component with the differential induced by removing the last component.Moreover,the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis.These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.展开更多
基金supported by National Natural Science Foundation of China(Grant No.71873128).
文摘Achieving higher true positive rate when decreasing false positive rate is always a great challenge to the imbalance learning community.This work combines penalized empirical likelihood method,lower bound algorithm and Nyströmmethod and applies these techniques along with kernel method to density ratio model.The resulting classifier,density ratio classifier(DRC),is a combination of kernelization,regularization,efficient implementation and threshold moving,all of which are critical to enable DRC to be an effective and powerful method for solving difficult imbalance problems.Compared with other methods,DRC is competitive in that it is widely applicable and it is simple and easy to use without additional imbalance handling skills.In addition,the convergence rate of the estimate of log density ratio is discussed as well.And the results of numerical analysis also show that DRC outperforms other methods in AUC and G-mean score.
文摘Let pod_(3)(n)denote the number of 3-regular partitions with distinct odd parts(and even parts are unrestricted)of a non-negative integer n.In this paper,we present infinite families of Ramanujan-type congruences modulo 2 and 3 for pod_(3)(n).
文摘In this article,we address two issues related to the perturbation method introduced by Zhang and Lu(J Comput Phys 194:773-794,2004),and applied to solving linear stochastic parabolic PDE.Those issues are the construction of the perturbation series,and its convergence.
文摘Let f be a formation function and G an A-group.It is said that A acts f-hypercentrally on G if A acts f-centrally on every A-composition factor of G.In this paper,groups are investigated by f-hypercentral actions.In particular,some well-known results,including a theorem of Huppert and a theorem of Hall-Higman,are generalized.
文摘In this paper,we introduce a new averaging rule,the nonlinear weighted averaging rule.As an application,this averaging rule is used to replace the midpoint averaging in the de Casteljau evaluation algorithm and with this scheme we can also generate transcendental functions which cannot be generated by the classical de Casteljau algorithm.We also investigate the properties of the curves of the functions generated by blossoming,where the results show that these curves and the classical Bézier curves have some similar properties,including variation diminishing property and endpoint interpolation.However,the curves obtained by blossoming using nonlinear weighted averaging rules induced by certain functions violate some properties like convex hull property.
基金funding provided by Max Planck Societysupported by a stipend from the InternationalMax Planck Research School(IMPRS)“Mathematics in the Sciences.”。
文摘Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a(simplicial,polyhedral or cellular)complex without closed orbits,where each cell may either have no arrows,receive a single arrow from one of its facets,or conversely,send a single arrow into a cell of which it is a facet.By following arrows,one can then construct a natural Floer-type boundary operator.Here,we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows(but not both),again in the absence of closed orbits.Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.
基金supported by NSFC (Grant Nos.11971358,11571259,11771339)Hubei Provincial Natural Science Foundation of China (No.2021CFB400)+1 种基金Fundamental Research Funds for the Central Universities (No.2042019kf0198)the Youth Talent Training Program of Wuhan University。
文摘In this article,we prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from an f-harmonic map.
文摘In the present paper,we study the structure of cyclic DNA codes of even length over the ring F_(2)+uF_(2)+u^(2)F_(2)where u^(3)=0.We investigate two presentations of cyclic codes of even length over F_(2)+uF_(2)+u^(2)F_(2)satisfying the reverse constraint and the reverse-complement constraint.
基金The work of the second author is partially supported by the National Natural Science Foundation of China(Grant No.10871023).
文摘Let R be a prime ring of characteristic different from 2,Q_(r) be its right MartindalequotientringandC beitsextendedcentroid,G beanonzero X-generalized skew derivation of R,and S be the set of the evaluations of a multilinear polynomial f(x_(1),...,x_(n))over C with n non-commuting variables.Let u,v∈R be such that uG(x)x+G(x)xv=0 for all x∈S.Then one of the following statements holds:(a)v∈C and there exist a,b,c∈Q_(r) such that G(x)=ax+bxc for any x∈R with(u+v)a=(u+v)b=0;(b)f(x_(1),...,x_(n))2 is central-valued on R and there exists a∈Q r such that G(x)=ax for all x∈R with ua+av=0.
基金is supported by the National Natural Science Foundation of China(Nos.11571338,11626253)Postdoctoral Science Foundation of China(2015M571931).
文摘Recently,it was found that during the process of certain refinement of hier-archical T-meshes,some basis functions of PHT-splines decay severely,which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned.In this paper,we present a method to modify the basis functions of PHT-splines in the case that the sup-ports of the original truncated basis functions are rectangular domains to overcome the decay problem.The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity,local support,linear independency.Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions.
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The first author was partially supported by NSF grants DMS-0900938 and DMS-1162309The second author was partially supported by NSF grant DMS-1001625.
文摘We undertake a local analysis of combinatorial independence as it connects to topological entropy within the framework of actions of sofic groups.
基金I thank Professors Rahul Roy,Jacob van den Berg,Anish Sarkar,Federico Camia and the referees for crucial comments that led to an improvement of the paper.I also thank Professors Rahul Roy,Federico Camia and IMSc for my fellowships。
文摘Consider n nodes{X_(i)}_(1≤i≤n) independently and identically distributed(i.i.d.)across N cities located within the unit square S.Each city is modelled as an r_(n)×r_(n)square,and MSTC_(n)denotes the weighted length of the minimum spanning tree containing all the n nodes,where the edge length between nodes X_(i)and X_(j)is weighted by a factor that depends on the individual locations of X_(i)and X_(j).We use approximation methods to obtain variance estimates for MSTC_(n)and prove that if the cities are well connected in a certain sense,then MSTC_(n)appropriately centred and scaled converges to zero in probability.Using the above proof techniques we also study MST_(n),the length of the minimum weighted spanning tree for nodes distributed throughout the unit square S with location-dependent edge weights.In this case,the variance of MST_(n)grows at most as a power of the logarithm of n and we use a subsequence argument to get almost sure convergence of MST_(n),appropriately centred and scaled.
基金The first author was supported by the research fund of Facultad de Ciencias,Code HERMES 41535,Universidad Nacional de Colombia,Bogota,Colombia。
文摘The aim of this paper is to investigate different radicals(Wedderburn radical,lower nil radical,Levitzky radical,upper nil radical,the set of all nilpotent elements,the sum of all nil left ideals)of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions.We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions.Finally,we establish the transfer of several ring-theoretical properties(reduced,symmetric,reversible,2-primal)from the coefficients ring of a skew PBW extension to the extension itself.
基金work was supported by National Natural Science Foundation of China(Grant No.11771309)Natural Science and Engineering Research Council of Canada(Grant No.311945-2013)the Fundamental Research Funds for the Central Universities of China。
文摘Which Levy processes satisfy Hunt’s hypothesis(H)is a long-standing open problem in probabilistic potential theory.The study of this problem for one-dimensional Levy processes suggests us to consider(H)from the point of view of the sum of Levy processes.In this paper,we present theorems and examples on the validity of(H)for the sum of two independent Levy processes.We also give a novel condition on the Levy measure which implies(H)for a large class of one-dimensional Levy processes.
文摘This paper presents exponential-type ratio and product estimators for a finite population mean in double sampling using information on several auxiliary variates.The proposed estimators can be viewed as a generalization over the estimators suggested by Singh and Vishwakarma(Austrian J Stat 36(3):217–225,2007).The expressions for biases and mean square errors(MSEs)of the proposed estimators have been derived to the first degree of approximation.In addition,the expressions for minimum attainable MSEs are also investigated using the criterion for optimality of the weights.An empirical study is carried out in the support of the present study.Both theoretical andempirical findings are encouraging and support thesoundness that the proposed procedures for mean estimation perform better than the usual unbiased estimators and other well-known estimators under some realistic conditions.
基金supported by National Natural Science Foundation of China (Grant Nos.11801102,11861017)Beijing Natural Science Foundation (Z180006)the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development of the National Institutes of Health.
文摘We investigated the false-negative,true-negative,false-positive,and true-positive predictive values from a general group testing procedure for a heterogeneous population.We show that its false(true)-negative predictive value of a specimen is larger(smaller),and the false(true)-positive predictive value is smaller(larger)than that from individual testing procedure,where the former is in aversion.Then we propose a nested group testing procedure,and show that it can keep the sterling characteristics and also improve the false-negative predictive values for a specimen,not larger than that from individual testing.These characteristics are studied from both theoretical and numerical points of view.The nested group testing procedure is better than individual testing on both false-positive and false-negative predictive values,while retains the efficiency as a basic characteristic of a group testing procedure.Applications to Dorfman’s,Halving and Sterrett procedures are discussed.Results from extensive simulation studies and an application to malaria infection in microscopy-negative Malawian women exemplify the findings.
基金supported by the National Scientific Foundation of China Nos.11431010 and 11571329“the Fundamental Research Funds for the Central Universities.”。
文摘We give a combinatorial characterization of upward planar graphs in terms of upward planar orders,which are special linear extensions of edge posets.
基金supported by SFB/Transregio 45 Periods,Moduli Spaces and Arithmetic of Algebraic Varieties of DFG,by NSF of China Grant Nos.11771203,11231003,11301495Fundamental Research Funds for the Central Universities,Nanjing University,No.0203-14380009by the Science Foundation of Shanghai(No.13DZ2260400).
文摘We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound.As an application we show that the Coleman–Oort conjecture holds for Shimura curves associated with partial corestriction upon a suitable choice of parameters,which generalizes a construction due to Mumford.
基金The authors would like to thank Joan Birman and Haynes Miller for their encouragements and helpful suggestions on this project.Fuquan Fang and Fengchun Lei supported in part by a Key Grant(No.11431009)an Overseas-Collaboration Grant(No.11329101)of NSFC of ChinaResearch is supported by the Singapore Ministry of Education research Grant(AcRF Tier 1 WBS No.R-146-000-190-112)and a Grant(No.11329101)of NSFC of China.
文摘A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator subgroup of the normal closures in the link group of themeridians excluding themeridian of the last component with the differential induced by removing the last component.Moreover,the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis.These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.
