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In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Z-eigenvalue and Z-eigenvector. We also prove results on convergence of the cyclic Jacobi (or Jacobi CoM2) algorithm. [ABSTRACT FROM AUTHOR]

NIDDM is characterized by progressive insulin resistance and the failure of insulin-producing pancreatic beta cells to compensate for this resistance. Hyperinsulinemia, inflammation, and prolonged activation of the insulin receptor (INSR) have been shown to induce insulin resistance by decreasing INSR substrate (IRS) protein levels. Here we describe a role for SOCS7 in regulating insulin signaling. Socs7-deficient mice exhibited lower glucose levels and prolonged hypoglycemia during an insulin tolerance test and increased glucose clearance in a glucose tolerance test. Six-month-old Socs7-deficient mice exhibited increased growth of pancreatic islets with mildly increased fasting insulin levels and hypoglycemia. These defects correlated with increased IRS protein levels and enhanced insulin action in cells lacking SOCS7. Additionally, SOCS7 associated with the INSR and IRS1--molecules that are essential for normal regulation of insulin action. These data suggest that SOCS7 is a potent regulator of glucose homeostasis and insulin signaling. [ABSTRACT FROM AUTHOR]