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161.
We prove a uniform bound for the density, p
t
(x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p
t
(x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al.
(5) 相似文献
162.
We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators. 相似文献
163.
The radial wave functions of inner electron shell and outer electron shell of a Ne atom were obtained by the approximate analytical
method and tested by calculating the ground state energy of the Ne atom. The equivalent volume of electron cloud and the refractive
index of Ne were calculated. The calculated refractive index agrees well with the experimental result. Relationship between
the refractive index and the wave function of Ne was discovered.
Supported by the New Star Program of Beijing Science and Technology, China (Grant No. 952870400), the Beijing Municipal Commission
of Education, and the Excellent Young Teachers Program of Ministry of Education, China 相似文献
164.
165.
Twenty high energy nuclear interactions produced in the graphite units of an emulsion chamber were recorded. The emulsion
chamber was exposed to cosmic rays at an atmospheric depth of 10 g cm−2 for about 7 hr over Hyderabad, India. Fourteen interactions which radiated energyΣ E
r⩾1000 GeV in the form ofγ-rays were analysed in detail. The median energy 〈Σ E
r〉 of the interactions was 1600 GeV. Results concerning the multiplicity, the transverse and longitudinal momentum distributions,
and the fractional energy distribution ofγ-rays in these interactions are presented. The average transverse momentum ofπ
0—mesons <pt
π
0> is found to increase very slowly with the primary energyE
0 and it can be approximated by the function <pt
π
0>=0·238E
0
0.06
. 相似文献
166.
Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous
and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal
subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate
framework. A new calculus calledF
α-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledF
α-integral andF
α-derivative respectively. TheF
α-integral is suitable for integrating functions with fractal support of dimension α, while theF
α-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function
occur naturally as solutions ofF
α-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems.
We discuss construction and solutions of some fractal differential equations of the form
whereh is a vector field andD
F,t
α
is a fractal differential operator of order α in timet. We also consider some equations of the form
whereL is an ordinary differential operator in the real variablex, and(t,x) ∈F × Rn whereF is a Cantor-like set of dimension α.
Further, we discuss a method of finding solutions toF
α-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed
back to get those of the former. This is illustrated with a couple of examples. 相似文献
167.
A high-performance liquid chromatographic method with photodiode array detection has been developed for the determination of squalene. After treated by extraction and fractional crystallization, squalene was analyzed on a C18 column (150 × 3.9 mm, 5 m) with acetonitrile as mobile phase. Excellent linearity of the calibration curve was observed in the range of 100–40000 gL–1 and the detection limit was 40 gL–1. The recoveries were from 89.6% to 100.5% and the relative standard deviations were from 0.5% to 1.4%. The method was successfully applied to the determination of squalene in squalene capsules, olive oil, algal lipids and algal cells. 相似文献
168.
In this paper we show that the Cahn–Hilliard stochastic PDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. We study the regularity of the trajectories of the solution and the absolute continuity of its law at some given time and position. This is done by showing a priori estimates which heavily depend on the specific equation, and by proving general results on stochastic and deterministic integrals involving general operators on smooth domains of d which are parabolic in the sense of Petrovskii, and do not necessarily define a semi-group of operators. These last estimates might be used in a more general framework. 相似文献
169.
In this paper we study the rotational form of the pressure-correction method that was proposed by Timmermans, Minev, and Van De Vosse. We show that the rotational form of the algorithm provides better accuracy in terms of the -norm of the velocity and of the -norm of the pressure than the standard form.
170.
For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators
for some self-adjoint operator
and some *-representation of the underlying *-algebra. 相似文献