Higher rank numerical ranges of normal matrices

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank numerical range $\Lambda_k(A)$ is the intersection of convex polygons with vertices $a_{j_1}, \..., a_{j_{n-k+1}}$, where $1 \le j_1 < \... < j_{n-k+1} \le n$. In this paper, it is shown that the higher rank numerical range of a normal matrix with $m$ distinct eigenvalues can be written as the intersection of no more than $\max\{m,4\}$ closed half planes. In addition, given a convex polygon ${\mathcal P}$ a construction is given for a normal matrix $A \in M_n$ with minimum $n$ such that $\Lambda_k(A) = {\mathcal P}$. In particular, if ${\mathcal P}$ has $p$ vertices, with $p \ge 3$, there is a normal matrix $A \in M_n$ with $n \le \max\left\{p+k-1, 2k+2 \right\}$ such that $\Lambda_k(A) = {\mathcal P}$.Comment: 12 pages, 9 figures, to appear in SIAM Journal on Matrix Analysis and Application

Predicting the outcome of conservative (non-surgical) voice therapy for adults with laryngeal pathologies associated with hyperfunctional voice use

Health Services Research Fund & Health Care and Promotion Fund: Research Dissemination Reports (Series 8)published_or_final_versio

Recursive Encoding and Decoding of Noiseless Subsystem and Decoherence Free Subspace

When the environmental disturbace to a quantum system has a wavelength much larger than the system size, all qubits localized within a small area are under action of the same error operators. Noiseless subsystem and decoherence free subspace are known to correct such collective errors. We construct simple quantum circuits, which implement these collective error correction codes, for a small number $n$ of physical qubits. A single logical qubit is encoded with $n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive relations among the subspaces employed in noiseless subsystem and decoherence free subspace play essential r\^oles in our implementation. The recursive relations also show that the number of gates required to encode $m$ logical qubits increases linearly in $m$.Comment: 9 pages, 3 figure

A New Hybrid Descent Method with Application to the Optimal Design of Finite Precision FIR Filters

In this paper, the problem of the optimal design of discrete coefficient FIR filters is considered. A novelhybrid descent method, consisting of a simulated annealing algorithm and a gradient-based method, isproposed. The simulated annealing algorithm operates on the space of orthogonal matrices and is used tolocate descent points for previously converged local minima. The gradient-based method is derived fromconverting the discrete problem to a continuous problem via the Stiefel manifold, where convergence canbe guaranteed. To demonstrate the effectiveness of the proposed hybrid descent method, several numericalexamples show that better discrete filter designs can be sought via this hybrid descent method

A note on the realignment criterion

For a quantum state in a bipartite system represented as a density matrix, researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state. We obtain bounds for elementary symmetric functions of singular values of realignment matrices. This answers some open problems proposed by Lupo, Aniello, and Scardicchio. As a consequence, we show that the proposed scheme by these authors for testing separability would not work if the two subsystems of the bipartite system have the same dimension.Comment: 11 pages, to appear in Journal of Physics A: Mathematical and Theoretica

Barriers and strategies for building information modelling implementation: a comparative study between New Zealand and China

Building Information Modelling (BIM) is a sharing platform that can present a parametric 3 D model with various project information in the form of a digital display. In recent years, BIM adoption has become increasing globally as the Architectural, Engineering, Construction (AEC) industry has recognised its benefits. Meanwhile, many challenges of BIM adoption in different countries have been well documented. To address the gap in literature, this study examines the differences and similarities of BIM adoption between New Zealand and China. A questionnaire was conducted across the two countries to investigate the barriers and strategies for the implementation of BIM. Data from 146 respondents was collected in New Zealand and China. The result shows that there is a difference in the perception of Knowledge Barrier, Technology Barrier, Internal Strategy and External Strategy (Legal/Technology viewpoint) between New Zealand professionals and Chinese professionals. The differences identified offer important implications for government agencies to promote BIM implementation and for BIM service providers to better target the end-users.Publishe

An Efficient Alignment Algorithm for Searching Simple Pseudoknots over Long Genomic Sequence

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