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We investigate minimal, compact, square-closed subsets of the unit circle by identifying them with subsets of [0,1] and with sets of infinite words on two symbols; in particular, such sets arising from substitution maps. We consider problems on finiteness, recognizability, square roots, density and measure. Properties of the free semigroup on two symbols play a significant role in the analysis.

Let S be the semigroup with identity, generated by x and y, subject to y being invertible and yx = xy2. We study two Banach algebra completions of the semigroup algebra ℂS. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that ℂS is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for ℂS is finite dimensional and hence that ℂS has a separating family of such modules.

It is shown that the complex semigroup algebra of a free monoid of rank at least two is *-primitive, where * denotes the involution on the algebra induced by word-reversal on the monoid.

Reworking of Devonian, Carboniferous and Permian palynomorphs into the Lower Triassic of western Canada, Yukon, the Canadian Arctic Archipelago and Alaska has been documented by a number of workers. The phenomenon occurs in other northern circumpolar localities, such as East Greenland and the Barents Sea, and was probably widespread. Examples are given from western and eastern Europe, Russia, the Middle East, Pakistan, China, Brazil and Australia. The abundance and diversity of reworked palynomorphs from a number of stratigraphic units of different ages into the Lower Triassic is an important palynostratigraphic phenomenon of chronostratigraphic value. It may be the result of a major regional regression in the late Permian, followed by a widespread marine transgression in the Early Triassic, or it may be due to tectonic activity resulting in eustatic sea-level rise. Whatever the mechanism, the advancing transgression resulted in erosion of exposed Devonian, Carboniferous and Permian rocks, but by...

The statement of the inequality is actually a little more precise, but we shall discuss that later. Carleman used the inequality as part of a new proof of a delightful theorem of Denjoy about quasi-analytic functions. We shall not pursue that application here since we are interested in the inequality as a theorem about infinite series. We wish to expound in detail some of the many proofs of this theorem. We remark in passing, with approval, that editors of mathematical journals in the 1920s were happy to publish many different proofs of the same theorem. There are many known results about the convergence (or divergence) of series obtained by linear transformations of a nonnegative sequence {an }. For example, if Ain is the arithmetic mean of a1, a2, ... , an, then Y An, always diverges unless {an } is the zero sequence (for the sum dominates part of the harmonic series). The arithmeticgeometric mean inequality (AGM) asserts that y, < An, with equality if and only if the sequence {an} is constant. So the AGM-inequality must be strict when E an is convergent and nonzero. It is perhaps surprising a priori that the AGM-inequality is so

Fifteen U–Pb (zircon) radiometric age determinations have been made on igneous rocks of Middle Devonian to Early Carboniferous age from the southern margin of the Magdalen basin in Cape Breton Island and northern mainland Nova Scotia. Volcanic rocks interbed with early rift-basin sedimentary rocks with some palynological biostratigraphy; dated intrusive rocks cut these sedimentary units. Our biostratigraphically constrained ages are in close agreement with the current Devonian time scale. Combined with previously published data, the age determinations show that igneous activity occurred in four pulses: Middle Devonian (390–385 Ma), early Late Devonian (375–370 Ma), latest Devonian to early Tournaisian (365–354 Ma), and late Tournaisian to early Visean (ca. 339 Ma). Middle Devonian (385–389 Ma) volcanic rocks are confined to the Guysborough Group. The Fisset Brook Formation (basalt and minor rhyolite) in the type area and elsewhere in Cape Breton Island and northern mainland Nova Scotia is Late Devonian (ca. 373 Ma), whereas the biostratigraphically distinct succession at Lowland Cove is younger (365 Ma). These Late Devonian rocks are synchronous with plutonism in the Cape Breton Highlands and the Meguma terrane. In the Cobequid Highlands, rhyolite of the Fountain Lake Group was synchronous with Horton Group deposition and with widespread granite plutons (362–358 Ma) emplaced during shear on the Cobequid fault zone. The overlying Diamond Brook Formation basalts are slightly younger (355 Ma). Late Tournaisian – early Visean mafic intrusions and minor basalt occur along the Cobequid – Chedabucto fault zone and in a belt from southern New Brunswick through Prince Edward Island to southwestern Cape Breton Island.

We study three extremal Banach algebras: (a) generated by two hermitian unitaries; (b) generated by an element of norm 1 all of whose odd positive powers are hermitian; (c) generated by an element of norm 1 all of whose even positive powers are hermitian. In all three cases the numerical range is found for various elements. The second algebra is shown to be isometrically isomorphic to a subalgebra of the first. The third algebra is identified with a space of functions.

Let G be the free product of groups A and B, where |A|≥3 and |B|≥2. We construct faithful, irreducible *-representations for the group algebras ℂ[G] and ℓ1(G). The construction gives a faithful, irreducible representation for F[G] when the field F does not have characteristic 2.

Let denote a subring of the complex field that contains 1 and is closed under complex conjugation. It is shown that, with respect to the involution induced by word-reversal, the algebra over of a free monoid admits a trace and a separating family of star matrix representations. From the existence of a trace it is deduced that the aforementioned involution is special, in the sense of Easdown and Munn. Similar results hold for the algebra over of a free monoid with involution.

Late Devonian (late Famennian) miospores have been found in the lowermost 7.3 m of the Horton Group on Harding Brook in the type area, Windsor Subbasin, Nova Scotia, below Tournaisian miospores of the Emphanisporites rotatus – Indotriradites explanatus Zone. Extending the age of the Horton Group in the type area down into the latest Devonian shows that the lowest beds of the ~ 1000 m thick group are coeval with latest Devonian rocks elsewhere in the Maritimes Basin that have been excluded from the Horton Group by some authors. Evidence presented here favours the argument that Late Devonian rocks lithologically similar to the Horton Group, deposited on the Acadian unconformity, should be included in that group. Miospore evidence indicates a minimum age of about 355 Ma for exhumation of the northeastern part of the South Mountain Batholith.

A class of extremal Banach algebras has Arens irregular multiplication.

Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u^2=v^2=1 . We show that: Ea(u,v)={f+gu:f,g\in C(T)} , where [ ] is the unit circle; Ea(u,v) is C^* -equivalent to C^*({\cal G}) , where {\cal G} is the infinite dihedral group; most of the hermitian elements k of Ea(u,v) have the property that k^n is hermitian for all odd n but for no even n ; any two hermitian words in {\cal G} generate an isometric copy of Ea(u,v) in Ea(u,v) .

Former general practitioner Westray, New Deer, and Turriff (b Aberdeen 1924; q Aberdeen 1949), d 12 September 2002. Archie entered singledhanded general practice in 1952 on the island of Westray, Orkney. He moved to New Deer in 1957, latterly assisted by his GP wife, Margaret. Both moved to a group practice in Turriff in 1966, where they helped secure the cottage hospital with the development of an attached health centre. Archie retired to Edinburgh in 1984, eventually moving to his native Deeside to enjoy his love of the mountains and pursue his interest in antiquarian books and local and family history. Predeceased by a son, he leaves Margaret; two children; and six grandchildren.

Two paleokarsts, different in age, character, and origin, occur in dolostones of the Lower Ordovician Beekmantown Group of eastern Ontario, well within the interior of the Laurentian paleoplatform. Surficial to shallow subsurface ( p CO 2 and H 2 S concentrations arising from mixing of continentally derived pore waters with brines derived from dissolution of Beekmantown evaporites. Compared to the region9s geologic history, a pre–late Paleozoic age for formation of the regional paleokarst and mineralization is likely. The two platform-interior paleokarsts demonstrate unexpected links between tectonism, changes in paleohydrological patterns, and porosity development well inboard of the Appalachian orogen.