scholarly journals INTEGRALS OVER PRODUCTS OF DISTRIBUTIONS FROM PERTURBATION EXPANSIONS OF PATH INTEGRALS IN CURVED SPACE

2002 ◽  
Vol 17 (15) ◽  
pp. 2019-2050 ◽  
Author(s):  
H. KLEINERT ◽  
A. CHERVYAKOV

We show that the requirement of coordinate invariance of perturbatively defined quantum-mechanical path integrals in curved space leads to an extension of the theory of distributions by specifying unique rules for integrating products of distributions. The rules are derived by using equations of motion and partial integration, while keeping track of certain minimal features stemming from the unique definition of all singular integrals in 1 - ∊ dimensions. Our rules guarantee complete agreement with much more cumbersome calculations in 1 - ∊ dimensions where the limit ∊ → 0 is taken at the end. In contrast to our previous papers where we solved the same problem for an infinite time interval or zero temperature, we consider here the more involved case of finite-time or temperature amplitudes.

1986 ◽  
Vol 33 (8) ◽  
pp. 2262-2266 ◽  
Author(s):  
J. Barcelos-Neto ◽  
Ashok Das

1987 ◽  
Vol 24 (1) ◽  
pp. 10-23 ◽  
Author(s):  
C. J. Bray ◽  
E. T. C. Spooner ◽  
C. M. Hall ◽  
D. York ◽  
T. M. Bills ◽  
...  

The McClean group of uranium deposits consists of elongate pods of high-grade uranium mineralization (width = ~ 15–40 m) tightly confined to within ±40 m of the basal unconformity. Uraninite–coffinite–sulphide/arsenide–chlorite–siderite mineralization at McClean is surrounded by a muscovite/illite ± haematite hydrothermal alteration halo,which can contain coffinite–pararammelsbergite (NiAs2) – muscovite/illite nodules. Ten laser probe 40Ar/39Ar dates, two of which are step-heat runs showing good plateaus, and 10 conventional K – Ar dates for this material show a distribution with asharp beginning at ~ 1320 Ma, a marked peak in the 1250–1200 Ma class interval, and a tail to dates as young as 1002 ± 33 (1σ) Ma. These determinations are in complete agreement with direct (U–Pb and Sm–Nd) dates on uraninite at the Midwest (e.g., 1328 ± 9 and 1110 ± 28 Ma), Key Lake (e.g., 1350 ± 4 and 1281 ± 6 Ma), and Collins Bay B deposits(e.g., 1281 ± 80 Ma). Since estimated depositional ages for the Athabasca sedimentary sequence are in the 1470 ± 15 to1428 ± 15 Ma range, uranium mineralization and associated hydrothermal alteration started ~ 100–150 Ma after Athabasca sedimentation, a result consistent with fluid-inclusion data, which indicate that mineralization took place at ~ 160–220 °C beneath ~ 3000 m of cover at a relatively advanced stage in the evolution of the basin. It is suggested that the similar initiation dates for uranium mineralization might reflect a widespread faulting event that affected the eastern part of the basin. A muscovite/illite closure temperature calculated from a measured argon diffusion activation energy of 36 ± 4 kcal/mol(1 kcal = 4.1868 kJ) indicates that the base of the Athabasca Basin in the McClean area has not been disturbed by temperatures greater than ~ 140 °C for 1.1–1.0 Ga. It is suggested that mineralization ceased when fracture permeability had been sealed by crystallization of secondary minerals. The duration of mineralization may have been ~ 150 Ma, a relatively long time interval not unreasonable for the base of a sedimentary basin.Secondary illites interstitial to quartz grains from the HLM1 stratigraphic borehole give 40Ar/39Ar ages of 1459 ± 4, 1341 ± 4, and 1113 ± 11 Ma, indicating that formation of diagenetic sheet silicates predated uranium mineralization. Recrystallization or formation of sheet silicates in relict sedimentary layers and in subunconformity altered basement referred to as "regolith" started at approximately the same time, since dates of 1484 ± 55 Ma (sedimentary layer), 1482 ± 49 Ma (regolith), and 1453 ± 49 Ma (regolith) have been obtained. Resetting of interstitial, sedimentary layer, and regolith sheet-silicate dates continued to ages of, for example, 1113 ± 11 Ma (interstitial) and 1038 ± 55 Ma (sedimentary layer), which exactly coincide with the youngest ages obtained for the alteration halo associated with mineralization.The youngest date obtained is a 40Ar/39Ar plateau age of 673 ± 3 Ma. The sample (2045-517) was obtained from within 2 mm of a concentrated pitchblende nodule and may have been disturbed in some way by its proximity to uranium.


1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.


1972 ◽  
Vol 45 ◽  
pp. 95-102
Author(s):  
E. I. Kazimirchak-Polonskaya

From the integration formulae of Numerov and Subbotin we have developed and programmed for an electronic computer a particular method for integrating the differential equations of cometary motion in special rectangular coordinates, with a variable step and allowing for all planetary perturbations and nongravitational effects over a time interval of 400 yr. Application of this method and our set of programmes to the investigation of the motion of P/Wolf permits us to eliminate the discontinuity that has hitherto existed in the theory on account of the comet's close approach to Jupiter in 1922.


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