Our aim in this paper is to study Musielak-Orlicz-Sobolev spaces on metric measure spaces. We consider a Hajłasz-type condition and a Newtonian condition. We prove that Lipschitz continuous functions are dense, as well as other basic properties. We study the relationship between these spaces, and discuss the Lebesgue point theorem in these spaces. We also deal with the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz spaces. As an application of the boundedness of the Hardy-Littlewood maximal operator, we establish a generalization of Sobolev’s inequality for Sobolev functions in Musielak-Orlicz-Hajłasz-Sobolev spaces., Takao Ohno, Tetsu Shimomura., and Obsahuje seznam literatury
We define and study Musielak-Orlicz-Sobolev spaces with zero boundary values on any metric space endowed with a Borel regular measure. We extend many classical results, including completeness, lattice properties and removable sets, to Musielak-Orlicz-Sobolev spaces on metric measure spaces. We give sufficient conditions which guarantee that a Sobolev function can be approximated by Lipschitz continuous functions vanishing outside an open set. These conditions are based on Hardy type inequalities., Takao Ohno, Tetsu Shimomura., and Obsahuje seznam literatury
The Chinese zither 古琴 guqin (Old zither 琴 qin zither), with strings stretched lengthwise on a flat wooden belly, is a pentatonic heptachord which can be played in three different ways: transversely on open strings, longitudinally by stopping, and also in harmonics (touched lightly on various marked points). Such an instrument, age-old in China, might be a theoretical archetype, which is able to make a contribution to the debate on issues concerning Greek music, something which has remained obscure in relation to the classical Western approach. Indeed, the assimilation “heptachord-heptatonic-diatonic” has ineluctably led most Western theorists to occasional misunderstandings of texts related to tuning, modulation, and the names of the modes. The demonstration is carried out in three steps: 1) a consideration of the seven tunings of the pentatonic guqin zither and their permutation through modulation on the diatonic scale; 2) an examination of the correspondence between musical systems and the Greater Perfect System; 3) a conclusion leading to the question as to whether the specific characteristics of the Chinese guqin were known or unknown by the Greeks. In order to maintain an effective methodological approach, we need to remember that Chinese writing gives different names to the notes, according to the system to which they belong. In conclusion, the hypotheses concerning the supposed exchanges and influences from Asia to Greece are strengthened anew.
We studied the demography and spatial structure of the muskrat population at the Nizhnekamsk Reservoir in central Russia. 45Ca radioactive prenatal labelling of muskrats was performed over a period of eight years. A total of 3,876 infividual muskrats were captured from a 12 km2 experimental plot and analysed on the basis of age, sex and presence of radioactivity. During the first year of life, the highest mortality was among muskrats from the latest litter, whilst during the second year of life it was amongst individuals from the earlier litters. Spring and autumn muskrat migration were studied and we observed a cyclical type of spatial structuring within the population. The proportion of individuals migrating in autumn varied from 18-57%, but significant differences in the migratory activities of different age groups were not observed. We found several tendencies throughout the eight years: a decrease in the proportion of the first litter (not overwintering), an increase in the proportion of third litters, and a decrease in the proportion of yearlings born in the first litter - we assume these represent responses to flooding.