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This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

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]

In this paper, we first give the definition of possibly non-unital function system, which is a characterization of the self-adjoint subspace of the space of continuous functions on a compact Hausdorff space with the induced order and norm structure. Similar to operator system case, we define the unitalization of a possibly non-unital function system. Then we construct two possibly non-unital operator system structures on a given possibly non-unital function system, which are the analogues of minimal and maximal operator spaces on a normed space. These two structures have many interesting relations with the minimal and maximal operator system structures on a given function system. [ABSTRACT FROM AUTHOR]

Stress is the imbalance of homeostasis, which can be sensed even at the subcellular level. The stress-sensing capability of various organelles including the endoplasmic reticulum (ER) has been described. It has become evident that acute or prolonged ER stress plays an important role in many human diseases; especially those involving organs/tissues specialized in protein secretion. This article summarizes the emerging role of ER stress in diverse human pathophysiological conditions such as carcinogenesis and tumor progression, cerebral ischemia, plasma cell maturation and apoptosis, obesity, insulin resistance, and type 2 diabetes. Certain components of the ER stress response machinery are identified as biomarkers of the diseases or as possible targets for therapeutic intervention. [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]