If p is an integrable function and positive almost everyshere on E,f and g are measurable function and positive almost everywhere on E,and a≤f≤A,b≤g≤B,then∫ Epf α d μ∫ Epg α d μ∫ Ep(fg) α2 d μ 2≤14ABab α4 +abAB α4 2.Also,the condition of equality holding is established.

In the paper,we prove an almost sure convergence for the maximum of stationary Gaussion vector sequencs under the conditons rn(p)log n(log log n)1+ε=O(1),rn(p,q)log n(log log n)1+ε=O(1),1≤p≠q≤d.

Let{ξ_i}_i~∞=1 be a stationary sequence and {u_n} the given sequence of real numbers. Under the con- ditions of D′(u_n)and D_2({u_k,u_n}),an almost sure central limit theorem for the maximum of weakly de- pendent sequences is obtained.

Let N_n=max {k≤n:X_(n,1)+X_(n,2)+…+X_(n,k)≤s_n}. The article discusses the almost sure convergency of N_n and the asymptotic normality of N_n,The results improved Bruss’s results.

In this paper, it is given a proof method about the almost sure central limit theorem in the separable metric space. Applying this method the almost sure central limit theorm for uniform empirical processes in(D[0,1],d) is obtained.

It is considered the minimum problem of the functionalJ(u)=integral from G f(|▽u|~2)dxfor the vector valued function u∈W_p~1(G,E~N),p≥2,N>1.The everywhereregularity in G for the gradient of solutions is proved.

本文考虑向量值函数的泛函J(u)=intergral from G f(|▽u|~2)dx u∈W_p~1(G,E_N),p≥2,N>1的极小问题,证明解梯度在 G 内的处处正则性.

Let N n+p be an n+p- dimensional locally symmetric manifolds, if12< δ≤K N≤1,M n be an n-dimensional compact minimal submanifolds of N n+p ,and sectional curvature of M n no less than K,S be the squre of the length of second fundamental form.

令N是n +p维局部对称空间 ,12 <δ≤KN≤ 1,M为n维紧致极小子流形 ,其截面曲率处处不小于 K ,S为第二基本形式的模长平方 .

There exists a continuous function whose Fourier sum, when taken in decreasing order of magnitude of the coefficients, diverges unboundedly almost everywhere.

Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of Lp functions converge to the functions almost everywhere and in Lp for 1>amp;lt;p>amp;lt;∞.

In L1 we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L1 in a certain Cesàro sense.

(1995), a nonlinear wavelet estimation of f(·) without restrictions of continuity everywhere on f(·) is given, and the convergence rate of the estimators in L2 is obtained.

It Tf (xo ) exists for a single point xo then Tf(x) exists everywhere for x?Rn and TF?Lipα(Rn).

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies.

The complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are also discussed for ?--mixing random fields.

[5], who indicated the necessity of introducing two zones of liquid-pres sure variation corresponding to the propagation of perturbation in porous blocks and cracks, respectively [6].

Refinement of the almost sure central limit theorem for associated processes

A stochastic estimate for the asymptotic distribution of normalized maxima of waiting times and an estimate for the upper limit almost sure are obtained.

In this paper an infinitely wide plate under pure plastic bending is discussed. The distribution ot stress and the relation between the couple acting on the plate and the corresponding deformation are found under the assumption that the relation between the intensity of shearing stress and shearing strain has the exponential form.The method suggested by the author is also compared with that proposed by R. Hill under the assumption that the plate is ideally plastic. Obviously the two have markeddifference. The...

In this paper an infinitely wide plate under pure plastic bending is discussed. The distribution ot stress and the relation between the couple acting on the plate and the corresponding deformation are found under the assumption that the relation between the intensity of shearing stress and shearing strain has the exponential form.The method suggested by the author is also compared with that proposed by R. Hill under the assumption that the plate is ideally plastic. Obviously the two have markeddifference. The latter gives rise to the discontinuity of stress in the neighbourhood ofthe "neutral layer", while the former always gives a continuous variation of stress.Also, the elastic recovery of plastic strain at the removal of the load is discussed. The result is compared with B. V. Ryabinin's experiment and is found to be in close agreement.It is believed that the present problem has its application in the cold working ofmetals.

The following fundamental theorem of the integral calculus is proved.Theorem Suppose that f(x)is a real function defined in the closed interval[a, b], and(ⅰ)its right upper derivate D~+f(x)>—∝, and right lower derivate D_+f(x)<∝, for every x except at most an enumerable set Γ in [a, b),(ⅱ)f(x) is left semicontinuous for every x∈(a, b],(ⅲ) for every x∈Γ(ⅳ)there exists a measurable function ψ(x)such that D~+f(x)≥ψ(x)≥D_+f(x) almost everywhere in [a, b) and at least one of the two functions max{ψ(x), 0}and min{ψ(x),...

The following fundamental theorem of the integral calculus is proved.Theorem Suppose that f(x)is a real function defined in the closed interval[a, b], and(ⅰ)its right upper derivate D~+f(x)>—∝, and right lower derivate D_+f(x)<∝, for every x except at most an enumerable set Γ in [a, b),(ⅱ)f(x) is left semicontinuous for every x∈(a, b],(ⅲ) for every x∈Γ(ⅳ)there exists a measurable function ψ(x)such that D~+f(x)≥ψ(x)≥D_+f(x) almost everywhere in [a, b) and at least one of the two functions max{ψ(x), 0}and min{ψ(x), 0}is integrable over [a, b],then ψ(x) is integrable, andWhere the integral is in the Perron as well as the Lebesgue sense.It may be mentioned that the preceding theorem is closely related to a recent result due to I. S. Gal ([1] Theorems 2 and 3). However there is some gap in his proof, since he has implicitly assumed that which is by no means evident. This gap is filled in the present note.

On the basis of new facts and laws revealed by modern astronomy and physics, it seems necessary and expedient to introduce the concept of "cos- moscopic" process, to stand side by side with macroscopic and microscopic processes. Cosmoscopic objects differ from macroscopic objects (things seen everday on the Earth as well as meteoric bodies, small asteroids and satellites) in mass and scale just as much as the difference between macroscopic objects and microscopic objects. Gigantic difference in quantity leads...

On the basis of new facts and laws revealed by modern astronomy and physics, it seems necessary and expedient to introduce the concept of "cos- moscopic" process, to stand side by side with macroscopic and microscopic processes. Cosmoscopic objects differ from macroscopic objects (things seen everday on the Earth as well as meteoric bodies, small asteroids and satellites) in mass and scale just as much as the difference between macroscopic objects and microscopic objects. Gigantic difference in quantity leads to marked difference in quality. The mechanical motion of celestial bodies, the dynamics of stellar systems, the condensation of self-gravitating gas mass, natural ther- monuclear reactions in stellar interior, the production of forbidden lines in nebulae and outer envelopes of stars, the strong coupling between hydrody- namic phenomena and electromagnetic phenomena, the existence of superdense matter, curvature of space in strong gravitational field, the evolution of celes- tial bodies, all these are examples of cosmoscopic phenomena and processes, and also form the basis on which the cosmoscopic concept is introduced, Stellar dynamics, cosmical electrodynamics, and general theory of relativity are examples of cosmoscopic laws. In cosmoscopic processes, gravitational interaction usually plays a dominant role, and plasma state is the state of matter most often met. The cosmoscopic concept will aid tn understanding more deeply material processes in the inorganic world. It will prevent us from applying without modification to cosmoscopic processes natural laws which strictly speaking applies only to macroscopic processes. Once the cosmoscopic law is understood, man can then create artificially cosmoscopic conditions on the Earth so that processes which only take place naturally in cosmoscopic processes, can then take place on the Earth. Thermonuclear reactions, forbidden lines (now applied so much in "Excited emission") are two examples; artificial cosmic rays, and artificial superdense matter might be realized later. In carrying out simulation experiments, the effect introduced by difference in scale and mass must be kept in mind. Differentiation among cosmocscopic, macroscopic, and microscopic processes shows that dialectical laws operate everywhere in Nature.