Search

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L\"owner and the minus partial orders. Motivated by applications in statistics we study these partial orders on $H_{n}^{+}(\mathbb{R})$. We describe the form of all surjective maps on $H_{n}^{+}(\mathbb{R})$, $n>1$, that preserve the L\"owner partial order in both directions. We present an equivalent definition of the minus partial order on $H_{n}^{+}(\mathbb{R})$ and also characterize all surjective, additive maps on $H_{n}^{+}(\mathbb{R})$, $n\geq3$, that preserve the minus partial order in both directions., Comment: 17 pages

The genus Archaeoellipsoides Horodyski & Donaldson comprises large (up to 135 micrometers long) ellipsoidal and rod-shaped microfossils commonly found in silicified peritidal carbonates of Mesoproterozoic age. Based on morphometric and sedimentary comparisons with the akinetes of modern bloom-forming Anabaena species, Archaeoellipsoides is interpreted as the fossilized remains of akinetes produced by planktic heterocystous cyanobacteria. These fossils set a minimum date for the evolution of derived cyanobacteria capable of marked cell differentiation, and they corroborate geochemical evidence indicating that atmospheric oxygen levels were well above 1% of present day levels 1,500 million years ago.

Populations of Polybessurus bipartitus Fairchild ex Green et al., a large morphologically distinctive microfossil, occur in silicified carbonates of the Upper Proterozoic (700-800 Ma) Limestone-Dolomite "Series," central East Greenland. Large populations of well-preserved individuals permit reconstruction of P. bipartitus as a coccoidal unicell that "jetted" upward from the sediment by the highly unidirectional secretion of extracellular mucopolysaccharide envelopes. Reproduction by baeocyte formation is inferred on the basis of clustered envelope stalks produced by small cells. Sedimentological evidence indicates that P. bipartitus formed surficial crusts locally within a shallow peritidal carbonate platform. Among living microorganisms a close morphological, reproductive, and behavioral counterpart to Polybessurus is provided by populations of an as yet underscribed cyanobacterium found in coastal Bahamian environments similar to those in which the Proterozoic fossils occur. In general morphology and "jetting" behavior, this population resembles species of the genus Cyanostylon, Geitler (1925), but reproduces via baeocyte formation. Polybessurus is but one of the more than two dozen taxa in the richly fossiliferous biota of the Limestone-Dolomite "Series." This distinctive population, along with co-occurring filamentous cyanobacteria and other microfossils, contributes to an increasingly refined picture of ecological heterogeneity in late Proterozoic oceans.

Diverse microorganisms ranging from cyanobacteria to eukaryotic algae and fungi live endolithically within ooids, hardgrounds and invertebrate shells on the present-day sea floor. These organisms are involved in the mechanical destruction of carbonates, and are useful ecological indicators of water depth and pollution. The Phanerozoic history of microbial endoliths has been elucidated through the study of microborings (the trace fossils of endolithic microorganisms) and rare cellularly preserved individuals, but nothing was known of the possible Precambrian evolution of comparable microorganisms until Campbell documented the occurrence of microborings in late Proterozoic ooids from central East Greenland. We now report the discovery of large populations of organically preserved endolithic microorganisms in silicified pisolites from 700-800-Myr-old Limestone-Dolomite Series of East Greenland. This fossil assemblage is significant for three reasons: (1) It confirms the prediction that oolites, pisolites and hardgrounds--the substrates for pre-Phanerozoic endoliths--provide a hitherto poorly explored but rewarding set of environments into which the search for early microfossils must be broadened; (2) the assemblage is diverse, containing about 12 taxa of morphologically distinct and previously unknown endolithic cyanobacteria, plus associated epilithic and interstitial populations; and (3) at least six of the fossil populations are indistinguishable in morphology, pattern of development, reproductive biology and inferred ecology from distinctive cyanobacterial species that bore ooids today in the Bahama Banks.

Distribution, external morphology, texture, and microbial composition of microbial mats in Hamelin Pool, Shark Bay, Western Australia, have been studied and reviewed along a composite representative profile starting from the permanently submerged zone, across the zones of periodic flooding, toward permanently emerged land and coastal dunes. The following nine types of algal mats have been recognized: colloform, gelatinous, smooth, pincushion, tufted, mamillate, film, reticulate, and blister. Solar ponds represent a particular environment. The mat types represent microbial communities that are characterized by one or more dominant microorganisms. The colonization and stabilization of loose sediment is carried out by a microbial assemblage of generalists that prepare the ground for later replacement and succession by specialized microflora. Lithification of microbial mats takes place periodically, mainly during the austral summer. This process is destructive for the microbial community but increases the preservation potential of the stromatolitic structures.

Microfossils which are preserved as shrivelled kerogenous residues provide little information about cellular organization and almost none about the metabolic properties of the organisms. The distinction between prokaryotic vs eukaryotic, and phototrophic vs chemo- and organotrophic fossil microorganisms rests entirely on morphological comparisons with recent counterparts. The residual nature of the microbial fossil record promotes the conclusion that it must be biased toward (a) most abundant organisms, (b) those most resistant to degradation, and (c) those inhabiting environments with high preservation potential e.g., stromatolites. These criteria support the cyanophyte identity of most Precambrian microbial fossils on the following grounds: (1) as primary producers they dominate prokaryotic communities in modern extreme environments, e.g., intertidal zone; (2) several morphological counterparts of modern cyanophytes and microbial fossils have been established based on structure, cell division patterns and degradation sequences. The impact of anaerobic and oxygenic microbial photosynthesis on the evolution of Precambrian environments is discussed.