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…     

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…     

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     

Degradation of 2, 4-D acid using Mn2+ as catalyst under UV irradiation

In this paper, the oxidative degradation of 2, 4-dichlorophenoxyacetic acid (2, 4-D) using Mn²⁺/H₂O₂ reagent under UV irradiation was studied. The results show that 2, 4-D was degraded more completely in Mn²⁺/H₂O₂ solution than traditional Fenton solutions. The effects of the concentration of Mn²⁺, H₂O₂ and pH were also investigated. And under the optimal condition of 1.48×10⁻⁴ mol/L, 8.99×10⁻⁵ mol/L and pH 3.38, the formation of ·OH was the most, both the decomposition rate of H₂O₂ and the degradation rate of 2, 4-D were the fastest. In addition, the photoreaction process was monitored using spin-trapping electron paramagnetic resonance (EPR), and the results indicated that the oxidative process was predominated mainly by the hydroxyl radical (·OH) gennerated in the system. Biography: HUANG Yingping (1964–), Professor, Ph. D., research direction: pollution ecology and water pollution control.     

Synthesis of tungstophosphates of heteropoly compound and its catalytic property in oxidative desulfurization of diesel

Several kinds of Tungstophosphates of Heteropoly Compounds (TPHC) with different compositions and the same structure were synthesized using the method of solid-phase reaction to change the mol ratio of Na₄P₂O₇ · 10H₂O, WO₃ and Na₂WO₄ · 2H₂O. TPHC, as a catalyst, was used in oxidative desulfurization of diesel. The results show that TPHC has catalytic activity. With THPC existing, the highest desulfurization rate reaches 51.85%, 32.6% higher than that yield without TPHC. And the desulfurization rate increases with decreasing WO₃. FT-IR spectra of TPHC shows that the catalytic activity is related to the activity of end oxygen(O_d).The more active O_d is, the stronger the capability of obtaining or losing electron is. It means that TPHC has a good catalytic activity. FT-IR spectra of TPHC indicates that desulfurization rate is in linear correlation with ( is intensity of W-O_d vibration band, is intensity of P-O_a vibration band) and correlation coefficient reaches 0.965. Biography: BIN Xiaobei (1965–), female, Ph.D. candidate, Associate professor, research direction: environment science and environmental materials.     

A topological approach to the correlation of standard formation enthalpy with path index of benzenes derivatives

A mathematical model formulated as ΔfHθm(g)=a+sum from i=1 to 5bipi(i≠2) was constructed for the relationshipbetween standard formation enthalpy ΔfHθm(g) and path index Pi of substituted benzenes derivatives. An empiric equation for the calculation of ΔfHθm(g) was worked out. The calculated values of standard formation enthalpy based on this model are excellently consistent with those from experimental for 55 organic compounds. The model is shown to be simple and of practical usefulness, particularly when required experimental data are unavailable.     

Strong stability of linear forms in φ -mixing random variables

In this paper some new results of strong stability of linear forms in φ-mixing random variables are given. It is mainly proved that for a sequence of φ-mixing random variables {x _n , n ⩾ 1} and two sequences of positive numbers {a _n , n ⩾ 1} and {b _n , n ⩾ 1} there exist d _n ∈ R, n = 1,2,L, such that a.s. under some suitable conditions. The results extend and improve the corresponding theorems for independent identically distributed random variables. Foundation item: Supported by the National Natural Science Foundation of China(10671149)     

The symmetric solutions of affiliated value model

0　IntroductionIntheprocessofR&Ditem ,thereisusuallycompetition .ManyfirmswanttoinvestR&Dtoobtainapatent.Lury[1 ] gaveanequilibriuminvestmentundertheconditionofone timeinvestmentandcontinuousinvestmentwhenthepayoffwasafixedprize.Ref.[2 ]analyzedtheeffectofinformationsharinginR&D .TherearemanyliteraturesaboutanalysisofR&DlevelswhenfirmscompeteinoutputmarketafterR&Dsucceeds[3 ,4] .Additionally ,todecideinvestmentlevel,therearemanyliteraturesaboutR&Ditem’spricing .MondherBellalah[5] ,K .…     

A sufficient condition fork′(z)(k(z)∈H q,q≥1) to be ofH 1

We proved ifk(z)∈H ^q (q≥1),g(z) is analytic on |z|=1, , thenk′(z)∈H ¹, especially, ifq=1, thenk(z) is an analytic function on the closed unit disk |z|≤1. Peng Zhigang: born in June 1970, Ph. D     

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     

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     

Integrability near the boundary of the Poisson integral of some singular measures

We consider the Poisson integral u = P＊μ on the half-space R＋^N＋1 （ N 〉 1 ） （or on the unit ball of the complex plane） of some singular measureμ. If μ is an s-measure （0 〈 s 〈 N）, then some sharp estimates of the integration of the harmonic function u near the boundary are given. In particular, we show that fpr p〉1,∫R^Nu^p（x,y）dx- y^-τ （y〉0,τ =（N-s）（p-1） ） （Given for p〉1, RN f 〉 0 and g 〉 0, ＂ f-g ＂ will mean that there exist constants C1 and C2, such that C1f ≤ g ≤ CEf ）.     

Synthesis of biodiesel from rapeseed oil using K2O/ γ -Al2O3 as nano-solid-base catalyst

Nano-solid-base catalyst K₂O/γ-Al₂O₃ was prepared and adopted for the synthesis of biodiesel by transesterification of rapeseed oil with methanol. The particle diameter of the catalyst was about 50 nm, which was measured by transmission electron microscopy (TEM). The variables affecting the yield of biodiesel during transesterification, such as mass ratio of KNO₃ to γ-Al₂O₃, calcination temperature, calcination time, and catalyst content, molar ratio of methanol to oil, reaction temperature and reaction time were investigated. The catalyst obtained by calcining a mixture of KNO₃ and γ-Al₂O₃ at 600 °C for 3 h, was found to be the optimum one, which gave the highest catalytic activity in reaction. With 3% (m _catalyst/m _oil) catalyst, when the transesterification carried out at a molar ratio of methanol to oil 12:1, a reaction temperature of 70 °C, and a reaction time of 3 h, yield of 94% was achieved. Foundation item: Supported by the National High-Technology Research and Development Program of China (863 Program) (2007AA100703)     

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.     

A class of complex quasi-Orlicz spaces and their analytic RN property

LetX be a complex quasi-Banach space and Φ:[0,∞)→[0,∞) an increasing convex function with Φ(0)=0, and Φ∈Δ₂. ThenL _Φ ^* (X) is a quasi-Banach space with continuous quasi-norm andL _Φ ^* (X) has the ARNP if and only ifX does. Supported by the National Natural Science Foundation of China Liu Peide: born in 1943, Professor     

Almost sure convergence and complete convergence for the weighted sums of martingale differences

Let {(D _n, FFF_n),n/->1} be a sequence of martingale differences and {a _ni, 1≤i≤n,n≥1} be an array of real constants. Almost sure convergence for the row sums are discussed. We also discuss complete convergence for the moving average processes underB-valued martingale differences assumption. Foundation item: Supported by the National Natural Science Foundation of China and the Doctoral Programme Foundation of China Biography: DENG Ai-jiao (1974-), female, Ph.D. candidate. Research interest is in stochastic processes and random fractal.     

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 .     

Application of thermogravimetry to study the kinetics and stability of porphyrin copper-sodium of ephedra

Thermogravimetry was used to study the non-isothermal decomposition kinetics and the stability of porphyrin copper-sodium from ephedra. The kinetic equation for the second step can be expressed as . The kinetic compensation effect is found to be: InA=0. 2468E _a −8. 0513.Ea is 206. 61 kJ/mol. The results show that porphyrin copper-sodium has a high activation energy and good stability. Biography: Zhang Lian-ru(1964), female, lecturer Research direction, applied biochemistry.     

Morita equivalent blocks in subgroups of finite groups

0 IntroductionLeRte(f K.[,1]O), ,ki) .e b.e, aOips- am ocdoumlaplre tseys dtiesmcre(tese eva lCuhatpito .nII Iri nignwith unique maxi mal idealJ(O) suchthatJ(O) =(π)forπ∈O,Kis the quotient field ofOcharacteristic 0 ,andkis theresidue fieldO/(π)of characteristicp, wherepis a fixedpri me number .Inthis paper , we fix a finite groupG,andletKcontains all the |G|-throots of unity,where |G|is the or-der of finite groupG.In particular bothKandkare splittingfields of every subgroup ofG.D…     

Asymptotic normality of multi-dimension quasi maximum likelihood estimate in generalized linear models with adaptive design

0Introduction Considerthemodel:y=h(x′β)+e wherey=y(1)y(q),h(t)=h(1)(t)h(q)(t),e=e(1)e(q),h(i),i=1,2,…,qisaknownboundedinjection:Rq→R1,h′isapositive matrixwith h(i)(t)tjasits(i,j)elementj=1,2,…,q,t=(t1,t2,…,tq),thepartialderivativesof2thorderofhexistandcontinuous.eistherandomerror.βisthep×1unknowparameterandxiisthefixedknownp×qmatrix,i=1,…,n,η=x′βiscalledpredictor,andthismodeliscalledgeneralizedlinearmodel.Supposetheindependentobservationsy1,…,ynaregetfromq dimensionrespons…