In this article we show that any embedded graph has a checkerboard colourable twual,which is equivalent to having a bipartite twual.We also obtain that any Eulerian embedded graph has a checkerboard colourable partial...In this article we show that any embedded graph has a checkerboard colourable twual,which is equivalent to having a bipartite twual.We also obtain that any Eulerian embedded graph has a checkerboard colourable partial Petrial,answering questions posed by Ellis-Monaghan and Moffatt[Trans.Amer.Math.Soc.,364,1529–1569(2012)].展开更多
In our earlier paper,we generalize the one-parameter(c)family of inhomogeneous firstorder differential operator representations of the orthogonal Lie algebras arising from conformal transformations to those of orthosy...In our earlier paper,we generalize the one-parameter(c)family of inhomogeneous firstorder differential operator representations of the orthogonal Lie algebras arising from conformal transformations to those of orthosymplectic Lie super algebras,and determine the irreducible condition.This paper deals with the cases when the irreducible condition fails.We prove that if n-m-1>0 and c is an integer satisfying 1≤c≤n-m-1,the representation of osp(2n+2|2m)has a composition series of length 2,and when n-m-1≥0 and c∈-N,the representation of osp(2n+2|2m)has a composition series of length 3,where N is the set of nonnegative integers.Moreover,we show that if c∈(max{n-m,0}-1/2-N)∪(-N),the representation of osp(2n+3|2m)has a composition series of length 2.In particular,we obtain an explicit presentation of the irreducible module with highest weight lλ2-λ1,where l is any positive integer and it is not a generalized Verma module.展开更多
Let X be a C^(1+)vector field on a compact Riemannian manifold M with dimension d≥3.LetΛbe a transitive singular hypebolic set with positive volume.We show thatΛ=M andΛis a uniformly hyperbolic set without singula...Let X be a C^(1+)vector field on a compact Riemannian manifold M with dimension d≥3.LetΛbe a transitive singular hypebolic set with positive volume.We show thatΛ=M andΛis a uniformly hyperbolic set without singularities.展开更多
In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extensi...In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.展开更多
In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusio...In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.展开更多
For double Walsh–Fourier series and with f∈L2([0,1)×[0,1))we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f(x,y).Assumptions are N(x,y)non-decrea...For double Walsh–Fourier series and with f∈L2([0,1)×[0,1))we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f(x,y).Assumptions are N(x,y)non-decreasing as a function of x and of y and,roughly speaking,partial derivatives with approximately constant ratio■≌2n0 for all x and y,where n0 is any fixed non-negative integer.Estimates,independent of N(x,y)and n0,are then extended to Lr,1<r<2.We give an application to the family N(x,y)=λxy on[0,1)×[0,1),anyλ>10.展开更多
Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient func...Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient function δ(x). In this way we obtain an estimated function with parameter β.Secondly, the empirical likelihood method is developed to estimate the parameter β. An estimated empirical log-likelohood ratio is proved to be asymptotically standard chi-squred, and the maximum empirical likelihood estimation(MELE) for β is shown to be asymptotically normal. Finally, based on the MELE of β, the empirical likelihood approach is again applied to reestimate the nonparametric part δ(x). The empirical log-likelohood ratio for δ(x) is proved to be also asymptotically standard chi-squred. Simulation study shows that the proposed method works better than the normal approximation method in terms of average areas of confidence regions for β, and the empirical likelihood confidence belt for δ(x) performs well.展开更多
Let R and S be associative rings and _SV_R a semidualizing(S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom_R(I_V(R),-) and Hom_R(-, I_V(R)) exact exact complex ···...Let R and S be associative rings and _SV_R a semidualizing(S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom_R(I_V(R),-) and Hom_R(-, I_V(R)) exact exact complex ···→ I_1 d_0→I_0→I^0 d_0→I^1→··· of V-injective modules I_i and I^i, i ∈ N_0, such that N≌Im(I_0→I^0). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A_V(R) which leads to the fact that V-Gorenstein injective modules admit exact right I_V(R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly VGorenstein injective if and only if N⊕E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext_(IV(R))^(≥n+1)(I, N) = 0 for all modules I with finite I_V(R)-injective dimension.展开更多
The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space.Motivated by the numerical results of Hairer and Voss,we prove that the Burgers ...The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space.Motivated by the numerical results of Hairer and Voss,we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.展开更多
Let G =(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the ...Let G =(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V(G) and for U■V(G), let g(U) =∑s∈Ug(s). A real-valued function g : V(G) → [0, 1] is a resolving function of G if g(RG{x, y}) ≥ 1 for any two distinct vertices x, y ∈ V(G). The fractional metric dimension dimf(G)of a graph G is min{g(V(G)) : g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ : V(G1) → V(G2) be a bijection. Then, a permutation graph Gσ =(V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = σ(u)}. First,we determine dimf(T) for any tree T. We show that 1 < dimf(Gσ) ≤1/2(|V(G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε > 0, there exists a permutation graph Gσ such that dimf(Gσ)- 1 < ε. We give examples showing that neither is there a function h1 such that dimf(G) < h1(dimf(Gσ)) for all pairs(G, σ), nor is there a function h2 such that h2(dimf(G)) > dimf(Gσ) for all pairs(G, σ). Furthermore,we investigate dimf(Gσ) when G is a complete k-partite graph or a cycle.展开更多
This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondenc...This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.展开更多
We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth.One fundamental in this topic is to extend the classical compactness results of theΣ-convergence method to ...We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth.One fundamental in this topic is to extend the classical compactness results of theΣ-convergence method to the Orlicz spaces.We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.展开更多
Let G be an arbitrary spanning subgraph of the complete graph Kr+1 on r+1 vertices and Kr+1-E(G) be the graph obtained from Kr+1 by deleting all edges of G.A non-increasing sequence π=(d1,d2,...,dn) of nonnegative in...Let G be an arbitrary spanning subgraph of the complete graph Kr+1 on r+1 vertices and Kr+1-E(G) be the graph obtained from Kr+1 by deleting all edges of G.A non-increasing sequence π=(d1,d2,...,dn) of nonnegative integers is said to be potentially Kr+1-E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr+1-E(G) as a subgraph.In this paper,a characterization of π that is potentially Kr+1-E(G)-graphic is given,which is analogous to the Erdo s–Gallai characterization of graphic sequences using a system of inequalities.This is a solution to an open problem due to Lai and Hu.As a corollary,a characterization of π that is potentially Ks,tgraphic can also be obtained,where Ks,t is the complete bipartite graph with partite sets of size s and t.This is a solution to an open problem due to Li and Yin.展开更多
Let U be a quantized enveloping algebra and its modified form.Lusztig gives some symmetries on U and.In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra,one can apply the BGP...Let U be a quantized enveloping algebra and its modified form.Lusztig gives some symmetries on U and.In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra,one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on U and their important properties,for instance,the braid relations.In this paper,we define a modified form ■ of the Ringel-Hall algebra and realize the Lusztig's symmetries on by applying the BGP-reflection functors to ■.展开更多
In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module l2 A, and give a one to one correspondence between the set of positive self-adjoint regular module operators on l2...In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module l2 A, and give a one to one correspondence between the set of positive self-adjoint regular module operators on l2A and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup {Tt }t∈R + L(l2A) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(l2A) is the von Neumann algebra of all adjointable module maps on l2A.展开更多
基金Supported by NSFC(Grant Nos.12171402,12101600)the Fundamental Research Funds for the Central Universities(Grant Nos.20720190062,2021QN1037)。
文摘In this article we show that any embedded graph has a checkerboard colourable twual,which is equivalent to having a bipartite twual.We also obtain that any Eulerian embedded graph has a checkerboard colourable partial Petrial,answering questions posed by Ellis-Monaghan and Moffatt[Trans.Amer.Math.Soc.,364,1529–1569(2012)].
基金Supported by National Key R&D Program of China(Grant No.2020YFA0712600)。
文摘In our earlier paper,we generalize the one-parameter(c)family of inhomogeneous firstorder differential operator representations of the orthogonal Lie algebras arising from conformal transformations to those of orthosymplectic Lie super algebras,and determine the irreducible condition.This paper deals with the cases when the irreducible condition fails.We prove that if n-m-1>0 and c is an integer satisfying 1≤c≤n-m-1,the representation of osp(2n+2|2m)has a composition series of length 2,and when n-m-1≥0 and c∈-N,the representation of osp(2n+2|2m)has a composition series of length 3,where N is the set of nonnegative integers.Moreover,we show that if c∈(max{n-m,0}-1/2-N)∪(-N),the representation of osp(2n+3|2m)has a composition series of length 2.In particular,we obtain an explicit presentation of the irreducible module with highest weight lλ2-λ1,where l is any positive integer and it is not a generalized Verma module.
基金support of Shanghai Key Laboratory of Pure Mathematics and Mathematical Practicethe Project funded by China Postdoctoral Science Foundation(Grant No.2020TQ0098)。
文摘Let X be a C^(1+)vector field on a compact Riemannian manifold M with dimension d≥3.LetΛbe a transitive singular hypebolic set with positive volume.We show thatΛ=M andΛis a uniformly hyperbolic set without singularities.
基金This work is supported by National Key R&D Program of China(Grant No.2018YFA0703900)National Natural Science Foundation of China(Grant Nos.11471079,11631004,11871163 and 11901302)。
文摘In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.
基金Supported in part by NSFC(Grant No.11771047)Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai(Grant No.2019RS1057)。
文摘In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.
基金Supported by MIUR Excellence Department Project awarded to the Department of Mathematics,University of Rome Tor Vergata(Grant No.CUP E83C18000100006)。
文摘For double Walsh–Fourier series and with f∈L2([0,1)×[0,1))we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f(x,y).Assumptions are N(x,y)non-decreasing as a function of x and of y and,roughly speaking,partial derivatives with approximately constant ratio■≌2n0 for all x and y,where n0 is any fixed non-negative integer.Estimates,independent of N(x,y)and n0,are then extended to Lr,1<r<2.We give an application to the family N(x,y)=λxy on[0,1)×[0,1),anyλ>10.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11571148 and 11731015)the Science and Technology Planning Project of Guangdong(Grant No.2017A030303085)the Natural Science Foundation of Guangdong(Grant No.2016A030307019)
文摘Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient function δ(x). In this way we obtain an estimated function with parameter β.Secondly, the empirical likelihood method is developed to estimate the parameter β. An estimated empirical log-likelohood ratio is proved to be asymptotically standard chi-squred, and the maximum empirical likelihood estimation(MELE) for β is shown to be asymptotically normal. Finally, based on the MELE of β, the empirical likelihood approach is again applied to reestimate the nonparametric part δ(x). The empirical log-likelohood ratio for δ(x) is proved to be also asymptotically standard chi-squred. Simulation study shows that the proposed method works better than the normal approximation method in terms of average areas of confidence regions for β, and the empirical likelihood confidence belt for δ(x) performs well.
基金supported in part by Guangdong Natural Science Funds for Distinguished Young Scholar(Grant No.2016A030306040)NSF of Guangdong(Grant No.2014A030313417)+2 种基金NNSF of China(Grant Nos.11471338 and 11622113)the third author is supported by the NNSF of China(Grant Nos.11371378 and 11521101)Guangdong Special Support Program
文摘Let R and S be associative rings and _SV_R a semidualizing(S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom_R(I_V(R),-) and Hom_R(-, I_V(R)) exact exact complex ···→ I_1 d_0→I_0→I^0 d_0→I^1→··· of V-injective modules I_i and I^i, i ∈ N_0, such that N≌Im(I_0→I^0). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A_V(R) which leads to the fact that V-Gorenstein injective modules admit exact right I_V(R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly VGorenstein injective if and only if N⊕E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext_(IV(R))^(≥n+1)(I, N) = 0 for all modules I with finite I_V(R)-injective dimension.
基金Supported by NSF of China(Grant Nos.11101427,11371367)Fundamental program of NUDT
文摘The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space.Motivated by the numerical results of Hairer and Voss,we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.
文摘Let G =(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V(G) and for U■V(G), let g(U) =∑s∈Ug(s). A real-valued function g : V(G) → [0, 1] is a resolving function of G if g(RG{x, y}) ≥ 1 for any two distinct vertices x, y ∈ V(G). The fractional metric dimension dimf(G)of a graph G is min{g(V(G)) : g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ : V(G1) → V(G2) be a bijection. Then, a permutation graph Gσ =(V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = σ(u)}. First,we determine dimf(T) for any tree T. We show that 1 < dimf(Gσ) ≤1/2(|V(G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε > 0, there exists a permutation graph Gσ such that dimf(Gσ)- 1 < ε. We give examples showing that neither is there a function h1 such that dimf(G) < h1(dimf(Gσ)) for all pairs(G, σ), nor is there a function h2 such that h2(dimf(G)) > dimf(Gσ) for all pairs(G, σ). Furthermore,we investigate dimf(Gσ) when G is a complete k-partite graph or a cycle.
基金Supported by China Scholarship Council(Grant No.201206125047)China Postdoctoral Science Foundation Funded Project(Grant No.2012M520715)the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.201462)
文摘This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.
文摘We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth.One fundamental in this topic is to extend the classical compactness results of theΣ-convergence method to the Orlicz spaces.We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.
基金Supported by National Natural Science Foundation of China(Grant No.11161016)
文摘Let G be an arbitrary spanning subgraph of the complete graph Kr+1 on r+1 vertices and Kr+1-E(G) be the graph obtained from Kr+1 by deleting all edges of G.A non-increasing sequence π=(d1,d2,...,dn) of nonnegative integers is said to be potentially Kr+1-E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr+1-E(G) as a subgraph.In this paper,a characterization of π that is potentially Kr+1-E(G)-graphic is given,which is analogous to the Erdo s–Gallai characterization of graphic sequences using a system of inequalities.This is a solution to an open problem due to Lai and Hu.As a corollary,a characterization of π that is potentially Ks,tgraphic can also be obtained,where Ks,t is the complete bipartite graph with partite sets of size s and t.This is a solution to an open problem due to Li and Yin.
基金Supported by National Natural Science Foundation of China(Grant No.11131001)the Fundamental Research Funds for the Central Universities(Grant No.BLX2013014)
文摘Let U be a quantized enveloping algebra and its modified form.Lusztig gives some symmetries on U and.In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra,one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on U and their important properties,for instance,the braid relations.In this paper,we define a modified form ■ of the Ringel-Hall algebra and realize the Lusztig's symmetries on by applying the BGP-reflection functors to ■.
基金supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(Grant No.10XNJ033,"Study of Dirichlet forms and quantum Markov semigroups based on Hilbert C-modules")
文摘In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module l2 A, and give a one to one correspondence between the set of positive self-adjoint regular module operators on l2A and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup {Tt }t∈R + L(l2A) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(l2A) is the von Neumann algebra of all adjointable module maps on l2A.
