On the Best Degree of Approximation by Euler (E,q)-Means

Let f(x)∈L_(2π) and its Fourier series by f(x)～α_0/2+sum from n=1 to ∞(α_ncosnx+b_nsinx)≡sum from n=0 to ∞(A_n(x)). Denote by S_n (f,x) its partial sums and by E_n~q(f,x) its Euler (E, q)-means, i. e. E_n~q(f,x)=1/(1+q)~π sum from m=0 to n((?)q~(n-m)S_m(f,x)), with q≥0 (E_n~0≡S_n). In [1] Holland and Sahney proved the following theorem. THEOREM A Ifω(f,t) is the modulus of continuity of f∈C_(2π), then the degree of approximation of f by the (E,q)-means of f is givens by##特殊公式未编改     

WEIGHTED BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL HARDY OPERATORS WITH BESOV-LIPSCHITZ FUNCTIONS

In this paper,we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions.The main result is that this kind of commutator,denoted by Hb α,is bounded from L x γp (R +) to Lx δ q (R +) with the bound explicitly worked out.     

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b∈Λβ0(Rn) is discussed from Lp(Rn) to Lq(Rn), 1/q=1/p-β0/n, and from Lp(Rn) to Triebel-Lizorkin space Fβ0,∞p. We also obtain the boundedness of generalized Toeplitz operatorθbα0 from LP(Rn) to Lq(Rn), 1/q =1/p-α0 β0/n. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on LP(Rn), 1 < p <∞.     

Characterizations of Weighted BMO Space and Its Application

In this paper, we prove that the weighted BMO space■is independent of the scale p ∈(0, ∞) in sense of norm when ω∈ A_1. Moreover, we can replace L~p(ω) by L~(p,∞)(ω). As an application, we characterize this space by the boundedness of the bilinear commutators [b, T ]_j(j = 1, 2), generated by the bilinear convolution type Calderón–Zygmund operators and the symbol b, from L~(p1)(ω) × L~(p2)(ω) to L~p(ω~(1-p)) with 1 p_1, p_2 ∞ and 1/p = 1/p_1 + 1/p_2.Thus we answer the open problem proposed by Chaffee affirmatively.     

On the Classification of Finite R-Trivial P-Semigroups

In this paper we discuss the classification of finite R-triVial P-semigroups (congruence permutable semigroup).It is clear that the monogenic semigroup C_(n,1)and c_(n,1)~1 are R-triVial P-semigYoups.F01"the Proper ideal R={a~k,a~(k ),…a~k of C_(n,1)~1,the branehextension[4]C_(n,1)~1×I/R is a P-sereigroup,denoted by C_(x,1,k)~1 It is a R-triVial semigroulp. Let N be the set of nattlral nulnbers,l≠s∈N,e_l,e_2,a are transfotmations of     

INCOMPLETE SEMI-ITERATIVE METHODS FOR SOLVING SINGULAR LINEAR OPERATOR EQUATIONS IN BANACH SPACE WITH APPLICATIONS IN MARKOV CHAIN MODELING

1　 IntroductionLet Cbe the open complex plane,let X be a complex Banach space.The set of allbounded linear operators from X into X is denoted by B[X] which is also a Banach space.If X=Cn,the n-dimensional Euclidean space,then B[X] is the set of all n×n matrices,denoted by Cn,n. We denote the spectrum of an operator T∈ B[X] byσ( T) and its resol-vent operator R( λ,T) =( λI-T) - 1 ,where I is the identity operator andλ∈C.The spectral radius of T is denoted by r( T) .N( T) and…     

On Sums of Exponents of Factoring Integers

Let n=p_1~(β_1)p_2~(β_2)…p_t~(β_t)be the prime factorization of n.Define h(n)=min(β_1,β_2,…,β_t)and H(n)=max(β_1,β_2,…,β_t).For convenience take h(1)=1 and H(1)=1.P.Erdos suggested that it is likely that sum from i=1 to n h(i)=n c n~(1/2) o(n)~(1/2),where c is a posi-tive constant.This conjecture was proved by Ivan Niven[1].In[1],it is proved that     

Gradient Estimates of Solutions to the Conductivity Problem with Flatter Insulators

We study the insulated conductivity problem with inclusions embedded in a bounded domain in R~n. When the distance of inclusions, denoted by ε, goes to 0, the gradient of solutions may blow up. When two inclusions are strictly convex, it was known that an upper bound of the blow-up rate is of order ε~(-1/2) for n = 2, and is of order ε~(-1/2+β)for some β 0 when dimension n ≥ 3. In this paper, we generalize the above results for insulators with flatter boundaries near touching points.     

Double Biproduct Hom-Bialgebra and Related Quasitriangular Structures

Let(H, β) be a Hom-bialgebra such that β~2= id_H.(A, α_A) is a Hom-bialgebra in the left-left Hom-Yetter-Drinfeld category (_H~H)YD and(B, α_B) is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category YD_H~H. The authors define the two-sided smash product Hom-algebra(A■H■B, α_A ? β ? α_B) and the two-sided smash coproduct Homcoalgebra(A◇H◇B, α_A ? β ? α_B). Then the necessary and sufficient conditions for(A■H■B, α_A ? β ? α_B) and(A◇H◇B, α_A ? β ? α_B) to be a Hom-bialgebra(called the double biproduct Hom-bialgebra and denoted by(A_◇~■H_◇~■B, α_A ? β ? α_B)) are derived. On the other hand, the necessary and sufficient conditions for the smash coproduct Hom-Hopf algebra(A◇H, α_A ? β) to be quasitriangular are given.     

LEAST SQUARES TYPE ESTIMATION FOR DISCRETELY OBSERVED NON-ERGODIC GAUSSIAN ORNSTEIN-UHLENBECK PROCESSES

In this article, we consider the drift parameter estimation problem for the nonergodic Ornstein-Uhlenbeck process defined as d X_t= θX_tdt + dG_t, t ≥ 0 with an unknown parameter θ 0, where G is a Gaussian process. We assume that the process {X_t, t ≥ 0} is observed at discrete time instants t_1 = ?_n, ···, t_n= n?_n, and we construct two least squares type estimators ■ and ■ for θ on the basis of the discrete observations {X_(t_i), i = 1, ···, n}as n →∞. Then, we provide sufficient conditions, based on properties of G, which ensure that ■ and ■ are strongly consistent and the sequences n?n~(1/2)(■-θ) and n?n~(1/2)(■-θ)are tight. Our approach offers an elementary proof of [11], which studied the case when G is a fractional Brownian motion with Hurst parameter H ∈(1/2, 1). As such, our results extend the recent findings by [11] to the case of general Hurst parameter H ∈(0, 1). We also apply our approach to study subfractional Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes.     

Resonance between automorphic forms and exponential functions

Let f be a holomorphic cusp form of weight k for SL2(Z) and λf(n) its n-th Fourier coefficient.In this paper,the exponential sum Xn 2X λf(n)e(αnβ) twisted by Fourier coefficients λf(n) is proved toh ave a main term of size |λf(q)|X3/4 when β = 1/2 and α is close to ±2√q,q ∈ Z,and is smaller otherwise for β 3/4.This is a manifestation of the resonance spectrum of automorphic forms for SL2(Z).     

AN ANZAHL THEOREM IN SINGULAR PSEUDO-SYMPLECTIC GEOMETRY OVER FINITE FIELDS

1 IntroductionLet F, be a fiuite field witl1 q elelllents. wl1ere q is a power of 2. We will follow tlie notationsand terntillologies in [11. LetThe singular pseudthsymplectic group with respect to Se,I over Fq denoted by PS=.+.+..=.+, (F,)is defiued to be tlie group fOrnled by all (2P + 6 + l) x (2v + b + l) nonsingUlar n1atrices T suchthat TSb.,T' == Sb,i. wl1ere TT denotes the transpose of T. If T 6 Ps=.+,+,.=.+, (Fq), then T isof the forlllwhere T1,S,T1,' = Se and T22 is nousin…     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

FREDHOLM OPERATORS ON THE SPACE OF BOUNDED SEQUENCES

Necessary and sufficient conditions are studied that a bounded operator T_x =(x_1~*x, x_2~*x,···) on the space ?_∞, where x_n~*∈ ?_∞~*, is lower or upper semi-Fredholm; in particular, topological properties of the set {x_1~*, x_2~*,···} are investigated. Various estimates of the defect d(T) = codim R(T), where R(T) is the range of T, are given. The case of x_n~*= d_nx_(tn)~*,where dn ∈ R and x_(tn)~*≥ 0 are extreme points of the unit ball B_?_∞~*, that is, t_n ∈βN, is considered. In terms of the sequence {t_n}, the conditions of the closedness of the range R(T)are given and the value d(T) is calculated. For example, the condition {n:0 |d_n| δ} = Φ for some δ is sufficient and if for large n points tn are isolated elements of the sequence {t_n},then it is also necessary for the closedness of R(T)(t_(n0) is isolated if there is a neighborhood U of t_(n0) satisfying t_n ■ U for all n ≠ n0). If {n:|d_n| δ} =Φ, then d(T) is equal to the defect δ{_tn} of {t_n}. It is shown that if d(T) = ∞ and R(T) is closed, then there exists a sequence {A_n} of pairwise disjoint subsets of N satisfying χ_(A_n)■R(T).     

BOUNDEDNESS OF THE HIGHER-DIMENSIONAL QUASILINEAR CHEMOTAXIS SYSTEM WITH GENERALIZED LOGISTIC SOURCE

This article considers the following higher-dimensional quasilinear parabolicparabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions■in a bounded domain??R~n(n≥2) with smooth boundary??, where the diffusion coefficient D(u) and the chemotactic sensitivity function S(u) are supposed to satisfy D(u)≥M1 (u+1)~(-α) and S(u)≤M2(u+1)~β, respectively, where M_1, M_2 0 andα,β∈R. Moreover, the logistic source f (u) is supposed to satisfy f (u)≤a-μu~γ with μ 0,γ≥1, and a≥0. As α+2βγ-1+2γ/n, we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.     

Jordanian Canonical Form Theory in an Infinite Dimensional Operator Algebra

Let B(H) be the algebra of bounded linear operators acting on a separable Hilbert space H. An operator T in B(H) is said to be strongly irreducible (see [1]), denoted by T (SI), if it is not similar to any reducible operator (see [2]). If the dimension of ?i is finite, B(H) may be regarded as an n × n matrix algebra. In this case, T (SI) if and only if T is similar to a Jordanian block. Jordanian canonical form theorem is one of the most important theorems in matrix theory. The Jordan…     

Characterization for commutators of n-dimensional fractional Hardy operators

In this paper, it was proved that the commutator Hβ,b generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from Lp1(Rn) to Lp2 (Rn) if and only if b is a C(M)O(Rn) function, where 1/p1 - 1/p2 = β/n, 1 ＜ p1 ＜∞, 0 ≤β＜ n. Furthemore,the characterization of Hβ,b on the homogenous Herz space (K)qα,p(Rn) was obtained.     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF INTEGRAL BOUNDARY VALUE PROBLEM

In this paper, the existence and multiplicity of positive solutions for a class of non-resonant fourth-order integral boundary value problem ■ with two parameters are established by using the Guo-Krasnoselskii's fixedpoint theorem, where f ∈ C([0, 1] × [0, +∞)×(-∞, 0], [0, +∞)), q(t) ∈ L1[0, 1]is nonnegative, α,β ∈ R and satisfy β 2π2, α 0, α/π4+ β/π2 1, λ1,2=(-β ?■)/2. The corresponding examples are raised to demonstrate the results we obtained.     

LIE-TROTTER FORMULA FOR THE HADAMARD PRODUCT

Suppose that A and B are two positive-definite matrices, then, the limit of(A~(p/2)B~pA~(p/2))~(1/p) as p tends to 0 can be obtained by the well known Lie-Trotter formula. In this article, we generalize the usual product of matrices to the Hadamard product denoted as * which is commutative, and obtain the explicit formula of the limit(A~p * B~p)~(1/p)as p tends to 0. Furthermore, the existence of the limit of(A~p * B~p)~(1/p)as p tends to +∞ is proved.