### Browsing by Author "Gavilanez, Franklin"

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Item Continuous and Discrete Inverse Conductivity Problems(2004) Gavilanez, Franklin; Baras, John S.; Berenstein, Carlos A.; ISRTomography using CT scans and MRI scans is now well-known as a medical diagnostic tool which allows for detection of tumors and other abnormalities in a noninvasive way, providing very detailed images of the inside of the body using low dosage X-rays and magnetic fields. They have both also been used for determination of material defects in moderate size objects. In medical and other applications they complement conventional tomography. There are many situations where one wants to monitor the electrical conductivity of different portions of an object, for instance, to find out whether a metal object, possibly large, has invisible cracks. This kind of tomography, usually called Electrical Impedance Tomography or EIT, has also medical applications like monitoring of blood flow. While CT and MRI are related to Euclidean geometry, EIT is closely related to hyperbolic geometry. A question that has arisen in the recent past is whether there is similar omographicmethod to monitor the ealthof networks. Our objective is to explain how EIT ideas can in fact effectively be used in this context.Item Local Monitoring of the Internet Network(2003) Gavilanez, Franklin; Baras, John S.; Berenstein, Carlos A.; ISRThe internet, as well as other freely evolving networks, has a topology that changes dynamically; therefore the topology is very complicated. Presently, there is an increase in dependency on the internet for several things such as communication, information traffic and many more, hence it is imperative that we prevent attackers from disrupting this network. To do this, it is essential to count on a mathematical model that can allow early detection (then ring the bell) of attacks to the network. The mathematical tool that we are looking for to accomplish that early detection is based on the use of Tomography ideas.The last statement is based on the April 28-29, 2000 workshop on the inter-face between the mathematical sciences and three areas of computer science:network traffic modeling, computer vision, and data mining and search. Thisworkshop was cosponsored by the Board on Mathematical Sciences andthe Computer Science and Telecommunications Board of the National ResearchCouncil.

Item Network Tomography(2005) Gavilanez, Franklin; Berenstein, Carlos A.; Baras, John S.; ISRWhile conventional tomography is associated to the Radon transform in Euclidean spaces, electrical impedance tomography or EIT is associated to the Radon transform in the hyperbolic plane. We discuss some recent work on network tomography that can be associated to a problem similar to EIT on graphs and indicate how in some sense it may be also associated to the Radon transform on trees.Item Network Tomography(2006-08-24) Gavilanez, Franklin; Berenstein, Carlos A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)While conventional tomography is associated to the Radon transform in Euclidean spaces, electrical impedance tomography, or EIT, is associated to the Radon transform in the hyperbolic plane. In this dissertation, we discuss some recent work on network tomography that can be associated to a problem similar to EIT on graphs and indicate how in some sense it may be also associated to the Radon transform on trees. We develop a strategy to determine the weight w for the case of general weighted graphs. We begin by considering relatively simple regions of interest in a graph and suitable choices for the data of the w-Neumann boundary value problem to produce a linear system of equations for the values of w.