On Tensor Spaces for Rook Monoid Algebras

Let m, n ∈ N, and V be an m-dimensional vector space over a field F of characteristic 0. Let U = F ＋ V and Rn be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of U^×n in FRn, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of U^×n in FR^n.

Glymphatic function and its influencing factors in different glucose metabolism states

BACKGROUND Dysfunction of the glymphatic system in the brain in different stages of altered glucose metabolism and its influencing factors are not well characterized.AIM To investigate the function of the glymphatic system and its clinical correlates in patients with different glucose metabolism states,the present study employed diffusion tensor imaging along the perivascular space(DTI-ALPS)index.METHODS Sample size was calculated using the pwr package in R software.This crosssectional study enrolled 22 patients with normal glucose metabolism(NGM),20 patients with prediabetes,and 22 patients with type 2 diabetes mellitus(T2DM).A 3.0T magnetic resonance imaging was used to evaluate the function of the glymphatic system.The mini-mental state examination(MMSE)was used to assess general cognitive function.The DTI-ALPS index of bilateral basal ganglia and the mean DTI-ALPS index was calculated.Further,the correlation between DTI-ALPS and clinical features was assessed.RESULTS The left-side,right-side,and mean DTI-ALPS index in the T2DM group were significantly lower than that in the NGM group.The right-side DTI-ALPS and mean DTI-ALPS index in the T2DM group were significantly lower than those in the prediabetes group.DTI-ALPS index lateralization was not observed.The MMSE score in the T2DM group was significantly lower than that in the NGM and prediabetes group.After controlling for sex,the left-side DTI-ALPS and mean DTI-ALPS index in the prediabetes group were positively correlated with 2-hour postprandial blood glucose level;the left-side DTI-ALPS index was negatively correlated with total cholesterol and low-density lipoprotein level.The right-side DTI-ALPS and mean DTI-ALPS index were negatively correlated with the glycosylated hemoglobin level and waist-to-hip ratio in the prediabetes group.The left-side,right-side,and mean DTI-ALPS index in the T2DM group were positively correlated with height.The left-side and mean DTIALPS index in the T2DM group were negatively correlated with high-density lipoprotein levels.CONCLUSION Cerebral glymphatic system dysfunction may mainly occur in the T2DM stage.Various clinical variables were found to affect the DTI-ALPS index in different glucose metabolism states.This study enhances our understanding of the pathophysiology of diabetic brain damage and provides some potential biological evidence for its early diagnosis.

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

Suppose that E and F are separable Banach spaces, X and Y are independent symmetric E and F-valued random vectors respectively. This paper is devoted to the study of the central limit theorem for X Y in the injective and projective tensor product spaces E F and E F. Special attention is paid to l2 l2. In addition, two counter-examples are given.

CANONICAL REPRESENTATIONS AND DEGREE OF FREEDOM FORMULAE OF ORTHOGONAL TENSORS IN N-DIMENSIONAL EUCLIDEAN SPACE

In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part ofn \2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.

A Cone Constrained Convex Program:Structure and Algorithms

In this paper,we consider the positive semi-definite space tensor cone constrained convex program,its structure and algorithms.We study defining functions,defining sequences and polyhedral outer approximations for this positive semidefinite space tensor cone,give an error bound for the polyhedral outer approximation approach,and thus establish convergence of three polyhedral outer approximation algorithms for solving this problem.We then study some other approaches for solving this structured convex program.These include the conic linear programming approach,the nonsmooth convex program approach and the bi-level program approach.Some numerical examples are presented.

A NUMERICAL BOUNDARY EIGENVALUE PROBLEM FOR ELASTIC CRACKS IN FREE AND HALF SPACE

We present in this paper a numerical method for hypersingular boundary integral equations. This method was developed for planar crack problems： additional edge singularities are known to develop in that case. This paper includes a rigorous error analysis proving the convergence of our numerical scheme. Three types of examples are covered： the Laplace equation in free space, the linear elasticity equation in free space, and in half space.