Minor stars in plane graphs with minimum degree five

The weight of a subgraph $H$ in $G$ is the sum of the degrees in $G$ of vertices of $H$. The {\em height} of a subgraph $H$ in $G$ is the maximum degree of vertices of $H$ in $G$. A star in a given graph is minor if its center has degree at most five in the given graph. Lebesgue (1940) gave an approximate description of minor $5$-stars in the class of normal plane maps with minimum degree five. In this paper, we give two descriptions of minor $5$-stars in plane graphs with minimum degree five. By these descriptions, we can extend several results and give some new results on the weight and height for some special plane graphs with minimum degree five.Comment: 11 pages, 3 figure

Singular Vector Filtering Method for Disturbance Enhancement Mitigation in Active Noise Control Systems

In multichannel active noise control systems, when reference signals are correlated, the disturbance enhancement phenomenon is likely to occur, i.e., the resulting sound is enhanced instead of being reduced in some frequency bands, if the filter is designed to minimize the total energy for all frequencies. In previous works, a truncated singular value decomposition method was applied to the system autocorrelation matrix to mitigate the disturbance enhancement due to the correlation of reference signals. Some small singular values and the associated singular vectors are removed, if they are responsible for unwanted disturbance enhancement in some frequency bands. However, some of these removed singular vectors may still contribute to noise control performance in other frequency bands, thus a direct truncation will degrade the noise control performance. In the present work, through an additional filtering process, the set of singular vectors that cause the disturbance enhancement are replaced by a set of new singular vectors whose frequency responses are attenuated in the frequency band where disturbance enhancement occurs, while the frequency responses in other frequency bands are unchanged. Compared with truncation, the proposed method can maintain the performance in the noise reduction bands, while mitigating the influence in disturbance enhancement bands