广义Hardy算子的加权Φ—不等式

In this paper,the conditions on pairs of weights(u,v)are given such that for the generalized Hardy operator Tf(x)=∫0^∞K（x,y)f(y)dy the following Φ-inequality holds:Φ2^-1(∫0^∞Φ2（Tf(x))V(x)dx≤CΦ1^-1(∫0^∞Φ1(f(x))U(x)dx)，where Φ1,Φ2 are Young function;the corresponding weak type Φ-inequality for T is characterized.     

LARGE DEVIATIONS FOR SYMMETRIC DIFFUSION PROCESSES

Let a(x)=(a_(ij)(x)) be a uniformly continuous, symmetric and matrix-valued function satisfying uniformly elliptic condition, p(t, x, y) be the transition density function of the diffusion process associated with the Diriehlet space (, H_0~1 (R~d)), where(u, v)=1/2 integral from n=R~d sum from i=j to d(u(x)/x_i v(x)/x_ja_(ij)(x)dx).Then by using the sharpened Arouson's estimates established by D. W. Stroock, it is shown that2t ln p(t, x, y)=-d~2(x, y).Moreover, it is proved that P_y~6 has large deviation property with rate functionI(ω)=1/2 integral from n=0 to 1<(t), α~(-1)(ω(t)),(t)>dtas s→0 and y→x, where P_y~6 denotes the diffusion measure family associated with the Dirichlet form (ε, H_0~1(R~d)).     

THE CLASSICAL SOLUTIONS TO A CAUCHY PROBLEM OF PARABOLIC TYPE COUPLED WITH OPERATORS

We consider the equation ut=Tf[B(x,t,Du,Φu)D^2u]+F(x,t,u,Du,Φu,Ψu) where Φand Ψ are vector-valued mappings.We obtain the existence anduniqueness of classical solution to the equation for a ε-periodic initial data.The problem is naturally arisen from image denoising.     

Oscillation Theorems for Nonlinear Second Order Elliptic Equations

Some oscillation theorems are given for the nonlinear second order elliptic equationsum from i,j=1 to N D_i[a_(ij)(x)Ψ(y)||▽y||~(p-2)D_(jy)] c(x)f(y)=0.The results are extensions of modified Riccati techniques and include recent results of Usami.     

孤立波数值方法综述(Ⅱ)

III Finite Element Methods We take element, I=sum from j=1 to N (I_j). Let Φ_h and Ψ_h be the trail function space and test function space respectively with basis (x) and (x). We suppose U~h(x,t)=sum from j=1 to N (U_i(t)(?)_j(x)), x∈I,t≥0. The usual Galerkin method is to find U~h∈L~∞(0,T;Φ_h) satisfying (21) In order to improve the stability and convergence, the dissipative finite element     

A NOTE ON AN INTEGRATED CAUCHY FUNCTIONAL EQUATION

In characterizing the semistable law, Shimizu reduced the problem to solving the equationH(x)=integral from n=1 to∞(H(x y)d(μ-ν)(y), x≥0) where μ andτ are given positive measures on [0,∞). In thisnote, we obtain a simple proof and show that some of his conditions can be weakened.     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

Strong skew commutativity preserving maps on rings with involution

Let R be a unital *-ring with the unit I.Assume that R contains a symmetric idempotent P which satisfies ARP = 0 implies A = 0 and AR(I-P) = 0 implies A = 0.In this paper,it is shown that a surjective map Φ:R→R is strong skew commutativity preserving(that is,satisfiesΦ(A)Φ(B)-Φ(B)Φ(A)~w= AB-BA~w for all A,B∈R) if and only if there exist a map f:R→Z_s(R)and an element Z∈Z_s(R) with Z~2=I such that Φ(A)=ZA +f(A) for all A∈R,where Z_s(R) is the symmetric center of R.As applications,the strong skew commutativity preserving maps on unital prime *-rings and von Neumann algebras with no central summands of type I_1 are characterized.     

Equivalent Characterization of Centralizers on B(H)

Let H be a Hilbert space with dim H≥2 and Z∈B(H) be an arbitrary but fixed operator.In this paper we show that an additive map Φ:B(H)→B(H) satisfies Φ(AB)=Φ(A)B=AΦ(B)for any A,B∈B(H) with AB=Z if and only if Φ(AB)=Φ(A)B=AΦ(B),A,B ∈B(H),that is,Φ is a centralizer.Similar results are obtained for Hilbert space nest algebras.In addition,we show that Φ(A~2)=AΦ(A)=Φ(A)A for any A∈B(H) with A~2=0 if and only if Φ(A)=AΦ(I)=Φ(I)A,A∈B(H),and generalize main results in Linear Algebra and its Application,450,243–249(2014) to infinite dimensional case.New equivalent characterization of centralizers on B(H) is obtained.     

SOME GENERALIZATION OF GRONWALL-BIHARI INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

The interest of this paper lies in the estimates of solutions of the three kinds of Gronwail-Bihari integral inequalities:(Ⅰ) y(x)≤f(x) sum from i=1 to n(g_i(x)integral from n=0 to x(h_i(d)y(s)ds)),(Ⅱ) y(x)≤f(x) g(x)φ(integral from n=0 to x(h(s)w(y(s))ds))(Ⅲ) y(x)≤f(x) sum from i=1 to n(g_i(x)integral from n=0 to a(h_i(s)y(s)ds g_(n 1)φ(integral from n=0 to x(h_(n 1)(s)w(y(t))ds)).The results include some modifications and generalizations of the results of D. Willett, U. D. Dhongade and Zhang Binggen. Furthermore, applying the conclusion on the above inequalities to a Volterra integral equation and a differential equation, the authors obtain some new better results.     

DOUBLE φ-INEQUALITIES FOR BANACH-SPACE-VALUED MARTINGALES

Let B be a Banach space, Φ1 , Φ2 be two generalized convex Φ-functions and Ψ 1 , Ψ 2 the Young complementary functions of Φ1 , Φ2 respectively with ∫t t 0 ψ2 (s) s ds ≤ c 0 ψ1 (c 0 t) (t > t 0 ) for some constants c 0 > 0 and t 0 > 0, where ψ1 and ψ2 are the left-continuous derivative functions of Ψ 1 and Ψ 2 , respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f = (f n ) n ≥ 0 , ‖f*‖Φ1 ≤ c‖S (p) (f ) ‖Φ2 (or ‖S (q) (f )‖Φ1 ≤ c‖f*‖Φ2 , respectively), where f and S (p) (f ) are the maximal function and the p-variation function of f respec- tively; (ii) If B is a UMD space, T v f is the martingale transform of f with respect to v = (v n ) n ≥ 0 (v*≤ 1), then ‖(T v f )*‖Φ1 ≤ c ‖f *‖Φ2 .     

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: c(x, y) = inf x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation:V_t(t, x) + sup u∈UV_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t) = 0,V(0, x) = Φ0(x).     

Liouville type theorems for Schrdinger systems

We study positive solutions to the following higher order Schr¨odinger system with Dirichlet boundary conditions on a half space:(-△)α2 u(x)=uβ1(x)vγ1(x),in Rn+,(-)α2 v(x)=uβ2(x)vγ2(x),in Rn+,u=uxn==α2-1uxnα2-1=0,onRn+,v=vxn==α2-1vxnα2-1=0,onRn+,(0.1)whereαis any even number between 0 and n.This PDE system is closely related to the integral system u(x)=Rn+G(x,y)uβ1(y)vγ1(y)dy,v(x)=Rn+G(x,y)uβ2(y)vγ2(y)dy,(0.2)where G is the corresponding Green’s function on the half space.More precisely,we show that every solution to(0.2)satisfies(0.1),and we believe that the converse is also true.We establish a Liouville type theorem—the non-existence of positive solutions to(0.2)under a very weak condition that u and v are only locally integrable.Some new ideas are involved in the proof,which can be applied to a system of more equations.     

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x~2+α_2xy+α_5x~2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx~j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.     

The rate of the normal approximation for jackknifingU-statistics

Let U_n be a U-statistic with symmetric kernel h(x,y) such that Eh(X_1,X_2)=θ and Var E[h(X_1,X_2)-θ|X_j]>0.Let f(x) be a function defined on R and f″ be bounded.If f(θ) is the parameterof interest,a natural estimator is f(U_n).It is known that the distribution function of z_n=(n~(1/2){Jf(U_n)-f(θ)})/(S_n~*) converges to the standard normal distribution Φ(x) as n→∞,where Jf(U_n) isthe jackknife estimator of f(U_n),and S_n~(*2) is the jackknife estimator of the asymptotic variance ofn~(1/2) Jf(U_n).It is of theoretical value to study the rate of the normal approximation of the statistic.In this paper,assuming the existence of fourth moment of h(X_1,X_2),we show that(?)|P{z_n≤x}-Φ(x)|=O(n~(-1/2)log n).This improves the earlier results of Cheng(1981).     

Primitive Roots and Linearized Polynomials

Let f(x) = sum from t=0 to n α_ix~i∈GF(p)[x],we associate it with a ploynomial f~*(x)=sum from i=0 to n α_ix~(p~i),f(x) and f~*(x)are called p-associates of each other. f~*(x) is called a p-ploynomial,customary to speak of linearized polynomial. Let f(x)=x~m- 1/g(x), m = m_1~r, q = p~m, g(x)∈GF(p)[x],r be the order of g(x). Cohen and the author observed that if m_1≥2, there alwaysexsists a primitive roots ζ∈GF(q) suck that f~*(ζ) = f~*(c), here f~*(c)≠0. In fact     

Dimension of divergence sets for dispersive equation

Consider the generalized dispersive equation defined by{iδtu+Ф(√-△)u=0,(x,t)∈R^n×R,u(x,0)=f(x),F∈F(R^n),(*)whereФ(√-△)is a pseudo-differential operator with symbolФ(|ζ|).In the present paper,assuming thatФsatisfies suitable growth conditions and the initial data in H^s(R^n),we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations(*)fails.These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.     

BMO ESTIMATES FOR MULTILINEAR FRACTIONAL INTEGRALS

In this paper,the authors prove that the multilinear fractional integral operator T A 1,A 2 ,α and the relevant maximal operator M A 1,A 2 ,α with rough kernel are both bounded from L p (1 p ∞) to L q and from L p to L n/(n α),∞ with power weight,respectively,where T A 1,A 2 ,α (f)(x)=R n R m 1 (A 1 ;x,y)R m 2 (A 2 ;x,y) | x y | n α +m 1 +m 2 2 (x y) f (y)dy and M A 1,A 2 ,α (f)(x)=sup r0 1 r n α +m 1 +m 2 2 | x y | r 2 ∏ i=1 R m i (A i ;x,y)(x y) f (y) | dy,and 0 α n, ∈ L s (S n 1) (s ≥ 1) is a homogeneous function of degree zero in R n,A i is a function defined on R n and R m i (A i ;x,y) denotes the m i t h remainder of Taylor series of A i at x about y.More precisely,R m i (A i ;x,y)=A i (x) ∑ | γ | m i 1 γ ! D γ A i (y)(x y) r,where D γ (A i) ∈ BMO(R n) for | γ |=m i 1(m i 1),i=1,2.     

A BOUNDARY VALUE PROBLEM FOR A NONLINEAR ORDINARY DIFFERENTIAL EQUATION INVOLVING A SMALL PARAMETER—The Riemann problem for a Generalized Diffusion Equatiois

This paper studies the boundary value problem involving a small parameter $$((k(V(t))+s)|V'(s)|^{N-1}V'(s))'+(sg(V(s))+f(V(s)))V'(s)=0 for s\in R$$, $$V(-\infty)=A,V(+\infty)=B;A0$$, $$U(x,0)=A for x<0,U(x,0)=B for x>0$$ under the hypotheses H1—H4 . The author's aim is not only to determine explicitly the discontinuous solution ,to the reduced problem；and the form and the number of its curves of discontinuity, but also to present, in an extremely natural way, the jump conditions which it must satisfy on each of its curves of diseontinuity. It is proved that the problem has a unique solution $U_{\varepsilon}(x,t)=V_{\varepsilon}(s),s=x/p(t),s\geq0,V_{\varepsilon}$pointwise converges to $V_{0}(s)$ as $s\downarrow0,V_{0}(s)$ has at least one jump point if and only if k(y) possesses at least one interval of degeneracy in [A-B], and there exists a one-to-one correspondence between the collection of all intervals of degeneracy of k(y) in [A-B] and the set of all jump points of $V_{0}(s)$     

SPECTRAL RADIUSES OF THE GALTON-WATSON BRANCHING PROCESSES

§1IntroductionSuppose that(X,Px)is a Markov chain on a countable(or finite)state space E.Givenany x,y∈E,we say that y can be reached from x and write x y,if there is n≥1suchthat Px(Xn=y)>0.If x y and y z,then x z.The markov chain X is said to beirreducible if any two states can be reached from each other.(See[1]or[2]).If X isirreducible,then there is a number r,with0≤r≤1,such that lim supn→∞[Px(Xn=y)]n1=r forany x,y∈E.The number r is called the spectral radius of X(refer to[3]).…