Weighted weak type (1,1) estimate for the Christ-Journé type commutator

Let K be the Calderón-Zygmund convolution kernel on R~d(d≥2).Christ and Journé defined the commutator associated with K and a∈L~∞(R~d)by T_af(x)=p.v.∫_(R~d)K(x-y)m_x,y~a·f(y)dy,which is an extension of the classical Calderón commutator. In this paper, we show that T_a is weighted weak type(1,1) bounded with A,1 weight for d≥2.     

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.     

STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE:EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

In this paper we study a fractional stochastic heat equation on R~d(d≥1) with additive noise ?/?t u(t,x) = Dα/δu(t,x) + b(u(t,x)) +W~H(t,x) where D α/δ is a nonlocal fractional differential operator and W~H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition,in the case of space dimension d=1,we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.     

DIRICHLET FORMS AND SYMMETRIC DIFFUSIONS ON A BOUNDED DOMAIN IN R~d

Let D be a bounded C~3-domain in R~d and(a_(ij))be a bounded symmetric matrixdefined on D.Consider the symmetric form(u,v)=1/2∫_D a_(ij)(x)(u(x))/(x_i) (v(x))/(x_j)dx,u,v∈H~1(D).Under some assumptions it is shown that the diffusion process associated with the regularDirichlet space(,(H~1(D))on L~2(D)can be characterized as a unique solution of acertain stochastic differential equation.     

Λ-模的σ-子模的标准表示式

Let A be a commutative ring with unit element, and let M be a Λ-module and σ∈Hom_Λ (M, M). Then a non-empty subset N of M is called a σ-submodule of the Λ-module M, if (1) a-b∈N for all a, bg∈N, and (2) λσ(α)∈N and x-σ(x)∈N for all λ∈Λ, α∈N, x∈M. Let N be a σ-submodule of M. N is said to be a primary σ-submodule of the Λ-module M, if (1) N≠M, and (2) whenever λ∈Λ, x∈M and λσ(x) ∈N, then either x∈N or λ^kσ(M)?N for some positive integer h. This paper is intended to show (1) that if M satisfies maximal condition of σ-submodule, and K is a σ-submodule of M, then K is a finite intersection of primary σ-submodules, and (2) that the uniqueness on the normal expression of σ-submodule of the Λ-module. Also, some results of fractional module have been obtained.     

Derivative formulas and Poincaré inequality for Kohn-Laplacian type semigroups

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator L :=1/2 sum from i=1 to m X_i~2 on R~(m+d):= R~m× R~d is investigated, where X_i(x, y) = sum (σki?xk) from k=1 to m+sum (((A_lx)_i?_(yl)) from t=1 to d,(x, y) ∈ R~(m+d), 1 ≤ i ≤ m for σ an invertible m × m-matrix and {A_l}_1 ≤ l ≤d some m × m-matrices such that the Hrmander condition holds.We first establish Bismut-type and Driver-type derivative formulas with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.     

HAUSDORFF DIMENSION OF THE DOUBLE POINT SET OF THE WESTWATER PROCESS

Let X={X_i}_(t∈[0,1])be the Westwater process which is the coordinate process under3-dimensional polymer measure v(g)constructed by J.Westwater.In this paper,theHausdorff dimension problem for the double point set of X is investigated.As a result,it is proved thatdim D_2=1,v(g)-a.e.,where D_2={x∈R~3:X=X_(?)=x for some s相似文献   

QUASI-SURE CONVERGENCE RATE OF EULER SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS

Let Xt(x) be the solution of stochastic dierential equations with smooth and bounded derivatives coeffcients. Let Xnt(x) be the Euler discretization scheme of SDEs with step 2-n. In this note, we prove that for any R 0 and γ∈(0, 1/2), supt∈[0,1],|x|≤R |Xnt(x, ω)- Xt(x, ω)|≤ξR,γ(ω)2-nγ, n≥1, q.e., where ξR,γ(ω) is quasi-everywhere finite.     

Sufficient And Necessary Condition For Convergence of Conditional Error Probability in NN-Pattern Discrimination

Let (θ_1,X_1),…, (θ_n,X_n), (θ, X) be iid random vectors ,where θ∈{0,1},X∈R~d Denote by θ′_n the nearest neighbour discriminator of θ based on the training samples (θ_1,X_1),…, (θ_n,X_n) and the observed X; put and This paper gives a sufficient and necessary condition for as n→∞, namely (P(θ=0, X=x)-P(θ=1, X=x))~2·P(θ=0, X=x)·P(θ=1, X=x)=0 for every x∈R~d.This generalizes a previous result of the authors [5] and improves a result of Wagner, T.J. [2].     

任意长圈覆盖指定的独立的点

An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k.
The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.     

$C(X)$上的开点与紧开拓扑

In this note we define a new topology on C(X),the set of all real-valued continuous functions on a Tychonoff space X.The new topology on C(X) is the topology having subbase open sets of both kinds:[f,C,ε[={g E C(X):|f(x)-g(x)| ε for every x∈C} and[U,r]~-={g∈C(X):g~(-1)(r)∩U≠φ},where f∈C(X),C∈KC(X)={nonempty compact subsets of X},ε 0,while U is an open subset of X and r∈R.The space C(X) equipped with the new topology T_(kh) which is stated above is denoted by C_(kh)(X).Denote X_0={x∈X:x is an isolated point of X} and X_c={x∈X:x has a compact neighborhood in X}.We show that if X is a Tychonoff space such that X_0=X_c,then the following statements are equivalent:(1) X_0 is G_δ-dense in X;(2) C_(kh)(X) is regular;(3) C_(kh)(X) is Tychonoff;(4) C_(kh)(X) is a topological group.We also show that if X is a Tychonoff space such that X_0=X_c and C_(kh)(X) is regular space with countable pseudocharacter,then X is σ-compact.If X is a metrizable hemicompact countable space,then C_(kh)(X) is first countable.     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

Morphic环的扩张

A ring R is called left morphic, if for any a ∈ R, there exists b ∈ R such that 1R(a) = Rb and 1R(b) = Ra. In this paper, we use the method which is different from that of Lee and Zhou to investigate when R[x, σ]/(xn) is (left) morphic and when the ideal extension E(R, V) is (left) morphic. It is mainly shown that: (1) If σis an automorphism of a division ring R, then S = R[x,σ]/(xn) (n ＞ 1) is a special ring. (2) If d, m are positive integers and n = dm, then E(/n, mZn) is a morphic ring if and only if gcd(d, m) = 1.     

On a Class of Infinite-Dimensional Hamiltonian Systems with Asymptotically Periodic Nonlinearities

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on R × RN:{■tu-xu + b ·▽xu + au + V(t,x)v = Hv(t,x,u,v),-■tv-xv-b·▽xv + av + V(t,x)u = Hu(t,x,u,v),where z =(u,v):R × RN → Rm × Rm,a > 0,b =(b1,···,bN) is a constant vector and V ∈ C(R × RN,R),H ∈ C1(R × RN × R2m,R).Under suitable conditions on V(t,x) and the nonlinearity for H(t,x,z),at least one non-stationary homoclinic solution with least energy is obtained.     

Almansi decomposition for Dunkl operators

Let Ωbe a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ωwhich are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x) |x|2f1(x) … |x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.     

Polar Sets and Relative Dimension for Generalized Brownian Sheet

Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.     

PHASE RETRIEVAL OF TIME-LIMITED SIGNALS

The problem of reconstructing a signal ψ(x) from its magnitude |ψ(x)| is of considerable interest to engineers and physicists.This article concerns the problem of determining a time-limited signal f with period 2π when |f(eix)| is known for x ∈ [-π,π].It is shown that the conditions |g(eix)| = |f(eix)| and |g(ei(x+b))-g(eix)| = |f(ei(x+b))-f(eix)|,b = 2π,together imply that either g = wf or g = vf,where both w and v have period b.Furthermore,if 2bπ is irrational then the functions w and v reduce to some constants c1 and c2,respectively;if 2bπ is rational then w takes the form w=eiαB1(eix)B2(eix) and v takes the form ei(x2πN/b+α)B1(eix)B2(eix),where B1 and B2 are Blaschke products.     

Existence of multiple solutions for a fractional p-Laplacian system with concave-convex term

In this article, we study the existence of multiple solutions for the following system driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions{(-?)_p~su = a(x)|u|~(q-2) u +2α/α + βc(x)|u|~(α-2) u|v|~β, in ?,(-?)_p~sv = b(x)|v|~(q-2) v +2β/α + βc(x)|u|α|v|~(β-2) v, in ?,u = v = 0, in Rn\?,(0.1) where Ω is a smooth bounded domain in Rn, n ps with s ∈(0,1) fixed, a(x), b(x), c(x) ≥ 0 and a(x),b(x),c(x) ∈L∞(Ω), 1 q p and α,β 1 satisfy pα + βp*,p* =np/n-ps.By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity of solutions to problem(0.1).?????     

Quasi-periodic Solutions of Nonlinear SchrSdinger Equations of Higher Dimension

We are concerned with the existence of the quasi-periodic solutions of the nonlinear Schrodinger(NLS) equation + (-△ + Mσ)u + ε|u|2u = 0, x ∈Td where △ is the d-Laplace and Mσ is a Fourier multiplier, i.e.,Mσe -1<,x> = σne -1<,x>, σn ∈ R. Regarding (1) as a Hamiltonian system and using the well-known infinite dimensional KAM theorem developed by them, Kuksin and Poschel[4] showed that there are invariant tori (thus quasi-periodic solutions) for Eq.(1) subject to Dirichlet boundary with d = 1.     

ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Suppose that X is a right process which is associated with a semi-Dirichlet form(ε,D(ε)) on L~2(E;m).Let J be the jumping measure of(ε,D(ε)) satisfying J(E×E-d) ∞.Let u ∈ D(ε)_b:= D(ε)∩ L~∞(E;m),we have the following Fukushima's decomposition u(X_t)-u(X_0) =M_t~u+N_t~u.Define P_t~uf(x)=E_x[e~(N_t~u)f(X_t)].Let Q~u(f,g) =ε(f,g)+ε(u,fg)for f,g∈ D(ε)_b.In the first part,under some assumptions we show that(Q~u,D(ε)_b) is lower semi-bounded if and only if there exists a constant α_0≥0 such that ‖P_t~u‖2≤e~(α_0~t) for every t0.If one of these assertions holds,then(P_t~u)t≥0 is strongly continuous on L~2(E;m).If X is equipped with a differential structure,then under some other assumptions,these conclusions remain valid without assuming J(E×E-d)∞.Some examples are also given in this part.Let A_t be a local continuous additive functional with zero quadratic variation.In the second part,we get the representation of A_t and give two sufficient conditions for P_t~A f(x) = E_x[e~(A_t) f(X_t)]to be strongly continuous.