New PDF release: A Bernstein theorem for special Lagrangian graphs

By Jost J., Xin Y. L.

We receive a Bernstein theorem for distinct Lagrangian graphs in for arbitrary simply assuming bounded slope yet no quantitative limit.

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The sorting of graph properties into those determined by the eigenvalues or not continues to be of research interest. 3. A little matrix theory Since the adjacency matrix of a graph is real and symmetric, we may use known results from the theory of matrices to our advantage. We will state some of them without proofs, which may be found in [14] or [20]. The first is sometimes called the principal axis theorem. 1 (Principal axis theorem) If A is a real symmetric matrix of order n, then A has n real eigenvalues and a corresponding orthonormal set of eigenvectors.

Vr are eigenvectors associated with distinct eigenvalues λ1 , λ2 , . . , λr , the set {v1 , v2 , . . , vr } is linearly independent. Eigenvectors associated with the same eigenvalue λ may be linearly dependent or independent. If we can find a basis of eigenvectors, and if P is the matrix with these eigenvectors as columns, then P−1 AP is a diagonal matrix. Conversely, if A is any square matrix and if we can find a matrix P for which P−1 AP is a diagonal matrix, then there is a basis of eigenvectors, and these eigenvectors form the columns of P; in this case, we say that A is diagonalizable.

24 Lowell Beineke, Robin Wilson and Peter Cameron r The automorphism group of the complete graph K or the null graph N is the n n symmetric group Sn . r The 5-cycle C has ten automorphisms, realized geometrically as the rotations 5 and reflections of a regular pentagon. This last group is the dihedral group D10 . More generally, Aut(Cn ) is the dihedral group D2n , for n ≥ 3. We can describe the automorphism group of a disconnected graph in terms of the group products introduced in the last section.

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A Bernstein theorem for special Lagrangian graphs by Jost J., Xin Y. L.


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