• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.1069-1105
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• 2001

Regular maps and hypermaps are cellular decompositions of closed surfaces exhibiting the highest possible number of symmetries. The five Platonic solids present the most familar examples of regular maps. The gret dodecahedron, a 5-valent pentagonal regular map on the surface of genus 5 discovered by Kepler, is probably the first known non-spherical regular map. Modern history of regular maps goes back at least to Klein (1878) who described in [59] a regular map of type (3, 7) on the orientable surface of genus 3. In its early times, the study of regular maps was closely connected with group theory as one can see in Burnside’s famous monograph [19], and more recently in Coxeter’s and Moser’s book [25] (Chapter 8). The present-time interest in regular maps extends to their connection to Dyck\`s triangle groups, Riemann surfaces, algebraic curves, Galois groups and other areas, Many of these links are nicely surveyed in the recent papers of Jones [55] and Jones and Singerman [54]. The presented survey paper is based on the talk given by the author at the conference “Mathematics in the New Millenium”held in Seoul, October 2000. The idea was, on one hand side, to show the relationship of (regular) maps and hypermaps to the above mentioned fields of mathematics. On the other hand, we wanted to stress some ideas and results that are important for understanding of the nature of these interesting mathematical objects.

• ALGAE
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• v.25 no.3
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• pp.133-140
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• 2010

Using chlorophyll fluorescence, the vertical migration of benthic diatoms responding to light intensity and affected by sediment grain size was studied. Minimal fluorescence ($F_o$) of surface sediment was measured by imaging pulse amplitude modulated (Imaging-PAM) fluorometer, and used to monitor diatom biomass variation in surface sediments. The test diatoms, Amphora coffeaeformis (C. Agardh) K$\ddot{u}$tzing and Cylindrotheca closterium (Ehrenberg) Reimann & Lewin, migrated to the sediment surface under irradiance from 50 to 500 ${\mu}mol$ photons $m^{-2}s^{-1}$. However, the diatoms exhibited no evident increase of surface biomass under dark conditions, and even showed slightly decrease of surface biomass under irradiances over 1,000 ${\mu}mol$ photons $m^{-2}s^{-1}$. The light intensity inducing the maximum surface migration of A. coffeaeformis was 100 ${\mu}mol$ photons $m^{-2}s^{-1}$, while the light intensity producing the same effect for C. closterium was 250 ${\mu}mol$ photons $m^{-2}s^{-1}$. C. closterium showed higher motility than A. coffeaeformis. Faster diatom surfacing was observed in larger grain size sediments (125-335 ${\mu}m$) than smaller ones (63-125 ${\mu}m$). This study confirmed the significant influence of light as a main triggering factor behind migration, indicated the distinct effect of different sediment grain size, and highlighted the species-specific migratory ability.