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Notes: Abstract In this paper, we describe the symbol calculus and index theorem for Wiener-Hopf operators on the group of complex 2×2 unitary matrices U2.

Notes: Summary The Drontermeer (600 ha) and the Veluwemeer (3400 ha) were separated from Lake IJssel by dikes and locks in 1957. The Wolderwijd (2600 ha), the Eemmeer (1900 ha), and the Gooimeer (2400 ha) were formed in the same way in 1967. The Drontermeer and the Veluwemeer are in open connection with each other. Shiplocks separate these lakes from the adjacent lakes. The Wolderwijd is separated from the Veluwemeer and the Eemmeer by locks. The Eemmeer and the Gooimeer are in open connection with Lake IJssel. Average depth of the lakes is about 1.50 m. A ship canal with a width of 100 m and a depth of 4 m has been dug through the lakes. Refreshment time for all lakes is about two months. The lakes are polluted by domestic and dairy wastes. The Wolderwijd is polluted only by dairy wastes. The phosphorus load of the Drontermeer and the Veluwemeer is about 4 to 5 g P/m2/year; of the Wolderwijd, less than 3 g P; and of the Eemmeer and Gooimeer, 20 to 30 g P/m2/year. Chemical investigations were begun in 1969. Phytoplankton has been investigated since 1970. Primary production has also been measured since 1970. At five depth levels (10, 20, 30, 50 and 100 cm), oxygen was measured in ten light bottles (250 ml) and two dark bottles (20 and 100 cm). Incubation time was one and one-half hour. The investigations are being continued. The chemical composition (Ptotal, Nkjehldahl, COD, BOD5, chlorophyll a) is strongly influenced by the biomass of the phytoplankton: Ptotal varies from 0.2 to 1.0; Nkjehdahl from 2.0 to 6.0; COD from 35 to 120; and BOD5 from 8.0 to 25.0 mg/l. Chlorophyll a varies from 80 to 400 ug/l and is not related with the number of algae individuals nor with dry of fresh weight. These values vary from lake to lake. The variation within the same lake is small. In June 1971, the phytoplankton species composition in the Drontermeer and the Veluwemeer changed suddenly:Oscillatoria agardhii Gom. replacedScenedesmus quadricauda div. var. as the dominant algae. Up to now,O.agardhii is dominant in the lakes bordering on Flevoland. Only in February and March do diatoms occur in the majority in the Eemmeer and Gooimeer. A phenomenon occurring in each lake almost every year is that of an increasing number ofO.agardhii filaments in May coupled with a majority ofOscillatoria redekei v. Goor,Oscillatoria limnetica Lemn., andAphanizomenon flos aquae (L) Ralfs. Dominance ofOscillatoria agardhii as referred to here means that more than 80% of the individuals consist of these alga from June to February. The Wolderwijd has been anOscillatoria agardhii lake from the beginning of our investigations in 1970. In the Eemmeer and the Gooimeer,O.agardhii disappeared suddenly in July 1972 and was replaced by green algae. In May 1974,O.agardhii reappeared in these lakes, which are influenced by varying discharges of the river Eem. Replacement of green algae byO.agardhii in the Drontermeer in 1971 was coupled with increasing Nkjehldahl, COD, BOD5, and pH. However, in the presence ofO.agardhii, inorganic nitrogen is depleted and phosphorus is sometimes depleted. Neither nitrogen nor phosphorus seem to be the limiting factor. There is a “level” of 40.000 filaments/ml in all of the lakes. This level is present in the Drontermeer and Veluwemeer where inorganic nitrogen is depleted from March to November and where PO4 −P is increasing from 0.00 in April up to 0.20 mg/l in August. In the Wolderwijd, the same inorganic nitrogen values and PO4 −P concentrations varying from 0.00 up to 0.05 mg/l are reached during summer. In the Eemmeer and Gooimeer, where the level of 40.000 individuals per ml is less constant, there is abundant inorganic nitrogen and PO4 −P. The characteristicOscillatoria lakes-the Drontermeer, the Veluwemeer and the Wolderwijd-had low concentrations of clay particles from January 1973 to August 1974. The standard deviation of the clay content values was small in all of these lakes. Of the lakes Eemmeer, Gooimeer, IJmeer, and IJsselmeer,O.agardhii occurred only in the Eemmeer and Gooimeer in the autumn during this period. All of these lakes have a greater clay content with strong fluctuations. Transparency in these lakes is related to the clay content. It is possible that a fluctuating light intensity is more harmuful to the oligophoteO.agardhii than to other species of phytoplankton. WhenO.agardhii is dominant, secchi disk values are always smaller than 40 cm. InOscillatoria water, Pmax, is found at 10 cm; the gross production in summer varies from 5 to 18 g O2/m2/day and the gross production is primarily related to the temperature and secondary to the illumination. The tentative conclusions of our investigations can be summarized as follows: discharge of phosphorus and nitrogen leads to increasing production of algae. At a certain time, light intensity is diminished to a level at which the oligophoteO.agardhii can grow. Organic wastes, especially dairy wastes, play an important, but unknown role. Light intensity fluctuations must be small to observeO.agardhii as the dominant alga in a lake. Lakes with a low and constant clay particle content are favourable for the development ofO.agardhii. Light intensity in these lakes is regulated by the density of theO.agardhii population itself. The excretion of toxic products in lakes with a prolonged persistence ofO.agardhii may be important.