Download Triangular Products of Group Representations and Their by Samuel M. Vovsi (auth.) PDF

By Samuel M. Vovsi (auth.)

The building thought of in those notes relies on an easy thought. permit (A, G ) and (B, G ) be crew representations, for definiteness devoted and finite­ 1 2 dimensional, over an arbitrary box. we will say trustworthy illustration (V, G) is an extension of (A, G ) through (B, G ) if there's a G-submodule W of V such that 1 2 the clearly coming up representations (W, G) and (V/W, G) are isomorphic, modulo their kernels, to (A, G ) and (B, G ) respectively. 1 2 query. between all of the extensions of (A, G ) by way of (B, G ), does there exist 1 2 one of these "universal" extension which includes an isomorphic reproduction of the other one? the answer's within the affirmative. rather, permit dim A = m and dim B = n, then the teams G and G could be regarded as matrix teams of levels m and n 1 2 respectively. If (V, G) is an extension of (A, G ) by way of (B, G ) then, below yes 1 2 collection of a foundation in V, all components of G are represented by way of (m + n) x (m + n) mat­ rices of the shape (*) ~1-~ ~-J lh I g2 I .

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Sometimes the heavy metals come from the corrosion of internally unprotected metallic water pipelines. Heavy metals can cause a variety of adverse health effects when people are exposed to it at levels above maximum contamination level for relatively short period of time (Isaac et al. 1997). Corrosion Due to Microorganisms The role of microorganisms in the corrosion of metals in the pipeline systems has been recognized in recent times. Microorganisms may influence corrosion by affecting the rate of anodic or cathodic activity producing corrosive end product and making the water become contaminated.

But mercury organic complexes can be attacked by bacteria to give methyl mercuric compound (pH 7) (CH3 -HgCH3 ). We still have much to learn about quantitative aspects of the mercury cycle. Methyl mercury can enter lake and river food chain, from water to plants to fish, to bird and to human beings, becoming concentrated at each step. Methyl mercury is particularly toxic to animals, because it can readily pass the blood–brain barrier, causing injury to cerebellum and cortex.

4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Water is one of the essential companion of life on earth and the nature has iterated the vitality of water for life. It is very appropriate to say that existence of life on Earth is largely owed to the presence of water, but it is fast becoming a scare commodity in most part of the world.

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