Free vibrations of a semi-infinite cylindrical shell, localized near the edge of the shell are investigated. The dynamic equations in the Kirchhoff-Love theory of shells are subjected to asymptotic analysis. Three types of localized vibrations, associated with bending, extensional, and super-low-frequency semi-membrane motions, are determined. A link between localized vibrations and Rayleigh-type bending and extensional waves, propagating along the edge, is established. Different boundary conditions on the edge are considered. It is shown that for bending and super-low-frequency vibrations the natural frequencies are real while for extensional vibrations they have asymptotically small imaginary parts. The latter corresponds to the radiation to infinity caused by coupling between extensional and bending modes.

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S u m m a r yThe migratory open air nesting Apis dorsata and Apis laboriosa honeybees migrate at least twice a year. DNA genotyping showed that the same swarms return to their natal nesting sites. We examined 23 nesting sites in Nepal, India and Bhutan, on which 587 colonies of A. dorsata and A. laboriosa nested. The results showed that the frequency of the periodic mass flights (PMF) performed by the colonies is a good indicator of the status of current colony performance. During the swarming period, both, A. dorsata and A. laboriosa issue several swarms. In some colonies, so many bees swarmed out, that those remaining in the maternal colonies did not cover the combs. After the rest of the brood emerged, all the bees of such colonies abscond during the swarming period. Thus, absconding appeared in results of total out swarming. The swarms do not migrate directly to the seasonal alternative nesting sites, but establish new colonies in the areas around. After environmental conditions deteriorate, all the bees with their queens abscond and migrate to alternate seasonal nesting sites. The next season, the swarms do not return to their original reproductive natal sites, but to those sites they occupied the previous season lately, where from they absconded.

-The investigations were conducted in the Himalayas in Nepal. Two cliff sites, one with 16 A. laboriosa nests and the other with 53 nests were observed. Together 120 periodic mass flights (PMF) performed by A. laboriosa worker bees during 8 days were recorded. Workers from 50-81% of colonies performed PMF, during days of adequate weather conditions. The flights were not performed during overcast days. Worker bees from different colonies perform PMF at different times of the day. Workers from the same colonies perform PMF at different times in subsequent days. While workers from some colonies did not perform any PMF during several days, others perform 2 or 3 PMF per day. Apis laboriosa / periodic mass flights / Nepal / Himalayas

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