On the period of the periodic orbits of the restricted three body problem

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).     

Impacts of primary deforestation upon limestone slopes in northern Vancouver Island, British Columbia

Effects of deforestation upon slopes in limestones and in volcanic rocks in the Benson River valley, northern Vancouver Island, have been investigated quantitatively. Postlogging soil erosion and vegetal regeneration success were assessed by measuring soil depth, percent bare rock and moss cover, and the numbers and diversity of trees, shrubs, and plants on 25 sampling sites, each containing ten measuring quadrats selected at random. Sixteen sites were on the Quatsino Formation, a well-karstified limestone, and nine on the Karmutsen Formation of basaltic lavas. Eight sites were of virgin forest, 16 were logged between 1970 and 1983, and one (on limestone) was logged in 1911. Both bedrock types were significantly affected by the cutting. There was greater loss of soil and an increase in bare rock on the limestones. Erosion was increased significantly by burning on the limestones but not on the volcanics. Within-group comparisons on the limestones determined that steeper slopes and harder burned areas suffered the most and are slowest to regenerate. Volume of timber on the 1911 site was 19 percent of that in similar uncut forest sites. It appears that complete recovery on the barren limestone slopes will require at least some centuries.     

Distinct metamorphic evolution of alternating silica-saturated and silica-deficient microdomains within garnet in ultrahigh-temperature granulites: An example from Sri Lanka

Here we report the occurrence of garnet porphyroblasts that have overgrown alternating silica-saturated and silica deficient microdomains via different mineral reactions. The samples were collected from ultrahigh-temperature (UHT) metapelites in the Highland Complex, Sri Lanka. In some of the metapelites, garnet crystals have cores formed via a dehydration reaction, which had taken place at silica-saturated microdomains and mantle to rim areas formed via a dehydration reaction at silica-deficient microdomains. In contrast, some other garnets in the same rock cores had formed via a dehydration reaction which occurred at silica-deficient microdomains while mantle to rim areas formed via a dehydration reaction at silica-saturated microdomains. Based on the textural observations, we conclude that the studied garnets have grown across different effective bulk compositional microdomains during the prograde evolution. These microdomains could represent heterogeneous compositional layers (paleobedding/laminations) in the precursor sediments or differentiated crenulation cleavages that existed during prograde metamorphism. UHT metamorphism associated with strong ductile deformation, metamorphic differentiation and crystallization of locally produced melt may have obliterated the evidence for such microdomains in the matrix. The lack of significant compositional zoning in garnet probably due to self-diffusion during UHT metamorphism had left mineral inclusions as the sole evidence for earlier microdomains with contrasting chemistry.     

THREE-WAY METHODS FOR THE CALIBRATION OF CHROMATOGRAPHIC SYSTEMS:COMPARING PARAFAC AND THREE-WAY PLS

For the calibration of chromatographic systems,different methods can be used.One class of methodsutilizes three-way approaches.The calibration problem is stated in such a way that the decompositionof a three-way array can serve for the prediction of retention on new stationary phases.Two three-way approaches are presented:the Unfold-PCA and PARAFAC models.The theory ofboth methods is presented and the differences are highlighted,the main difference being that PARAFACis a trilinear decomposition whereas Unfold-PCA is not.Both three-way methods are evaluated on asmall data set consisting of retention measurements of eight solutes at six mobile phase compositions onsix stationary phases.The differences in performance of the two models are minor,For calibration purposes,two variants of the methods are discussed:three-way PLS and an extensionof PARAFAC.Again the theory and differences between the two methods are explained.The predictiveperformance of the two methods is compared using the same data set as earlier.The differences inpredictive performance,however,are minor.Both methods are capable of predicting 98% of thevariation in the test sets.Yet,there are other considerations when comparing methods than predictiveperformance,e.g.the quality of the predictions.     

Biogeochemical records of paleoenvironmental changes in Nainital Lake,Kumaun Himalayas,India

Rapid urbanization and increased tourism around Nainital Lake in the Kumaun Himalayan region in north India has raised concerns about sediment and water pollution. Lead-210 dated sediment cores from the lake represent ~95 years of accumulation and yield a mean sedimentation rate of ~4.7 mm year⁻¹. Total organic carbon (TOC), percent N and S and their atomic C/N and C/S ratios, stable isotopes (δ¹³C, δ¹⁵N, and δ³⁴S), and specific biomarkers (n-alkanes and pigments) were measured in the core. Organic matter is primarily derived from in-lake algal production and TOC flux varies from 1.0 to 3.5 g m⁻² year⁻¹. Sediments are anoxic (Eh −328 to −187 mV) and have low (0.10–0.30 g m⁻² year⁻¹) N, but high (0.37–1.0 g m⁻² year⁻¹) S flux. Shifts in δ¹³C, δ¹⁵N, and δ³⁴S suggest in-lake microbial processes dominated by denitrification and sulfate reduction. The sediments are dominated by short-chain hydrocarbons with low Carbon Preference Index values. The pigments indicate a gradual shift to cyanobacterial domination of the phytoplankton community in recent years. Despite an increase in external input of nutrients, the trophic state of the lake has remained largely unchanged, and the perceived human-induced impacts are limited.     

The development of three-dimensional current system during a magnetospheric substorm

It is suggested that the pattern of three-dimensional substorm current circuit varies significantly even during the lifetime of a single substorm. This gives rise to quite complex time variations of the magnetic field at low latitude stations even for relatively isolated substorms. To verify this, three-dimensional current models with time dependent spatial variations are used to simulate one type of complex low-latitude “substorm signature”. It is shown that the utmost care should be exercised in determining different substorm phases on the basis of such a signature. The results indicate also that, in certain longitudes in the evening sector, one should expect distinct differences in characteristics between positive bays observed on the ground and at the synchronous distance.     

Geochemical reconstruction of late Holocene drainage and mixing in Kluane Lake,Yukon Territory

The level of Kluane Lake in southwest Yukon Territory, Canada, has fluctuated tens of metres during the late Holocene. Contributions of sediment from different watersheds in the basin over the past 5,000 years were inferred from the elemental geochemistry of Kluane Lake sediment cores. Elements associated with organic material and oxyhydroxides were used to reconstruct redox fluctuations in the hypolimnion of the lake. The data reveal complex relationships between climate and river discharge during the late Holocene. A period of influx of Duke River sediment coincides with a relatively warm climate around 1,300 years BP. Discharge of Slims River into Kluane Lake occurred when Kaskawulsh Glacier advanced to the present drainage divide separating flow to the Pacific Ocean via Kaskawulsh and Alsek rivers from flow to Bering Sea via tributaries of Yukon River. During periods when neither Duke nor Slims river discharged into Kluane Lake, the level of the lake was low and stable thermal stratification developed, with anoxic and eventually euxinic conditions in the hypolimnion.