Skin cancer is one of the most common cancers in the United States. As technological advancements are made, algorithmic diagnosis of skin lesions is becoming more important. In this paper, we develop algorithms for segmenting the actual diseased area of skin in a given image of a skin lesion, and for classifying different types of skin lesions pictured in a given image. The cores of the algorithms used were based in persistent homology, an algebraic topology technique that is part of the rising field of Topological Data Analysis (TDA). The segmentation algorithm utilizes a similar concept to persistent homology that captures the robustness of segmented regions. For classification, we design two families of topological features from persistence diagrams-which we refer to as persistence statistics and persistence curves, and use linear support vector machine as classifiers.

Classical Ramsey theory(at least in its simplest form) is concerned with problems of the following kind: given a setXand a colouring of the set[X]nof unorderedn-tuples fromX, find a subsetY ⊆ Xsuch that all elements of[Y]nget the same colour. Subsets with this property are calledmonochromaticorhomogeneous, and a typical positive result in Ramsey theory has the form that whenXis large enough and the number of colours is small enough we can expect to find reasonably large monochromatic sets. Polychromatic Ramsey theoryis concerned with a “dual” problem, in which we are given a colouring of[X]nand are looking for subsetsY ⊆ Xsuch that any two distinct elements of[Y]ngetdifferentcolours. Subsets with this property are calledpolychromaticorrainbow. Naturally if we are looking for rainbow subsets then our task becomes easier when there are many colours. In particular given an integerkwe say that a colouring isk-boundedwhen each colour is used for at mostkmanyn-tuples. At this point it will be convenient to introduce a compact notation for stating results in polychromatic Ramsey theory. We recall that in classical Ramsey theory we write to mean “every colouring of[κ]ninkcolours has a monochromatic set of order type α”. We will write to mean “everyk-bounded colouring of[κ]nhas a polychromatic set of order type α”. We note that whenκis infinite andkis finite ak-bounded colouring will use exactlyκcolours, so we may as well assume thatκis the set of colours used.

In this paper we deal with codes identifying sets of vertices in random graphs, that is l-identifying codes. These codes enable us to detect sets of faulty processors in a multiprocessor system, assuming that the maximum number of faulty processors is bounded by a fixed constant l. The l-identifying codes or simply identifying codes are of special interest. For random graphs we use the model G(n,p), in which each one of the (/sub 2//sup n/) possible edges exists with probability p. We give upper and lower bounds on the minimum cardinality of an l-identifying code in a random graph, as well as threshold functions for the property of admitting such a code. We derive existence results from probabilistic constructions. A connection between identifying codes and superimposed codes is also established.

We show how to construct an arrangement of n lines having a monotone path of length O(n2-(d/vlog n)), where d>0 is some constant, and thus nearly settle the long standing question on monotone path length in line arrangements.

We prove that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard. The best known algorithm (Krivelevich, Nathaniel, and Sudakov, 2001)colors such a graph using O(n/sup 1/5/) colors. Our result immediately implies that for any constants k > 2 and c/sub 2/ > c/sub 1/ > 1, coloring a k-uniform c/sub 1/-colorable hypergraph with c/sub 2/ colors is NP-hard; leaving completely open only the k = 2 graph case. We are the first to obtain a hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k /spl ges/ 4 such a result has been shown by Guruswami et al. (2000), who also discussed the inherent difference between the k = 3 case and k /spl ges/ 4. Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph (Kneser, 1955; Lovasz, 1978) and the Schrijver graph (Schrijver, 1978). We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has 'many' non-monochromatic edges.

(MATH) Let f: {0,1}n → {0,1} be a boolean function. For ε≥ 0 De(f) be the minimum depth of a decision tree for f that makes an error for ≤ε fraction of the inputs χ ϵ {0,1}n. We also make an appropriate definition of the approximate certificate complexity of f, Cε(f). In particular, D0(f) and C0(f) are the ordinary decision and certificate complexities of f. It is known that $D_0(f) \leq (C_0(f))^2$. Answering a question of Tardos from 1989, we show that for all $\Ge > 0$ there exists a $\Gd' > 0$ such that for all $0 \leq \Gd < \Gd'$, we have $D_\Ge(f) \leq K (C_\Gd(f))^2$ where $K = K(\Ge,\Gd) > 0$ is a constant independent of f. The algorithm used in the proof is modeled after those developed by R. Impagliazzo and S. Rudich for use in other problems.

We prove a dual version of the celebrated inequality of D. Reimer (a.k.a. the van den Berg-Kesten conjecture). We use the dual inequality to prove a combinatorial conjecture of S. Rudich motivated by questions in cryptographic complexity. One consequence of Rudich's Conjecture is that there is an oracle relative to which one-way functions exist but one-way permutations do not. The dual inequality has another combinatorial consequence which allows R. Impagliazzo and S. Rudich to prove that if P=NP then NP/spl cap/coNP/spl sube/i.o.AvgP relative to a random oracle.