On Pointwise Product Vector Measure Duality

This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.

A Fast Estimation Method for Direction of Arrival Using Tripole Vector Antenna

The tripole vector antenna comprises three orthogonal dipole antennas, so it could completely capture all the electric field of the incident electromagnetic (EM) wave. Then, the electric field information could be used to estimate the direction of arrival (DOA) of the EM wave if two conditions are satisfied. One is that there exists only one single EM wave in space. The other is that the EM wave is elliptically or circularly polarized. The new estimation method obtains two snapshot vectors through the output of a tripole antenna and computes their cross-product vector. Furthermore, the direction of the cross-product vector is used to estimate the DOA of the EM wave directly. We analyze the statistical characteristics of the DOA estimation error to prove that the new scheme is an asymptotic unbiased estimation. Furthermore, unlike the existing Multiple Signal Classification (MUSIC)-based algorithms, the proposed approach only need one tripole vector antenna instead of an antenna array. Meanwhile, the new method also outperforms existing MUSIC-based algorithms in the term of computational complexity. Finally, the performance and advantages of the proposed method are verified by numerical simulations.

Vector-Valued Forms

This chapter studies vector-valued forms. Ordinary differential forms have values in the field of real numbers. This chapter allows differential forms to take values in a vector space. When the vector space has a multiplication, for example, if it is a Lie algebra or a matrix group, the vector-valued forms will have a corresponding product. Vector-valued forms have become indispensable in differential geometry, since connections and curvature on a principal bundle are vector-valued forms. All the vector spaces will be real vector spaces. A k-covector on a vector space T is an alternating k-linear function. If V is another vector space, a V-valued k-covector on T is an alternating k-linear function.

Hierarchical approach for deriving a reproducible unblocked LU factorization

We propose a reproducible variant of the unblocked LU factorization for graphics processor units (GPUs). For this purpose, we build upon Level-1/2 BLAS kernels that deliver correctly-rounded and reproducible results for the dot (inner) product, vector scaling, and the matrix-vector product. In addition, we draw a strategy to enhance the accuracy of the triangular solve via iterative refinement. Following a bottom-up approach, we finally construct a reproducible unblocked implementation of the LU factorization for GPUs, which accommodates partial pivoting for stability and can be eventually integrated in a high performance and stable algorithm for the (blocked) LU factorization.

Characterization and properties of weakly optimal entanglement witnesses

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of $W^{wopt}=\sigma-c_{\sigma}^{max} I$, where $c_{\sigma}^{max}$ is a non-negative number and $I$ is the identity matrix. We show the relation between the weakly optimal witness $W^{wopt}$ and the eigenvalues of the separable states $\sigma$. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R) (2013)], and we examine their geometric properties.