Tensor product q-Bernstein polynomials

An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B _n,m (f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed. AMS subject classification (2000)  65D17     

A scalable optimization approach for fitting canonical tensor decompositions

Tensor decompositions are higher‐order analogues of matrix decompositions and have proven to be powerful tools for data analysis. In particular, we are interested in the canonical tensor decomposition, otherwise known as CANDECOMP/PARAFAC (CP), which expresses a tensor as the sum of component rank‐one tensors and is used in a multitude of applications such as chemometrics, signal processing, neuroscience and web analysis. The task of computing CP, however, can be difficult. The typical approach is based on alternating least‐squares (ALS) optimization, but it is not accurate in the case of overfactoring. High accuracy can be obtained by using nonlinear least‐squares (NLS) methods; the disadvantage is that NLS methods are much slower than ALS. In this paper, we propose the use of gradient‐based optimization methods. We discuss the mathematical calculation of the derivatives and show that they can be computed efficiently, at the same cost as one iteration of ALS. Computational experiments demonstrate that the gradient‐based optimization methods are more accurate than ALS and faster than NLS in terms of total computation time. Copyright © 2011 John Wiley & Sons, Ltd.     

欧拉运动学方程的另一种推导方法

欧拉运动学方程的推导,关键是确定本体坐标系与空间坐标系之间的转换关系.通过引进方位矢量可以较轻易地得到反映两坐标系转换关系的转换张量,从而能清晰直观地推导出欧拉运动学方程,而这是有别于传统的推导方法的.     

Relative weak injectivity of operator system pairs

The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg [11] proved that the C^?${\mathrm{C}}^{?}$-algebra C^?(_F∞)${\mathrm{C}}^{?}({\mathbb{F}}_{\infty })$ of the free group _F∞${\mathbb{F}}_{\infty }$ on countably many generators characterises relative weak injectivity for pairs of C^?${\mathrm{C}}^{?}$-algebras by means of the maximal tensor product. One of the main results of this paper shows that C^?(_F∞)${\mathrm{C}}^{?}({\mathbb{F}}_{\infty })$ also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products  and .     

Controllability of Boolean control networks with impulsive effects and forbidden states

This paper investigates the controllability of Boolean control networks (BCNs) with impulsive effects while avoiding certain forbidden states. Using semi‐tensor product of matrices, the BCNs with impulsive effects can be converted into impulsive discrete‐time systems. Then, some necessary and sufficient conditions for the controllability are obtained. It is interesting to find that impulsive effects play an important role in the controllability of BCNs. Finally, an example is given to show the efficiency of the obtained results. Copyright © 2013 John Wiley & Sons, Ltd.     

Prescribing Curvature Problems on the Bakry-Emery Ricci Tensor of a Compact Manifold with Boundary

t The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Amp~re type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.     

NOTES ON THE RESCALED SASAKI TYPE METRIC ON THE COTANGENT BUNDLE

Let （M, g） be an n-dimensional Riemannian manifold and T＊M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compati-ble paracomplex structures on T＊M. Second, we construct locally decomposable Golden Riemannian structures on T＊M . Finally we investigate curvature properties of T＊M .     

A Note on n-Perinormal Operators

The study of operators satisfying

σ_ja(T)=σa(T)${\sigma }_{ja}\left(T\right)={\sigma }_a\left(T\right)$

is of significant interest. Does

σ_ja(T)=σa(T)${\sigma }_{ja}\left(T\right)={\sigma }_a\left(T\right)$

for n-perinormal operator

T∈B(H)?$T\in B\left(H\right)?$

This question was raised by Mecheri and Braha [Oper. Matrices 6 (2012), 725–734]. In the note we construct a counterexample to this question and obtain the following result: if T is a n-perinormal operator in B(H), then

σ_ja(T)\{0}=σa(T)\{0}.${\sigma }_{ja}\left(T\right)\backslash \left\{0\right\}={\sigma }_a\left(T\right)\backslash \left\{0\right\}.$

We also consider tensor product of n-perinormal operators.     

An Ore-type condition for arbitrarily vertex decomposable graphs

Let G be a graph of order n and r, 1≤r≤n, a fixed integer. G is said to be r-vertex decomposable if for each sequence (n₁,…,n_r) of positive integers such that n₁+?+n_r=n there exists a partition (V₁,…,V_r) of the vertex set of G such that for each i∈{1,…,r}, V_i induces a connected subgraph of G on n_i vertices. G is called arbitrarily vertex decomposable if it is r-vertex decomposable for each r∈{1,…,n}.In this paper we show that if G is a connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n−3, then G is arbitrarily vertex decomposable or isomorphic to one of two exceptional graphs. We also exhibit the integers r for which the graphs verifying the above degree-sum condition are not r-vertex decomposable.     

共形平坦黎曼流形上的Schouten张量

M是一个紧致的局部共形平坦黎曼流形,其上定义的Schouten张量是一个Codazzi张量.本文借助这个Codazzi张量引入Cheng和Yau的自伴算子,从而获得了局部共形平坦流形上的一些性质,改进了已有的结论.