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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.
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.
Proof of Conway's Lost Cosmological Theorem
, 1997
"... John Horton Conway’s Cosmological Theorem about sequences like 1, 11, 21, 1211, 111221, 312211,..., for which no extant proof existed, is given a new proof, this time hopefully for good. ..."
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John Horton Conway’s Cosmological Theorem about sequences like 1, 11, 21, 1211, 111221, 312211,..., for which no extant proof existed, is given a new proof, this time hopefully for good.
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 Simon’s prediction that a computer would discover and prove an important mathematical theorem. 1
Annals of the Japan Association for Philosophy of Science Vol.21 (2013) 21～35 21 Mathematical Knowledge: Motley and Complexity of Proof
"... Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and procedures would be patently unrecognizable a century, certainly ..."
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Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and procedures would be patently unrecognizable a century, certainly two centuries, ago. What has been called “classical mathematics ” has indeed seen its day. With its richness, variety, and complexity any discussion of the nature of modern mathematics cannot but accede to the primacy of its history and practice. As I see it, the applicability of mathematics may be a driving motivation, but in the end mathematics is autonomous. Mathematics is in a broad sense selfgenerating and selfauthenticating, and alone competent to address issues of its correctness and authority. What brings us mathematical knowledge? The carriers of mathematical knowledge are proofs, more generally arguments and constructions, as embedded in larger contexts.1 Mathematicians and teachers of higher mathematics know this, but it should be said. Issues about competence and intuition can be raised as well as factors of knowledge involving the general dissemination of analogical or inductive reasoning
The Reduced Genus of a Multigraph
, 1999
"... . We dene here the reduced genus of a multigraph as the minimum genus of a hypergraph having the same adjacencies with the same multiplicities. Through a study of embedded hypergraphs, we obtain new bounds on the coloring number, clique number and point arboricity of simple graphs of a given reduced ..."
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. We dene here the reduced genus of a multigraph as the minimum genus of a hypergraph having the same adjacencies with the same multiplicities. Through a study of embedded hypergraphs, we obtain new bounds on the coloring number, clique number and point arboricity of simple graphs of a given reduced genus. We present some new related problems on graph coloring and graph representation. 1 Introduction Graph Coloring is a central topic in Graph Theory and numerous studies relate coloring properties with the genus of a graph. The maximum chromatic number among all graphs which can be embedded in a surface of genus g is given by Heawood's formula [14], as established by Ringel and Youngs for g > 0 [22]; the case g = 0, which is the Four Color Theorem, has been established by Appel and Haken [1][2] (see [24] for a simpler proof). Other approaches have related the chromatic number with other graph invariants: For instance, Szekeres and Wilf [28] gave the simple upper bound (G) sw(G) + 1...
Lecture Series on Computer and Computational Sciences
, 2006
"... Abstract: A theory about the implication structure in graph coloring is presented. Discovering hidden relations is one of the most relevant activities in every scientific discipline. The development of mathematical models to study and discover such hidden relations is of the most highest interest. T ..."
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Abstract: A theory about the implication structure in graph coloring is presented. Discovering hidden relations is one of the most relevant activities in every scientific discipline. The development of mathematical models to study and discover such hidden relations is of the most highest interest. The main contribution presented in this work is a model of hidden relations materialized as implicit edges in graph coloring problems, these relations can be interpreted in physical and chemical models as hidden forces, hidden interactions, hidden reactions or hidden variables. Also, a set of novel results in kcolorability, the four color theorem and Chromatic Polynomials are derived from the application of the concept of implicit edge.
A Search of The FourColor Theorem and its Higher Dimensional Generalization
"... Abstract: FourColor Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of FourColor Theorem based on Kuratowski’s Theorem using some induction argument and give a description of the most complicated coloring map, a simple proof of Kuratowski’s Theorem ..."
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Abstract: FourColor Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of FourColor Theorem based on Kuratowski’s Theorem using some induction argument and give a description of the most complicated coloring map, a simple proof of Kuratowski’s Theorem using Euler charateristic is also presented. We also conjecture the higher dimensional generalization of FourColor Theorem.