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Quantity II Geometry.- a few Algebro-Geometrical points of the Newton allure Theory.- Smoothing of a hoop Homomorphism alongside a Section.- Convexity and Loop Groups.- The Jacobian Conjecture and Inverse Degrees.- a few Observations at the Infinitesimal interval relatives for normal Threefolds with Trivial Canonical Bundle.- On Nash Blowing-Up.- preparations of strains and Algebraic Surfaces.- average capabilities on definite Infinitedimensional Groups.- Examples of Surfaces of common variety with Vector Fields.- Flag Superspaces and Supersymmetric Yang-Mills Equations.- Algebraic Surfaces and the mathematics of Braids, I.- in the direction of an Enumerative Geometry of the Moduli house of Curves.- Schubert kinds and the range of Complexes.- A Crystalline Torelli Theorem for Supersingular K3 Surfaces.- Decomposition of Toric Morphisms.- an answer to Hironaka’s Polyhedra Game.- at the Superpositions of Mathematical Instantons.- what number Kahler Metrics Has a K3 Surface?.- at the challenge of Irreducibility of the Algebraic procedure of Irreducible airplane Curves of a Given Order and Having a Given variety of Nodes.

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**Sample text**

2) where f E OG and PTL(T) is the orthogonal projection of L(G) onto L(T). The norm is that corresponding to the G-invariant metric on L( G). Then the required mapping is f --. (p(f), E(f)), and will be called the moment map, for reasons to be explained later. Our main result is then: The image of OG under the moment map is the convex Theorem 1. II 11 2 restricted to the integer lattice in L(T). Moreover, we will show that the image of the closure of each Bruhat cell in OG is a compact convex sub-polyhedron of the image OG.

Using the almost eomplex structure J on 11 1 , we obtain an action of the complexification 1~ X 0* on 11 1 . We can make this adion explicit on the algebraic loops, if we use the identiftcation natg = M~ 1 a jP. In fact, 0* acts on itself by multiplication, and 1~ acts on G c by conjugation. Thus, both groups act naturally on M~lg and the two actions commute and preserve P. 1. The action of Tc X 0* on 11 1 preserves each Bruhat cell C>.. It therefore also preserves the Bruhat manifolds P>.. To obtain the corresponding facts for the Birkhoff decomposition, observe first that from their description in' §2, it is clear that the Birkhoff manifolds 1>.

Under the moment map. It is Lhc convex hull of Lhe graph of %II 11 2 restricted to CONVU:XlTY AND LOOP GROUPS 41 a finite subset of A. Every integer lattice point occurs so it is obvious that the image of naly is contained in the polyhedron described in theorem 1. To prove equality, it is enough to prove that the image is convex, and this follows from a simple fact about the Bruhat decomposition, namely that for any two closed Bruhat cells there is another which contains them both. G) of each stratum of the Birkhoff decomposition.