8028128
Maximal Sets of Integers with Distinct Divisors
A set of positive integers is said to have the distinct divisor property if there is an injective map that sends every integer in the set to one of its proper divisors. In 1983, P. Erdős and C. Pomerance showed that for every $c>1$, a largest subset of $[N,cN]$ with the distinct divisor property has cardinality $\sim \delta(c)N$, for some constant $\delta(c)>0$. They conjectured that $\delta(c)\sim c/2$ as $c \to \infty$. We prove their conjecture. In fact we show that there exist positive absolute constants $D_1,D_2$ such that $D_1\le c^{\beta}(c/2-\delta(c))\le D_2$ where $\beta = \log 2/\log (3/2)$.
7924557
The Improbability That an Elliptic Curve Has Subexponential Discrete Log Problem under the Menezes—Okamoto—Vanstone Algorithm
Abstract. The security of elliptic curve cryptosystems is based on the presumed intractability of the discrete logarithm problem on the curve. Other than algorithms that work in an arbitrary group and are exponential in the general case, the only general-purpose algorithm that has ever been proposed for the elliptic curve discrete logarithm is that of Menezes—Okamoto—Vanstone (MOV). The MOV algorithm, which embeds an elliptic curve group of prime order l in the multiplicative group of a field Fqk, is subexponential only under special circumstances, however. In this paper we first prove that, under a mild condition that always holds in practical applications, the condition that l|(qk-1) , which is obviously necessary for realizing the MOV algorithm, is also sufficient. We next give an improved upper bound for the frequency of occurrence of pairs of primes l, p such that l|(pk-1) for k small, where l is in the Hasse interval 
 $[p+1-2\sqrt{p},p+1+2\sqrt{p}]$ .
1445498
Density of Carmichael numbers with three prime factors
We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors.
38155628
Tight Bounds for Finding Degrees from the Adjacency Matrix
King and Smith-Thomas[5] have shown that to find a sink (a vertex with outdegree 0 and indegree n−1) in an n vertex directed graph, 3n-⌊log n⌋-3 probes to its adjacency matrix are necessary and sufficient in the worst case. We first generalize this result to show that for any integer k between 0 and n−1, 
 
 
to find a vertex of outdegree k in a simple n vertex directed graph, Θ(n(n−k)) probes to its adjacency matrix are necessary and sufficient in the worst case, and 
 
 
to find a vertex of degree k in an n vertex undirected graph, Θ(n2) probes to its adjacency matrix are necessary and sufficient in the worst case.
27124060
The Complexity of Finding Certain Trees in Tournaments
A tournament Tn is an orientation of the complete graph on n vertices. We continue the algorithmic study initiated by Hell and Rosenfeld[5] of recognizing various directed trees in tournaments. Hell and Rosenfeld considered orientations of paths, and showed the existence of oriented paths on n vertices finding which in Tn requires Θ(n lgα n) “edge probes” where α ≤ 1 is any fixed non-negative constant. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree). In particular, we show, by proving upper and lower bounds, that the complexity of finding a vertex of outdegree k(n−1)/2) in Tn is Θ(nk). We also establish an Ω(n2) lower bound for finding a vertex of maximum outdegree in Tn. These bounds are in sharp contrast to the O(n) bounds for selection in the case of transitive tournaments.
26352609
On the pseudoachromatic number of join of graphs
Given an undirected graph G = (V, E), with vertex set V and edge set E, the pseudoachromatic number ψ(G) of the graph G is the maximum number of colors used to color the vertices in such a way that, for any given pair of colors i, j there exists an edge e = (u, v) ∈ E(G) such that u is colored i and v is colored j. In this paper we give a complete characterization of when the ψ of the join of any two graphs is the sum of the ψ of the two graphs.
28520158
Blended Learning - A Trend in e-Learning
A method, apparatus, and article of manufacture for a computer-implemented random reliability engine for computer-implemented dimension reduction using association rules for data mining application. The data mining is performed by the computer to retrieve data from a data store stored on a data storage device coupled to the computer. The data store has records that have multiple attributes. The multiple attributes of a table are clustered to produce a plurality of sets of attributes. Each set of attributes is clustered to obtain data mining attributes.
13457273
A new Contrast Based Image Fusion using Wavelet Packets
Image Fusion, a technique which combines complimentary information from different images of the same scene so that the fused image is more suitable for segmentation, feature extraction, object recognition and Human Visual System. In this paper, a simple yet efficient algorithm is presented based on contrast using wavelet packet decomposition. First, all the source images are decomposed into low and high frequency sub-bands and then fusion of high frequency sub-bands is done by the means of Directive Contrast. Now, inverse wavelet packet transform is performed to reconstruct the fused image. The performance of the algorithm is carried out by the comparison made between proposed and existing algorithm.