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If $$z\in {\mathbb {C}}$$ and $$1\le n$$ is a natural number then $$\begin{aligned} \sum _{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_{1}+\cdots +q_i e_{i} }=1, \end{aligned}$$ where $$d_1=p_1^{r_1}\dots p_m^{r_m }$$ , $$d_2=q_1^{e_1}\dots q_i^{e_i }$$ are the prime decompositions of $$d_1, d_2$$ . This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3].

The sharp upper and lower bounds on the Hermitian-Toeplitz determinant of third order are computed for the classes of strongly starlike functions, lemniscate starlike functions and lune starlike functions. Moreover, a non-sharp upper bound on the fourth Hankel determinant for the lemniscate starlike functions is also obtained. Relevant connections of our results with the existing ones are also pointed out.

Both the Chang’E-1 and SELenological and ENgineering Explorer (SELENE) lunar missions, launched in 2007, provide an opportunity for significant advances in lunar science. In particular, both orbiters provide refined lunar topography models with unprecedented finer resolution and improved accuracy, especially for the far side and the polar areas of the Moon where fewer valid measurements have been available to date. An evaluation of the derived topography models is essential for the improved interpretation of selenophysics, including a knowledge of the interior of the Moon. This study provides both external and internal accuracy assessments for the derived topographic models (note that different data sets are used, albeit independent). The external comparison, which consists of comparing the topographic models with landmarks established with lunar laser ranging and radio tracking, yields an accuracy estimate of 157 m for the Change’E-1 model and 58 m for the SELENE model (150 m and 55 m if the newly recovered Lunakhod-1 site is included). The internal comparison, consisting of crossover analysis of the altimeter data after the removal of once-per-orbital-revolution errors, yields an accuracy estimate of 206 m and 68 m, respectively. These comparisons allow the establishment of conservative estimates of accuracy of 200 m and 70 m for the Chang’E-1 and SELENE models, respectively. Given the conservative estimates on the accuracy, both models yield significant improvement by factors of 2.5 and 8, respectively, when compared with a contemporary lunar topography model, i.e., the Unified Lunar Control Network 2005 (ULCN2005).

The effects of temperature, water level and moon phase on the movements of pikeperch (Sander lucioperca) were studied during pre-spawning and spawning season from April to June. Trends in catches and temperature were removed with linear regression. Cross-correlation analysis showed that temperature change one and two days earlier had the strongest effect on catches. Close to the new moon, pikeperch catches increased, whereas changes in water level had no effects on movements.

The Russian Luna-Glob project has been conceived with a view to understand the origin of the Earth-Moon system. The objectives and main features of the Luna-Glob mission, which will mainly study the internal structure of the Moon by seismic instruments, are described in the context of the past and current program of lunar exploration in Russia.

The Moon-to-Earth low energy trajectories of ‘detour’ type are found and studied within the frame of the Moon-Earth-Sun-particle system. These trajectories use a passive flight to the Earth from an initial elliptic selenocentric orbit with a high aposelenium. The Earth perturbation increases the particle selenocentric energy from a negative value first to zero and then to a positive one and therefore leads to a passive escape of the particle motion from the Moon attraction near the translunar libration pointL 2. This results in the particle flight to a distance of about 1.5 million km from the Earth where the Sun gravitation decreases the particle orbit perigee distance to a small value that leads to the particle approaching the Earth vicinity in about 100 days of the flight. A set of the Moon-to-Earth ‘detour’ trajectories is defined numerically. Characteristics of these trajectories are presented. The ‘detour’ trajectories give essential economy of energy (about 150 m/s in Delta V) relative to the usual ones.

The Indian lunar mission Chandrayaan-1 will have a mass of 523 kg in a 100 km circular polar orbit around the Moon. The main factors that dictate the design of the Indian Moon mission are to use the present capability of launch vehicles and to achieve the scientific objectives in the minimum development time and cost. The detailed mission planning involves trade-off studies in payload optimization and the transfer trajectory determination that accomplishes these requirements. Recent studies indicate that for an optimal use of the existing launch vehicle and space-craft systems, highly elliptical inclined orbits are preferable. This indeed is true for the Indian Moon mission Chandrayaan-1. The proposed launch scenario of the Indian Moon mission program and capabilities of this mission are described in this paper, highlighting the design challenges and innovations. Further, to reach the target accurately, appropriate initial transfer trajectory characteristics must be chosen. A numerical search for the initial conditions combined with numerical integration produces the near accurate solution for this problem. The design of such transfer trajectories is discussed in this paper.

The escape rate of atmospheric molecules on the Moon is calculated. Based on the assumption that the rates of emission and escape of gases attain equilibrium, the ratio of molecular number densities during day and night, n0d/n0n, can be explained. The plausible emission rate of helium and radioactive elements present in the Moon has also been calculated.

The Clementine spacecraft orbited the Moon and acquired science data for 10 weeks in the Spring of 1994. During this time it collected global 11-band multispectral images and near global altimetry. Select areas of the Moon were imaged at 25 m/pixel in visible light and 60 m/pixel in thermal wavelengths. From these datasets a new paradigm for the evolution of the lunar crust emerged. The Moon is no longer viewed as a two-terrane planet, the Apollo samples were found not to represent the lunar crust as a whole, and the complexity of lunar crustal stratigraphy was further revealed. More than ten years later the Clementine datasets continue to significantly advance lunar science and will continue to do so as new measurements are returned from planned missions such as Chandrayaan, SELENE, and Lunar Reconnaissance Orbiter. This paper highlights the scientific research conducted over the last decade using Clementine data and summarizes the influence of Clementine on our understanding of the Moon.

China plans to implement its first lunar exploration mission Chang'E-1 by 2007. The mission objectives are

The correlation between the b-values of acoustic emissions (AEs) and the phase of the moon was investigated at the Underground Research Laboratory (URL) in Canada. The same data as those used in Iwata (2002) were examined, which showed that the occurrence of AEs is correlated with the phase of the moon. It was expected, therefore, that the b-value of the AEs would also be sensitive to tidal stress/strain fluctuations. We investigated the variation of the b-values as a function of the phase of the moon. Results show that b-values immediately following the times of full/new moon are higher than those at other times. Using AIC (Akaike Information Criterion) and random (Monte Carlo) simulations, it was confirmed that this feature is statistically significant. We also investigated whether or not there was a change in the b-values immediately before the times of full/new moon, but no statistically significant change was observed. The results suggest that the effect of stress/strain fluctuations on AE occurrences at the URL is asymmetric to the times of full/new moon.

L'astronome espagnol Alfonso de Cordoba a publie differents travaux dont une collection de tables astronomiques, Tabule astronomice Elisabeth Regine redigees en latin et dediees a la royaute espagnole des XV et XVI e siecles

We have estimated a criterion for escape of silicate vapor from the disk formed by a giant impact. Escape from the disk affects the mass distribution and specific angular momentum of the disk. We applied the results to the disk formed by the Moon-forming impact. In the case of a hot (>6000 K) and highly vaporized (>63%) disk, the material outside of the Roche radius escapes. This indicates that the formation of the Moon is influenced by the thermal state of the disk just after the giant impact.

Eighteen astronomical observations from Western Europe as recorded in several historical sources from the ninth century are confirmed by retrocalculation. Carolingian astronomy was based on the motions of sun and moon and served mainly as a means of calendar computation in the time of Charlemagne. Subsequently eclipses and comets (but not planets’ constellations) were interpreted in a specific astrological manner. A recent hypothesis of three fictitious centuries in the histo|riography of the early middle age is diproved

The Tablet Collection of the Oriental Institute of the University of Chicago contains several astronomical texts among the series bearing registration numbers A 3400ff. These tablets, all bought from dealers, apparently come from Uruk.1 The majority of the texts contain mathematical astronomy and have been published by Otto Neugebauer in his Astronomical Cuneiform Texts (Lund Humphries, London, 1955). One (A 3456) is a collection of observations of Mercury and the dates of solstices, equinoxes and Sinus phenomena for the years SE 116 to 132.2 A 3405, the text discussed here, contains a collection of the dates and longitudes of planetary phenomena and lunar eclipses for the period SE 60 to 70. In answer to a query by Neugebauer, A 3405 was identified by J. Schaumberger as containing planetary observations from the same period as those cited by Ptolemy in his Almagest. Working from a photograph, Neugebauer and Abraham Sachs studied A 3405, and in his 1948 classification article Sachs described it as follows

Il existe une ancienne valeur, fondee sur le Talmud et determinant la duree d'une annee a partir du calendrier juif et des cycles lunaires

A short general explanation of tidal forces and tidal effects is given. The influences of Earth tides and ocean tides on the Earth's rotation vector are presented. Today, the theoretical models for periodic variations in the Earth's rotation and in polar motion can be compared with precise measurements done by modern space techniques. Secular changes of the Earth's rotation due to interactions within the Earth-Moon-system are also discussed.

We analyse the reasons for systematic errors in the experimental curves obtained by Moon-reflected radar signals and propose corrected experimental angular spectra

Lunar patterns of settlement of 15 Caribbean reef fishes were assessed from daily collections of newly arrived fishes from each of two small (A=63 and B=28 m2) patch reefs at Punta de San Blas (Panama) during two consecutive 2 yr periods (1984, 1985 and 1986, 1987). In 12 species settlement was lunar-cyclic, with a broad peak of activity around the new moon in 11 cases, and at first quarter in 1 species. However, 3 of these “lunar-cyclic” species displayed intermittent semilunar periodicity in settlement, with peaks around the quarter moons. One other species had a semilunar cycle of settlement with peaks at both quarter moons. Two species apparently lacked lunar settlement cycles. In some, but not all, paris of congeners: (a) monthly variation in settlement intensity was positively correlated, (b) the duration of settlement pulses in the same month consistently differed, and (c) the timing of settlement pulses in the same month often differed (although one species did not consistently arrive before the other). Recruitment of juvenile fish that survived to the end of the lunar cycle in which they settled was monitored monthly, concurrently with settlement, at a number of large sites scattered up to 2 km apart. The levels of variation in the amount of settlement and recruitment each month differed in only 2 of 17 cases, with settlement variation exceeding recruitment variation in only one of these. The intensity of recruitment was positively correlated with the intensity of settlement in 16 of 17 cases. In 15 of those 16 cases, R 2 values for linear regressions of recruitment on settlement exceeded R 2 values for curvilinear regressions that would indicate either increasing or decreasing recruitment success with increasing intensity of settlement. Thus it appears that variation in recruitment can be used to estimate variation in settlement, and that recruitment success may be density-independent.

In this paper the problem of the classification of natural samples is discussed. An updated version of the G-mode multivariate statistical method for the classification of natural samples, applicable to a wide range of research fields, is discussed in this paper. This method allows an automatic classification in terms of homogeneous taxonomic units, without any a priori knowledge of the taxonomic structure of the natural observations; it provides informations on the different levels of classification present in the data set under study (classes and subclasses), on the level of information residing in each variable, on the level of similarity and/or difference among homogeneous classes.

The oscillating peak model of basin and crater formation proposed by Murray is analysed mathematically. The results could be compared with some basins in order to confirm the basic ideas of this model.

It is amazing to see how many webpages are devoted to the art of finding the date of Easter Sunday. Just for illustration, the reader may search for terms such as Gregorian calendar, date of Easter, or Easter algorithm. Sophisticated essays as well as less enlightening contributions are presented, and many a doubt is expressed about the reliability of some results obtained with some Easter algorithms. In short, there is still a great interest in those problems. Gregorian Easter algorithms exist for two centuries (or more?), but most of their history is rather obscure. Some reasons may be that some important sources are written in Latin or in the German of Goethe’s time, or they are difficult to discover. Without being complete, the following paper is intended to shed light on how those techniques emerged and evolved.1 Like a microcosm, the history of Easter algorithms resembles the history of any science: it is a story of trials, errors, and successes, and, last but not least, a story of offended pride.

Periodicity in predatory activity was observed in overwintered females of Typhlodromus pyri Scheuten (Acari: Phytoseiidae). Chronobiometrical analysis revealed that an approximately circabiseptan (about-14-day) rhythm is statistically significant, with a striking depression around the full moon.

The change in the eccentricity of the AES orbit of the type 1963-30D as a result of one passage through the Moon's shadow is estimated, and it has been shown that the effect is not negligible.

Вычuсляеmся nрецессuонно-нуmaцuонныŭ моменm nрuлuвообрaзующuх сuл ом Луны в зaвuсuмосmu оm nосmоянных Сmоксa Землu. Выясняеmся оmношенuе uн¶rt;uвu¶rt;уaльных членов эmого моменma к nрuлuвообрaзующеŭ сuловоŭ функцuu.

Diurnal variations of occurrence hours and the period of Pi2 pulsations at Choutuppal (India) for nearly half a solar cycle are presented. Maximum occurrence is noticed to be around local midnight in all the seasons. Shorter periods are observed in the late afternoon hours in E-and J-seasons. Lunar influence on the occurrence of Pi2 pulsations at this station is inferred with the occurrence due to this influence peaking in the vicinity of the lunar phase4.

The results of simultaneous and homogeneous reduction of 8 libration series are presented. On this basis all 6 free libration constants have been estimated using some new formulae.

This paper is a review of certain lunar features with specific regard to those found in the crater Tsiolkovsky.

Обсуж¶rt;aеmся свобо¶rt;ное u вынуж¶rt;енное ¶rt;вuженuе осu врaщенuя aбсолюmно mвер¶rt;оŭ Землu в случaе, ко ¶rt;a эквamорuaльные моменmы uнерцuu не рaвны. Дaюmся чuсленные резульmamы ¶rt;ля uзмененuя скоросmu врaщенuя Землu, вызвaнно о рaвumaцuонным воз¶rt;еŭсmвuем Луны u Солнцa, u maкже ¶rt;ля вынуж¶rt;енноŭ нуmaцuu осu врaщенuя оmнoсumельно осu лaвно о моменma uнерцuu.

An analysis of 1210 visual brightness estimations of the Moon's ashen light is presented, performed by a working group of amateur astronomers from June 1972 to December 1973. In the Moon phase interval 0.1 ⩽ Tb⩽ 0.7 the brightness expressed in a semi-empirical scale, SG, is found to be linearly related to the phase. Monthly deviations from the mean brightness show well defined winter maxima (January) and summer minima (July). Within the referenced period the brightness of the ashen light tends to increase, whereas the solar magnetic activity decreased. In addition, minor correlations and, respectively, anti-correlations are found at stratospheric temperature and, respectively density. On account of the nature of the ashen light its variations are regarded as fluctuations of the Earth's albedo.

The largest members in longitude and latitude of the theoretical movement of the earth's moon, developped byHill 1891 fromDelaunays theory of the moon, have been here calculated with new, now better known numerical values. Effectively a highly better accordance of the observed values with the theoretical ones has been obtained. This method will be able to give an important contribution to the determination of the figure of the earth with regard to the geometrical and the static flattening, specially by moons observation material of 1850–1950. Les membres les plus grands en longitude et latitude du mouvement theorique de la lune developpes parHill 1891, fondes sur la theorie deDelaunay, sont ici calcules avec de nouvelles valeurs numeriques de meilleure connaissance. C'est pourquoi une bonne accordance est obtenue des valeurs observees avec les valeurs theoriques. La methode donne une contribution importante a la determination de la figure de la terre par rapport a l'aplatissement geometrique et l'aplatissement statique, principalement par egard aux observations de la lune 1850–1950.

The artificial moon, projected from serious scientists, presents, whenever previously without reality, a field of interesting theoretical problems. At first the figure of movement of the body itself within the varying gravity field, further more its observation by angles and distances relative to earth's body and to sky and the possible reconclusions to earth's figure and to natural moons movement. The latter beeing in a too far distance from the earth, the former surely will help to solve different problems of geodesy and astronomy. La Lune artificielle, comme projet d'une science serieuse moderne, presente, quand meme prealablement sans realite, un grand nombre de problemes theoriques interessantes. Avant tout la figure elle meme de sa route dans le champs variant de la gravite et de la force centrifuge, de plus l'observation par angles et distances relatives a la terre et au ciel, et les reconclusions a la figure de la terre et au mouvement de la Lune naturelle. Cette derniere se trouvant perpetuellement dans une position de distance trop grande, la premiere peut aider sans doute la solution de problemes differents de geodesie et astronomie.

Hilbrecht and Kiiveler (1985) have presented the results of ‘possible’ variations of the Earth’s albedo which have been deduced from the ashen light of the Moon. The observations presented in the paper were made during the period of June 1972 to December 1973. While it is acknowledged that studies of the possible variations of the Earth’s albedo are an important part of investigations of climate variations, the approach adopted by Hilbrecht and Kiiveler (lot tit) and the results presented are not sufficiently accurate for this purpose. During the early part of this century a number of astronomers made estimates of the global mean albedo of the Earth using this approach (Hunt et al., 1985). In Table I we have summarised the values that were estimated during these previous studies. The wide variation in values of the Earth’s albedo would, at first, suggest a major climatic change! However, the differences between the individual results is a direct result of the difficulties in accurately interpreting the photometric observations as a consequence of the phase ventions of the total and local regions of the Moon. The variations of the brightness of the ashen light are shown in Figure 3 of Hilbrecht and Kiiveler (lop tit). There are differences of a factor of 6 in the observations during January, March 1973 and July 1973 (see Figure 1 of Hilbrecht and Kiiveler) which will drastically affect the statistical significance of the results. Furthermore, the error bars of

FOR the general use of amateur astronomers this is the best atlas and handbook we have yet seen. The sixteen maps are printed exceptionally clearly, and, while not overcrowded, show more than 7,000 objects. Each map is about 10 in. by 8 in., and is part of a lune, covering, exclusive of overlap, four hours of R.A., and 60° N. or S. of declination. The atlas opens out flat, and shows two maps joined together at the equator, so that about one-fifth of the whole sky is seen at once. Meridians and parallels mark every hour of R.A., and every tenth degree of declination, while marginal divisions enable a position to be fixed to the nearest 5m. or 1°. The polar regions are shown on two pairs of maps. A Star Atlas and Telescopic Handbook (Epoch 1920). For Students and Amateurs. By Arthur P. Norton. Pp. 19 + 16 star and 2 index maps. (London and Edinburgh: Gall and Inglis, 1910.) Price 5s. net.