On the periodic solutions of a particular Lame equation

We consider the Hill's equation: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqOVdGhabaGaamizaiaadsha% daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWGTbGaai% ikaiaad2gacqGHRaWkcaaIXaGaaiykaaqaaiaaikdaaaGaam4qamaa% CaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaaiykaiabe67a4jabg2% da9iaaicdaaaa!4973!\[\frac{{d^2 \xi }}{{dt^2 }} + \frac{{m(m + 1)}}{2}C^2 (t)\xi = 0\]Where C(t) = Cn (t, {frbuilt|1/2}) is the elliptic function of Jacobi and m a given real number. It is a particular case of theame equation. By the change of variable from t to defined by: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaOWaaiqaaq% aabeqaamaalaaajaaybaGaamizaGGaaiab-z6agbqaaiaadsgacaWG% 0baaaiabg2da9OWaaOaaaKaaGfaacaGGOaqcKbaG-laaigdajaaycq% GHsislkmaaleaajeaybaGaaGymaaqaaiaaikdaaaqcaaMaaeiiaiaa% bohacaqGPbGaaeOBaOWaaWbaaKqaGfqabaGaaeOmaaaajaaycqWFMo% GrcqWFPaqkaKqaGfqaaaqcaawaaiab-z6agjab-HcaOiab-bdaWiab% -LcaPiab-1da9iab-bdaWaaakiaawUhaaaaa!51F5!\[\left\{ \begin{array}{l}\frac{{d\Phi }}{{dt}} = \sqrt {(1 - {\textstyle{1 \over 2}}{\rm{ sin}}^{\rm{2}} \Phi )} \\\Phi (0) = 0 \\\end{array} \right.\]it is transformed to the Ince equation: (1 + · cos(2)) y + b · sin(2) · y + (c + d · cos(2)) y = 0 where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaiaadggacq% GH9aqpcqGHsislcaWGIbGaeyypa0JcdaWcgaqaaiaaigdaaeaacaaI% ZaGaaiilaiaabccacaWGJbGaeyypa0Jaamizaiabg2da9aaacaqGGa% WaaSaaaKaaGfaacaWGTbGaaiikaiaad2gacqGHRaWkcaaIXaGaaiyk% aaqaaiaaiodaaaaaaa!4777!\[a = - b = {1 \mathord{\left/{\vphantom {1 {3,{\rm{ }}c = d = }}} \right.\kern-\nulldelimiterspace} {3,{\rm{ }}c = d = }}{\rm{ }}\frac{{m(m + 1)}}{3}\]In the neighbourhood of the poles, we give the expression of the solutions.The periodic solutions of the Equation (1) correspond to the periodic solutions of the Equation (3). Magnus and Winkler give us a theory of their existence. By comparing these results to those of our study in the case of the Hill's equation, we can find the development in Fourier series of periodic solutions in function of the variable and deduce the development of solutions of (1) in function of C(t).     

Depositional environments and facies of the Late Triassic Abu Ruweis Formation, Jordan

The Abu Ruweis Formation is composed of carbonates, evaporites, and mudstones, with some locally developed pelletic, oolitic and stromatolitic limestones. The lateral persistence of bedding, the purity of the evaporite rocks, the alternating arrangement of marine carbonates and evaporites indicates periodic deposition in subaqueous conditions (salina). Petrographic investigations, X-ray diffraction analysis as well as chemical analysis have shown that the outcropping evaporite beds are mainly composed of secondary gypsum, with rare anhydrite relics. Five microfacies of gypsum were recognized according to their fabrics: porphyroblastic and granoblastic gypsum showing polarization texture, gypsum pseudomorph after anhydrite laths, and satin spar gypsum. The textures they display indicate a hydration origin of precursor anhydrite, which is in turn rehydrated from primary gypsum. Some of these anhydrites were formed as a result of replacement processes of the carbonate sediments associated with the evaporites, as evidenced from the textural relationships of the carbonate and sulfate minerals. The O¹⁸ content ranges from 1.45 to 8.38% PDB and the C¹³ content ranges from −1.52 to 4.73% PDB. Trace elements analysis has shown that the Abu Ruweis dolomites are rich in strontium (up to 600 ppm), and sodium (up to 835 ppm). The isotope composition and trace elements content, as well as the petrographic characteristics point to a penecontemporaneous hypersaline dolomitization origin for the Abu Ruweis dolomites. The evaporites were deposited during a regressive lowstand systems tract, whereas the carbonates were deposited under shallow water marine conditions during a highstand systems tract. The Abu Ruweis succession represents a relatively stable arid climate within a rapidly subsiding basin. Restricted conditions were provided by the development of beach barriers.     

Apport des isotopes stables dans l'estimation des altitudes de recharge de sources thermales du MarocRecharge altitude estimation of thermal springs using stable isotopes in Morocco

The recharge altitude estimation of thermal springs from northern and eastern Morocco using ¹⁸O and ²H contents requires the definition of regional isotopic altitudinal gradients (?0.25‰ for 100 m for the Rif and ?0.30‰ for the East) and the calculation of residence time using ¹⁴C. The altitudes of emergence vary widely between 170 and 1040 m under the altitude of the recharge areas. The ¹⁸O and ²H compositions of palaeowaters ($\text{>10}\text{000}$ yr BP) indicate two effects, altitude and palaeoclimate. To cite this article: A. Winckel et al., C. R. Geoscience 334 (2002) 469–474.     

Sedimentology and depositional environments of the Ordovician Umm Sahm Sandstone Formation in southern Jordan

The Ordovician Umm Sahm Sandstone Formation of Jordan comprises approximately 200-m-thick succession of fluvial quartzarenites with subordinate claystone and siltstone lithologies of shallow marine conditions. The Umm Sahm Formation is characterized by its dark brown color, frequent jointing, and steep scarps. The Umm Sahm Formation is bounded by the marine claystones of Hiswah Formation at the top and the fluvial sandstones of the Disi Formation at the bottom. The Umm Sahm Formation is composed of two main facies: fluvial facies and tidal facies. The fluvial facies constitutes about 93% of the total thickness. The lower few meters of the succession passes upward from the Disi Sandstone Formation into similar massive white sandstone facies exhibiting similar white color, fine- to coarse-grained sandstone, with round-shaped pebbles. Trough and planar cross-bedding show a northwest unidirectional palaeocurrent trend. Light brown colored quartzarenites similar to those of the Cambrian Umm Ishrin Sandstone Formation are most common in the upper part of the succession. The tidal facies occupies three intervals in the middle part of the succession. It is composed of laminated and thin-bedded sandstones, siltstones, and claystones. They are rippled and varicolored with abundant trace fossils (Cruziana, Harlania, ruzophycus). The presence of hummocky cross stratification indicates the earliest short-lived tempestite conditions during the Paleozoic erathem of Jordan. The first appearance of Graptolites in the Ordovician rocks of Jordan was recorded during this study in the tidal facies of the Umm Sahm Formation. The vertical arrangement of both fluvial and tidal facies indicates three successive short-lived transgressions and regressions. These marine incursions indicate the successive shoreline advances of the Tethys Ocean, which was located northward, and inundated the southern braid plain. The three short-lived transgressive events took place, and the Tethys marine margin was displaced southward, giving rise to deposition of tidal facies in an open coast tidal flat. Following the transgressive events, renewed progradation and strandline retreat took place, fed by large amounts of siliciclastics derived from the continent and transported by braided streams across the intertidal zone.     

Nutrient distribution and fluxes from three Mediterranean coastal rivers (NE Algeria) under large damming

Nutrient distribution and fluxes into and from dams and into coastal waters from three rivers (NE Algeria) were assessed during a one-year period in three stations for each river: at the entrance and the exit of dam and at the outlet. The main characteristics of the rivers were the high levels of NH₄ and PO₄, even in dam entrances, contrarily to SiO₄ levels that are still low upstream the dams. From the inorganic nutrient incoming fluxes, the dams trapped annually 42 to 93%, depending on the nutrient, but released in great levels dissolved organic forms at their exits. At catchment scale, dissolved nitrogen loadings reach 338 kg/km²/yr, in which the organic fraction forms up to 34%; while those of dissolved phosphorus reach 172 kg/km²/yr, with a great organic fraction. The Si:N ratios decreased while N:P ratios increased at river outlets, indicating large inputs of N over P in the lower catchments.     

Vulnerability assessment of coastal areas to sea level rise from the physical and socioeconomic parameters: case of the Gulf Coast of Bejaia,Algeria

The study area (the Gulf of Bejaia) is a coastal zone of about 70 km long in the eastern-central part of the Algerian coast. The coastline characterized by sandy beaches, hotels and tourist facilities, airport, port, villages and towns has known during these last decades several threats like storms, floods and erosion. The present work concerns the mapping of the physical and socioeconomic vulnerability of the Gulf Coast of Bejaia to sea level rise, using Coastal Vulnerability Index (CVI) and geospatial tools. The Physical CVI (CVI_Phys) is calculated from seven physical variables: geomorphology, coastal slope, coastal regional elevation, sea level rise rate, shoreline erosion/accretion rates, tidal range and significant wave height. On the other hand, the parameters population, cultural heritage, roads, railways, land use and conservation designation constitute, for their part, the socioeconomic CVI (CVI_eco). The values obtained from the calculation of CVI_Phys vary between 3.53 and 81.83. These results revealed that 22.42 km of the studied coastline has a low physical vulnerability, 21.68 km a high vulnerability and 15.83 km a very high vulnerability, indicating that the most part of the coastline (53.59%) is vulnerable to sea level rise. According to the obtained values of CVI_eco, the most vulnerable areas of high and very high risk represent 31.81 km of the total coastline. They were found along the western (Bejaia and Tichy) and eastern (Aokas, Souk El Tenine and Melbou) coast, while the least vulnerable stretches, covering 38.19 km of the total length of the coast, occupy the rest of the area. This study highlighted areas that will be most affected by future sea level rise (SLR) and storm events. It revealed that several development projects of Bejaia Gulf Coast, including tourist expansion areas, are planned in sites identified as very vulnerable. The results obtained from this assessment could guide local planners and decision-makers in developing coastal management plans in the most vulnerable areas.     

Assessment of raw clays from Maghnia (Algeria) for their use in the removal of Pb<Superscript>2+</Superscript> and Zn<Superscript>2+</Superscript> ions from industrial liquid wastes: a case study of wastewater treatment

Homogenized samples of raw clays resulting from two (2) different lots of natural clays from Maghnia (Algeria) have been assessed for their potential use in the removal of Pb²⁺ and Zn²⁺ ions from industrial liquid wastes (LW). Raw and acid-activated samples have been characterized by powder X-ray diffraction, FT-IR spectroscopy, electron microscopy (SEM), and X-ray fluorescence (XRF) and used as adsorbents for the removal of Pb²⁺ and Zn²⁺ ions from aqueous system using adsorption method under different conditions. The effect of factors including contact time, pH, and dosage on the adsorption properties of Pb²⁺ and Zn²⁺ ions onto clays was investigated at 25 °C. The obtained results revealed that the removal percentages of Pb²⁺ and Zn²⁺ ions, from both aqueous solution (AS) and LW, were varying between 90 and 98% for 40 min and optimal pH values ranged from 5 to 6 for Pb²⁺ and Zn²⁺ ions, respectively. The kinetics of both Pb²⁺ and Zn²⁺ ion adsorption fitted well with the pseudo-second-order model. Langmuir, Freundlich, and Temkin adsorption isotherms were used, and their constants were evaluated. The values of thermodynamic parameters, ΔH°, ΔS°, and ΔG° indicated that the adsorption of Pb²⁺ and Zn²⁺ ions was spontaneous and exothermic process in nature. The adsorption and desorption isotherms indicated that Pb²⁺ and Zn²⁺ adsorption to raw clays was reversible. The experimental results obtained showed that the raw clays from Maghnia (Algeria) had a great potential for removing Pb²⁺ and Zn²⁺ ions from industrial liquid wastes using adsorption method.     

Periodic orbits of the generalized Friedmann–Robertson–Walker Hamiltonian systems

The averaging theory of first order is applied to study a generalization of the Friedmann–Robertson–Walker Hamiltonian systems with three parameters. We provide sufficient conditions on the three parameters of the generalized system to guarantee the existence of continuous families of periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter.