Fast and Efficient Formulations for Electroencephalography-Based Neuroimaging Strategies

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Results on geometric networks and data structures

This thesis discusses four problems in computational geometry. In traditional colored range-searching problems, one wants to store a set of n objects with m distinct colors for the following queries: report all colors such that there is at least one object of that color intersecting the query range. Such an object, however, could be an `outlier' in its color class. We consider a variant of this problem where one has to report only those colors such that at least a fraction t of the objects of that color intersects the query range, for some parameter t. Our main results are on an approximate version of this problem, where we are also allowed to report those colors for which a fraction (1-epsilon)t intersects the query range, for some fixed epsilon > 0. We present efficient data structures for such queries with orthogonal query ranges in sets of colored points, and for point stabbing queries in sets of colored rectangles. A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. R-trees are box-trees with nodes of high degree. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. The geometric minimum-diameter spanning tree (MDST) of a set of n points is a tree that spans the set and minimizes the Euclidian length of the longest path in the tree. So far, the MDST can only be found in slightly subcubic time. We give two fast approximation schemes for the MDST, i.e. factor-(1+epsilon) approximation algorithms. One algorithm uses a grid and takes time O*(1/epsilon^(5 2/3) + n), where the O*-notation hides terms of type O(log^O(1) 1/epsilon). The other uses the well-separated pair decomposition and takes O(1/epsilon^3 n + (1/epsilon) n log n) time. A combination of the two approaches runs in O*(1/epsilon^5 + n) time. The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments connecting a pair of rectangles, and the distance measure we use is the L1-distance. We study the following problem: given n non-intersecting rectangles and a graph describing which pairs of rectangles are to be connected, we wish to place the connecting segments such that the dilation is minimized. We obtain the following results: for arbitrary graphs, the problem is NP-hard; for trees, we can solve the problem by linear programming on O(n^2) variables and constraints; for paths, we can solve the problem in time O(n^3 log n); for rectangles sorted vertically along a path, the problem can be solved in O(n^2) time

Spin Ice, Fractionalization and Topological Order

The spin ice compounds {\dys} and {\holm} are highly unusual magnets which epitomize a set of concepts of great interest in modern condensed matter physics: their low-energy physics exhibits an emergent gauge field and their excitations are magnetic monopoles which arise from the fractionalization of the microscopic magnetic spin degrees of freedom. In this review, we provide an elementary introduction to these concepts and we survey the thermodynamics, statics and dynamics---in and out of equilibrium---of spin ice from these vantage points. Along the way, we touch on topics such as emergent Coulomb plasmas, observable "Dirac strings", and irrational charges. We close with the outlook for these unique materials.Comment: (15 pages, 9 figures) see http://www.annualreviews.org/doi/abs/10.1146/annurev-conmatphys-020911-125058 for the published versio

Domain structure of bulk ferromagnetic crystals in applied fields near saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a branched domain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp-interface energy giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy, and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied fieldComment: 38 pages, 7 figures, submitted to J. Nonlin. Sci

Design and Implementation of Wireless Point-Of-Care Health Monitoring Systems: Diagnosis For Sleep Disorders and Cardiovascular Diseases

Chronic sleep disorders are present in 40 million people in the United States. More than 25 million people remain undiagnosed and untreated, which accounts for over $22 billion in unnecessary healthcare costs. In addition, another major chronic disease is the heart diseases which cause 23.8% of the deaths in the United States. Thus, there is a need for a low cost, reliable, and ubiquitous patient monitoring system. A remote point-of-care system can satisfy this need by providing real time monitoring of the patient\u27s health condition at remote places. However, the currently available POC systems have some drawbacks; the fixed number of physiological channels and lack of real time monitoring. In this dissertation, several remote POC systems are reported to diagnose sleep disorders and cardiovascular diseases to overcome the drawbacks of the current systems. First, two types of remote POC systems were developed for sleep disorders. One was designed with ZigBee and Wi-Fi network, which provides increase/decrease the number of physiological channels flexibly by using ZigBee star network. It also supports the remote real-time monitoring by extending WPAN to WLAN with combination of two wireless communication topologies, ZigBee and Wi-Fi. The other system was designed with GSM/WCDMA network, which removes the restriction of testing places and provides remote real-time monitoring in the true sense of the word. Second, a fully wearable textile integrated real-time ECG acquisition system for football players was developed to prevent sudden cardiac death. To reduce power consumption, adaptive RF output power control was implemented based on RSSI and the power consumption was reduced up to 20%. Third, as an application of measuring physiological signals, a wireless brain machine interface by using the extracted features of EOG and EEG was implemented to control the movement of a robot. The acceleration/deceleration of the robot is controlled based on the attention level from EEG. The left/right motion of eyeballs of EOG is used to control the direction of the robot. The accuracy rate was about 95%. These kinds of health monitoring systems can reduce the exponentially increasing healthcare costs and cater the most important healthcare needs of the society