Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity

Some multiplicity results are presented for the eigenvalue problem

(Pλ,μ)     

Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erd?s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles.In the plane, our main result is an O(n44/19)=O(n^2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n7/3) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω(nlogn) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most ; there exist n-element point sets (for arbitrarily large n) that span (6/π²−o(1))n² minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω(nlogn) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds).In 3-space we prove an O(n17/7β(n))=O(n^2.4286) upper bound on the number of unit-area triangles spanned by n points, where β(n) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O(n8/3), is an old result of Erd?s and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n²+O(n), and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω(n4/3) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O(n4/3+ε), for any ε>0. (iii) Every set of n points, not all on a line, determines at least Ω(n2/3/β(n)) triangles of distinct areas, which share a common side.     

Syntheses of vinca alkaloids and related compounds. 104. A concise synthesis of (-)-vincapusine

beta-Iodo-enamines with an eburnane skeleton (5a and 5c) were obtained with the aid of iodine from compounds 2a and 2c and were then transformed into hydroxyl lactams (6a and 6c) with CuSO4.5H2O in a mixture of DMF and water. Lactams (6a and 6c) were reduced selectively with BH3.SMe2 to result in the first synthesis of (-)-vincapusine (4a) as well as its natural 14-decarbomethoxy analogue (4c).     

A randomized on–line algorithm for the k–server problem on a line

The k–server problem is one of the most important and well‐studied problems in the area of on–line computation. Its importance stems from the fact that it models many practical problems like multi‐level memory paging encountered in operating systems, weighted caching used in the management of web caches, head motion planning of multi‐headed disks, and robot motion planning. In this paper, we investigate its randomized version for which Θ(log k)–competitiveness is conjectured and yet hardly any <k competitive algorithms are known, even for the simplest of metric spaces of O(k) size. We present a –competitive randomized k–server algorithm against an oblivious adversary when the underlying metric space is given by n equally spaced points on a line . This algorithm is <k competitive for . Thus, it provides a super–linear bound for n with o(k)–competitiveness for the first time and improves the best results known so far for the range on . © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Alternating Paths along Axis-Parallel Segments

It is shown that for a set S of n pairwise disjoint axis-parallel line segments in the plane there is a simple alternating path of length . This bound is best possible in the worst case. In the special case that the n pairwise disjoint axis-parallel line segments are protruded (that is, if the intersection point of the lines through every two nonparallel segments is not visible from both segments), there is a simple alternating path of length n. Work on this paper was partially supported by National Science Foundation grants CCR-0049093 and IIS-0121562. A preliminary version of this paper has appeared in the Proceedings of the 8th International Workshop on Algorithms and Data Structures (Ottawa, ON, 2003), vol. 2748 of Lecture Notes on Computer Science, Springer, Berlin, 2003, pp. 389–400.     

Distinct Distances in Homogeneous Sets in Euclidean Space

It is shown that every homogeneous set of n points in d-dimensional Euclidean space determines at least distinct distances for a constant c(d) > 0. In three-space the above general bound is slightly improved and it is shown that every homogeneous set of n points determines at least distinct distances.     

Purcell’s method, Egerváry and related results

After having been appeared, Egerváry was perhaps the first who responded to Purcell’s paper in 1957. Later in a posthumous paper he returned to the method in 1960, showing that it could be derived from his rank reduction procedure. We review here Purcell’s method in connection with Egerváry’s activity and also, we give a short survey on subsequent developments.     

Segment Orders

We study two kinds of segment orders, using definitions first proposed by Farhad Shahrokhi. Although the two kinds of segment orders appear to be quite different, we prove several results suggesting that the are very much the same. For example, we show that the following classes belong to both kinds of segment orders: (1) all posets having dimension at most 3; (2) interval orders; and for n≥3, the standard example S _n of an n-dimensional poset, all 1-element and (n−1)-element subsets of {1,2,…,n}, partially ordered by inclusion. Moreover, we also show that, for each d≥4, almost all posets having dimension d belong to neither kind of segment orders. Motivated by these observations, it is natural to ask whether the two kinds of segment orders are distinct. This problem is apparently very difficult, and we have not been able to resolve it completely. The principal thrust of this paper is the development of techniques and results concerning the properties that must hold, should the two kinds of segment orders prove to be the same. We also derive equivalent statements, one version of which is a stretchability question involving certain sets of pseudoline arrangements. We conclude by proving several facts about continuous universal functions that would transfer segment orders of the first kind into segments orders of the second kind.