We study the graph partitioning problem on ddimensional ball graphs in a geometric way. Let B be a set of balls in d-dimensional Euclidean space with radius ratio δ and λ-precision. We prove that it can be partitioned into three sets BS , BI , BE such that the intersection of BI and BE is empty, and for some constant α, the volume of BI and BE is less than α portion of volume of B, and the volu...

This paper presents a unified model for bi-directional DC-DC converters for both directions of power flow. The bi-directional DC-DC converter is analyzed using a seamless dynamic model with an independent voltage source and an independent current source, which polarity depends on the direction of the power flow. A small signal model is derived using a state space averaging method. Furthermore, ...

Let I be the ideal generated by alternating polynomials in two sets of n variables. Haiman proved that the q, t-Catalan number is the Hilbert series of the bi-graded vector space M(= ⊕ d1,d2 Md1,d2) spanned by a minimal set of generators for I. In this paper we give simple upper bounds on dim Md1,d2 in terms of number of partitions, and find all bi-degrees (d1, d2) such that dimMd1,d2 achieve t...

1 Homework Assignment # 1 1. Show that the -δ definition of continuity of a map f : X → Y between metric spaces is equivalent to the definition of continuity in terms of open subsets. Proof. Suppose f is -δ-continuous (i.e., continuous in the sense of the -δ definition of continuity) and that U is an open subset of Y . We need to show that f−1(U) is an open subset of X. The definition of an ope...

We prove that, if a compact subspace of a metric space can be covered by a family C of bi–Lipschitz local collars, then it admits also a bi–Lipschitz global collar, whose complexity, measured by the so–called weak bi–Lipschitz constant, depends only on C. If the compactness condition is dropped, a slightly weaker version of this result remains valid, provided C is uniform in a suitable natural ...

We present a scale-space with separate scales in the greyscale and spatial dimensions, using ideas from mathematical morphology. Dynamics are used for greyscale (luminance) filtering and openings or closings with scaled disks for spatial scaling. Dynamics cater naturally for the inclusion relationships between regions and give a very good measure of the salience (scale) of a region in greyscale...

Model selection is an important problem of costsensitive SVM (CS-SVM). Although using solution path to find global optimal parameters is a powerful method for model selection, it is a challenge to extend the framework to solve two regularization parameters of CS-SVM simultaneously. To overcome this challenge, we make three main steps in this paper. (i) A critical-regions-based biparameter space...