The boundedness of singular integral operator along an open curve

It is well known that, the singular integral operatorS defined as: ifL is a closed smooth contour in the complex plane C, thenS is a bounded linear operator fromH ^μ(L) intoH ^μ(L): ifL is an open smooth curve, thenS is just a linear operator fromH ^* intoH ^*. In this paper, we define a Banach space , and prove that is a bounded linear operator, then verify the boundedness of other kinds of singular integral operators. Supported by the National Science Foundation of China Wang Xiaolin: born in Aug. 1950, Professor     

On the boundary value problem for harmonic maps of the Poincaré disc

The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmüller space, defined as a set of normalized quasisymmetric homeomorphisms h of the unit circle S onto itself where h admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfying where ρ(z)|dz|2 is the Poincare metric of D. Let B . be a Banach space consisting of holomorphic quadratic differentials φ in D with norms It is shown that for any given quasisymmetric homeomorphism h : S1→S1∈ T . , there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding is h and the Hopf differential of such a harmonic map belongs to B .     

Generalized Galerkin approximations for pseudoinverses and operator equations of the first kind

0 IntroductionLeqtuiliHnebaer afo rHimlbaenrtd sap baoceun.dLeedtlianaenardfufnbceti oan baolu onndeHd ,serse--spectively. Our goal is to seekv∈Hsuchthata(u,v) =f(u) , u∈ H (1)As a rule, we call the expression (1) a variational equationandva solutiontothis equation.By virtue of the Riesz repre-sentationtheorem,there exists a uniquely determined boundedlinear operatorA:H→Hsuchthata(u,v) =〈u,Av〉, u,v∈ H,where〈·,·〉is the inner product onH.If the sesquilinearformais also with posit…     

The Hàjek-Rènyi inequlity for Banach space valued random variable sequences and its application

In this paper we prove the following Hàjek-Rènyi inequality: Let 0≤p≤1, then for any Banach spaceB, anyL ^p integrableB valued random variable sequence {D _n , n≥1}, any real number sequence {b _n , n≥1} with 0<b _n, ↑ ∞, any integern≥1, there exist a constantC=C _p>0 (only depending onp) such that

Population-size-dependent branching processes in Markovian random environments

A branching model {Z n} n≥0is considered where the offspring distribution of the population's evolution is not only dependent on the population size, but also controlled by a Markovian environmental process {ξ n} n≥0. For this model, asymptotic behaviour is studied such as limn→∞Z n and limn→∞Z n/m n in the case that the mean m k, θof the offspring distribution converges to m>1 as the population size k grows to ∞. In the case that {ξ n} n≥0is an irreducible positive recurrent Markov chain, certain extinction (i.e. P(Z n=0 for some n)=1) and noncertain extinction (i.e. P(Z n=0 for some n)<1) are studied.     

B-Valued Dyadic Derivative

The principles of the new maximal operator H＊ we defined are discussed. We prove that it is bounded from martingale Hardy-Lorentz L^Xp.q[0,1） to the Lorentz L^Xp.q[0,1） for 1/2〈 p〈∞, 0〈~ q ≤ ∞, where X is any Banach space. When the Banach space X has the RN property, the sequence dnHnf converges to f a.e. Meanwhile the convergence in L^Xp norm for 1≤p〈∞ is a consequence of that the family functions K （n∈N） is an approximate identity.     

A local asymptotic behavior for ruin probability in the renewal risk model

Let R（t）=u＋ct-∑ I=1^N（t） Xi,t≥0 be the renewal risk model, with Fx（x）being the distribution function of the claim amount X. Let ψ（u） be the ruin probability with initial surplus u. Under the condition of Fx（x） ∈ S^＊（γ）,y ≥ 0, by the geometric sum method, we derive the local asymptotic behavior for ψ（u,u ＋ z] for every 0 （ z （ oo, On one hand, the asymptotic behavior of ψ（u） can be derived from the result obtained. On the other hand, the result of this paper can be applied to the insurance risk management of an insurance company.     

Almost sure convergence of weighted sums of NA sequences

For double arrays of constants {a _ni, 1≤i≤k _n, n≥1} and NA r.v. 's {X _n, n≥1}, conditions for almost sure convergence of are given. Both casesk _n ↑ ∞ andk _n=∞ are treated. A Marcinkiewicz-type theorem for i. d. NA sequences is obtained as a special case. Supported by the National Natural Science Foundation of China Cheng Riyan: born in 1968, MS student     

Characterization of type<Emphasis Type="Italic">p</Emphasis> Banach spaces by the weak law of large numbers

For weighted sums of the form where {a _nj , 1 ⩽j⩽k _n ↑∞,n⩾1} is a real constant array and {X _aj , 1≤j≤k _{n, n}≥1} is a rowwise independent, zero mean, random element array in a real separable Banach space of typep, we establishL ^r convergence theorem and a general weak law of large numbers respectively, conversely, we characterize Banach spaces of typep in terms of convergence inr-th mean and probability for such weighted sums. Foundation item: Supported by the National Natural Science Foundation of China (No. 10071058) Biography: Gan Shi-xin (1939-), male, Professor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.     

Some limit theorems for weighted sums of random variable fields

1Preliminaries DuringthelastyearsthereappearedquiteafewpapersaboutlimittheoremsforBanachspacevaluedrandomvariables(r.v.’s)withmultidimensionalindex.Itistheaim ofthispapertogivecomparativeweakconditionswhichimply limittheoremsforweightedsumsofBanachspacevaluedr.v.fields.Thecurrentworkextendsthecorrespondingresultsof Refs.[15].Westartwithsomenotationsanddefinitions.LetNdbe thesetofalld dimensionallatticeofpointswithpositiveinte gersascomponents,whered≥1isafixedinteger.LetNdbe equippedwithco…     

Almost sure convergence of weighted sums of NA sequences

For double arrays of constants {a _ni, 1≤i≤k _n, n≥1} and NA r.v. 's {X _n, n≥1}, conditions for almost sure convergence of are given. Both casesk _n ↑ ∞ andk _n=∞ are treated. A Marcinkiewicz-type theorem for i. d. NA sequences is obtained as a special case. Supported by the National Natural Science Foundation of China Cheng Riyan: born in 1968, MS student     

Some polar sets for the generalized Brownian sheet

Let (t)(t ∈ R ₊ ^N ) be the d-dimensional N-parameter generalized Brownian sheet. We study the polar sets for (t). It is proved that for any a ∈ R ^d , P{ (t) = a, for some t ∈ R _> ^N } = and the probability that (t) has k-multiple points is 1 or 0 according as whether 2kN > d(k − 1)β or 2kN < d(k−1)α. These results contain and extend the results of the Brownian sheet, where R _> ^N = (0,+∞) ^N ,R ₊ ^N =[10,+∞) ^N ,0< α≤1 and β≥1. Biography: LI Huiqiong (1966–), female, Associate professor, research direction: stochastic process and random fractal.     

The generalized inverse of singular integral operators on an open arc and its applications

After choosing weight functions suitably, we define a Banach spaceH _ω ^μ (L) and discuss the generalized inverse of singular integral operators on an open arc. Using the generalized inverse, we obtain the solutions for the following singular integral equation

A matrix version of the Wielandt inequality and its application to statistics

Suppose that A is an n × n positive definite Hemitain matrix. Let X and Y be n × p and n × q matrices (p+ q≤n), such that X* Y = 0. The following inequality is proved X*AY( Y* AY)-Y Y*AX≤( (λ1-λn)/(λ1+λn))2 X*AX, where λ1 and λn are respectively the largest and smallest eigenvalues of A, and M- stands for a generalized inverse of M. This inequality is an extension of the well-known Wielandt inequality in which both X and Y are vectors. The inequality is utilized to obtain some interesting inequalities about covari-ance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlation.     

Composition operators with linear fractional symbols on vector-valued Bergman spaces

0　IntroductionLetDdenotetheopenunitdiskinthecomplexplaneC , DitsboundaryanddAtheLebesguemeasureonD ,normal izedsothatA(D) =1.ForaBanachspace (X ,‖·‖X) ,wewriteH(D ,X)fortheclassofallX valuedanalyticfunctionsonD .Let1≤ p<∞ ,theX valuedBergmanspaceBp(X)istheclassofallf∈H(D ,X) forwhich‖f‖Bp(X) =∫D‖f(z)‖pXdA(z) 1 /p <∞ (1)IfX =C,thenwewriteBp =Bp(C) fortheclassicalBergmanspaces.Letφ∶D→Dbeananalyticself mapofD .Thenthee quation Cφf=f φdefinesacompositionoperator…     

An extrapolation theorem on martingale spaces with Ap weights

In this paper, we prove an extrapolation theorem of operators in martingale spaces with A _p weights, which shows that for operators T defined on martingale space L _u ¹ , if T is bounded on martingale space L ^p0(w) for some 1 < p ₀ < ∞ and every, w∈ A _∞, so it is on L ^p,s (w) for every 1 < p, s < ∞, and w ∈ A _∞. We also get some properties of A _p weights and prove that if w ∈ A ₁, then the maximal operator M is bounded on L ^p,q (w) with 1 < p < ∞, 1 < q ≤ ∞.     

Multi-parameter infinite-dimensional (r,δ)-OU process

The definition of n-parameter infinite-dimensional (r,δ) -Ornstein-Uhlenbeck process {xt (·) } ((r, δ)-OUPn∞ for short) is given. The absolute continuity of distribution μt of Xt(·) and the limit of Xt(·) when |t| →∞ is discussed.     

Kinetics of C2(a3Πu) radical reactions with NO, N2O, O2, H2 and NH3

Pulsed laser photolysis/laser-induced fluorescence (LP-LIF) is utilized to measure rate constants for C₂(a³Π_u) reactions with NO, N₂O, O₂, H₂ and NH₃. Multiphoton dissociation of C₂Cl₄ at 266 nm is employed for the generation of C₂(a³Π_u) radicals. The C₂(a³Π_u) concentration is monitored by the fluorescence of the (0, 0) band of the (d³Π_g&#8596;a³Π_u) transition at 516.5 nm. C₂(a³Π_u) removal rate constants for the reactions are determined as k_NO = (5.46 ± 0.10) × 10^-11 cm³ molecule^-1 s^-1 , k_N2O = (1.63 ± 0.20) × 10^-13 cm³ molecule^-1 s^-1 , k_N2O = (1.58 ± 0.16) × 10^-11 cm³ molecule^-1 s^-1, k_O2 = (5.92 ± 1.00) × 10^-14 cm³ molecule^-1 s^-1, k_H2< 1.0× 10^-14 cm³ molecule^-1 s^-1. Based on the data analysis and theoretical calculation, we suggest that the C₂(a³Π_u) reactions with H₂ and NH₃ proceed via the hydrogen abstraction mechanism, barriers exist at the entrance channel of the reactions of C₂(a³Π_u) with H₂ and NH₃.     

The evolution model of the earth’s limited expanding

Based on the prerequisite that the earliest acid rock formed because of the initial global expanding tectonic processes, by combining it with other geological limitation conditions, and starting from the lithosphere elastic mechanics, the following is defined: the earth expansion started at about 4300 Ma ago, the average increasing rate of the earth radius is 0.40 mm/a, the initial expanding earth radius is 4651 km, and the initial global expansion of tectonic processes happened at 4183.7 Ma ago. From the natural and attenuation evolution condition of the earth and terrestrial planets, through the limit condition got from the combination of the mathematics logic and the geological significance, the limited expanding earth evolution model is established as follows: in whichR ₀= 4.651×10⁶ m,A =1.86×10⁶ m, β = -6.01×10⁻¹⁰/ a andt _s=3×10⁸ a. From this model the maximum radiusR _maxof the limited expanding earth evolution should be 6511 km. The earth’s radius increasing rate is about 0.1 mm/a at present.     

A note on Jacobi-Eisenstein series

The so_called Jacobi_Eisenstein series is defined by E k, S (z, w)=∑y∈J ∞\J (1, j) Z 1|k, Sy(z, w) . The Fourier coefficients of E k, S is determined completely. The theorem extends the results of Eichler and Zagier to the case of general index matrices.