The Projection Method for Reaching Consensus and the Regularized Power Limit of a Stochastic Matrix

In the coordination/consensus problem for multi-agent systems, a well-known condition of achieving consensus is the presence of a spanning arborescence in the communication digraph. The paper deals with the discrete consensus problem in the case where this condition is not satisfied. A characterization of the subspace $T_P$ of initial opinions (where $P$ is the influence matrix) that \emph{ensure} consensus in the DeGroot model is given. We propose a method of coordination that consists of: (1) the transformation of the vector of initial opinions into a vector belonging to $T_P$ by orthogonal projection and (2) subsequent iterations of the transformation $P.$ The properties of this method are studied. It is shown that for any non-periodic stochastic matrix $P,$ the resulting matrix of the orthogonal projection method can be treated as a regularized power limit of $P.$Comment: 19 pages, 2 figure

Petrography, mineralogy and SIMS U-Pb geochronology of 1.9–1.8 Ga carbonatites and associated alkaline rocks of the Central-Aldan magnesiocarbonatite province (South Yakutia, Russia)

The N-S trending Central-Aldan magnesiocarbonatite province is located in the Aldan-Stanovoy shield (South Yakutia, Russia). Several apatite-dolomitic carbonatite occurrences were studied: Seligdar, Muostalaah, Ust-Chulman and Birikeen. Mineralogical and petrographic investigations indicate intense hydrothermal-metasomatic alteration and metamorphism, which are reflected in the evolution of the mineral parageneses. The primary minerals are fluorapatite, magnetite, ilmenite, dolomite, K-feldspar, phlogopite and accessory zircon, titanite, baddeleyite and thorite. The hydrothermal-metasomatic minerals are quartz, calcite and siderite aggregates with haematite, monazite-(Ce), xenotime-(Y), rutile-(Nb), barite-(Sr), anhydrite, ancylite-(Ce) and rare sulphide mineral phases. Alkaline rocks associated with the Muostalaah complex, were also studied. The following U-Pb ages have been obtained (Ma): 1930 ± 7 for Muostalaah alkaline basic rocks, 1906 ± 6 for Muostalaah carbonatites, and 1880 ± 13 and 1878 ± 17 for Seligdar and Ust-Chulman carbonatites, respectively