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Invariant neutral subspaces for Hamiltonian matrices
No full textHamiltonian matrices with respect to a nondegenerate skewsymmetric or skewhermitian indefinite inner product in finite dimensional real, complex, or quaternion vector spaces are studied. Subspaces that are simultaneously invariant for the matrices and neutral in the indefinite inner product are of special interest. The dimension of maximal (by inclusion) such subspaces is identified in terms of the canonical forms and sign characteristics. Criteria for uniqueness of maximal invariant neutral subspaces are given. The important special case of invariant Lagrangian subspaces is treated separately. Comparisons are made between real, complex, and quaternion contexts; for example, for complex Hamiltonian matrices with respect to a nondegenerate skewhermitian inner product in a finite dimensional complex vector space, the (complex) dimension of (complex) maximal invariant neutral subspaces is compared to the (quaternion) dimension of (quaternion) maximal invariant neutral subspaces, and necessary and sufficient conditions are given for the two dimensions to coincide (this is not always the case)
Quadratic forms on graphs with application to minimizing the least eigenvalue of signless Laplacian over bicyclic graphs
No full textGiven a graph and a vector defined on the graph, a quadratic form is defined on the graph depending on its edges. In order to minimize the quadratic form on trees or unicyclic graphs associated with signless Laplacian, the notion of basic edge set of a graph is introduced, and the behavior of the least eigenvalue and the corresponding eigenvectors is investigated. Using these results a characterization of the unique bicyclic graph whose least eigenvalue attains the minimum among all non-bipartite bicyclic graphs of fixed order is obtained
Spatio-Temporal Ecological and Evolution Dynamics in Natural Butterfly Populations (2014 Field Season)
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Minimum vector rank and complement critical graphs
No full textGiven a graph G, a real orthogonal representation of G is a function from its set of vertices to Rd such that two vertices are mapped to orthogonal unit vectors if and only if they are not neighbors. The minimum vector rank of a graph is the smallest dimension d for which such a representation exists. This quantity is closely related to the minimum semidefinite rank of G, which has been widely studied. Considering the minimum vector rank as an analogue of the chromatic number, this work defines critical graphs as those for which the removal of any vertex decreases the minimum vector rank; and complement critical graphs as those for which the removal of any vertex decreases the minimum vector rank of either the graph or its complement. It establishes necessary and sufficient conditions for certain classes of graphs to be complement critical, in the process calculating their minimum vector rank. In addition, this work demonstrates that complement critical graphs form a sufficient set to prove the Graph Complement Conjecture, which remains open
Shadowing property of random linear cocyles
No full textIn this brief note, it is shown that the random hyperbolicity of a random linear cocycle is equivalent to having the Lipschitz shadowing property
