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Recent Excluded Minor Theorems
 Surveys in Combinatorics, LMS Lecture Note Series
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.
Functional Unit Maps for DataDriven Visualization of HighDensity EEG Coherence
"... Synchronous electrical activity in different brain regions is generally assumed to imply functional relationships between these regions. A measure for this synchrony is electroencephalography (EEG) coherence, computed between pairs of signals as a function of frequency. Existing highdensity EEG coh ..."
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Synchronous electrical activity in different brain regions is generally assumed to imply functional relationships between these regions. A measure for this synchrony is electroencephalography (EEG) coherence, computed between pairs of signals as a function of frequency. Existing highdensity EEG coherence visualizations are generally either hypothesisdriven, or datadriven graph visualizations which are cluttered. In this paper, a new method is presented for datadriven visualization of highdensity EEG coherence, which strongly reduces clutter and is referred to as functional unit (FU) map. Starting from an initial graph, with vertices representing electrodes and edges representing significant coherences between electrode signals, we define an FU as a set of electrodes represented by a clique consisting of spatially connected vertices. In an FU map, the spatial relationship between electrodes is preserved, and all electrodes in one FU are assigned an identical gray value. Adjacent FUs are visualized with different gray values and FUs are connected by a line if the average coherence between FUs exceeds a threshold. Results obtained with our visualization are in accordance with known electrophysiological findings. FU maps can be used as a preprocessing step for conventional analysis.
Computational Discovery in Pure Mathematics
"... Abstract. We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body ..."
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Abstract. We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body of knowledge in pure mathematics. We discuss to what extent the output from certain programs can be considered a discovery in pure mathematics. This enables us to assess the state of the art with respect to Newell and Simonâ€™s prediction that a computer would discover and prove an important mathematical theorem. 1