Frequent occurrence of electrooculography (EOG) artifacts leads to serious problems in interpreting and analyzing the electroencephalogram (EEG). In this paper, a robust method is presented to automatically eliminate eye-movement and eye-blink artifacts from EEG signals. Independent Component Analysis (ICA) is used to decompose EEG signals into independent components. Moreover, the features of topographies and power spectral densities of those components are extracted to identify eye-movement artifact components, and a support vector machine (SVM) classifier is adopted because it has higher performance than several other classifiers. The classification results show that feature-extraction methods are unsuitable for identifying eye-blink artifact components, and then a novel peak detection algorithm of independent component (PDAIC) is proposed to identify eye-blink artifact components. Finally, the artifact removal method proposed here is evaluated by the comparisons of EEG data before and after artifact removal. The results indicate that the method proposed could remove EOG artifacts effectively from EEG signals with little distortion of the underlying brain signals.
In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number.
More precisely, we prove the following statements by a unified approach: 1. Every graph $G$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $k$; in addition, if $G$ is $2$-connected and non-bipartite, then it contains cycles of all lengths modulo $k$. 2. For all $k\geq 3$, every $k$-connected graph contains a cycle of length zero modulo $k$. 3. Every $3$-connected non-bipartite graph with minimum degree at least $k+1$ contains $k$ cycles of consecutive lengths. 4. Every graph with chromatic number at least $k+2$ contains $k$ cycles of consecutive lengths. The 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are best possible.
Nikiforov conjectured that for a given integer k ≥ 2, any graph G of sufficiently large order n with spectral radius µ(G) ≥ µ(S n,k ) (or µ(G) ≥ µ(S + n,k )) contains C 2k+1 or C 2k+2 (or C 2k+2 ), unless G = S n,k (or G = S + n,k ), where C ℓ is a cycle of length ℓ and S n,k = K k ∨ K n−k , the join graph of a complete graph of order k and an empty graph on n − k vertices, and S + n,k is the graph obtained from S n,k by adding an edge in the independent set of S n,k . In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k ≥ 2, any graph G of sufficiently large order n with spectral radius µ(G) ≥ µ(S n,k ) (or µ(G) ≥ µ(S + n,k )) contains a cycle C ℓ with ℓ ≥ 2k + 1 (or C ℓ with ℓ ≥ 2k + 2), unless G = S n,k (or G = S + n,k ). These results also imply a result of Nikiforov given in [Theorem 2, The spectral radius of graphs without paths and cycles of specified length, LAA, 2010].
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