Two player game variant of the Erdos-Szekeres problem

The classical Erdos-Szekeres theorem states that a convex k-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erdos-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex k-gon problem, convex k-gon with specified number of interior points and the chromatic variant. In this paper, we introduce the following two player game variant of the Erdos-Szekeres problem: Consider a two player game where each player playing in alternate turns, place points in the plane. The objective of the game is to avoid the formation of the convex k-gon among the placed points. The game ends when a convex k-gon is formed and the player who placed the last point loses the game. In our paper we show a winning strategy for the player who plays second in the convex 5-gon game and the empty convex 5-gon game by considering convex layer configurations at each step. We prove that the game always ends in the 9th step by showing that the game reaches a specific set of configurations.

A Variant of the Hadwiger-Debrunner (p, q)-Problem in the Plane

Let X be a convex curve in the plane (say, the unit circle), and let be a family of planar convex bodies such that every two of them meet at a point of X. Then has a transversal of size at most . Suppose instead that only satisfies the following ``(p, 2)-condition'': Among every p elements of , there are two that meet at a common point of X. Then has a transversal of size . For comparison, the best known bound for the Hadwiger-Debrunner (p, q)-problem in the plane, with , is . Our result generalizes appropriately for if is, for example, the moment curve.

A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem

A reliable and efficient a posteriori error estimator is derived for a class of discontinuous Galerkin (DG) methods for the Signorini problem. A common property shared by many DG methods leads to a unified error analysis with the help of a constraint preserving enriching map. The error estimator of DG methods is comparable with the error estimator of the conforming methods. Numerical experiments illustrate the performance of the error estimator. (C) 2015 Elsevier B.V. All rights reserved.

AN INVESTIGATION OF CUTTING MECHANISMS AND STRAIN FIELDS DURING ORTHOGONAL CUTTING IN CFRPs

Fiber-reinforced plastics (FRPs) are typically difficult to machine due to their highly heterogeneous and anisotropic nature and the presence of two phases (fiber and matrix) with vastly different strengths and stiffnesses. Typical machining damage mechanisms in FRPs include series of brittle fractures (especially for thermosets) due to shearing and cracking of matrix material, fiber pull-outs, burring, fuzzing, fiber-matrix debonding, etc. With the aim of understanding the influence of the pronounced heterogeneity and anisotropy observed in FRPs, ``Idealized'' Carbon FRP (I-CFRP) plates were prepared using epoxy resin with embedded equispaced tows of carbon fibers. Orthogonal cutting of these I-CFRPs was carried out, and the chip formation characteristics, cutting force signals and strain distributions obtained during machining were analyzed using the Digital Image Correlation (DIC) technique. In addition, the same procedure was repeated on Uni-Directional CFRPs (UD-CFRPs). Chip formation mechanisms in FRPs were found to depend on the depth of cut and fiber orientation with pure epoxy showing a pronounced ``size effect.'' Experimental results indicate that in-situ full field strain measurements from DIC coupled with force measurements using dynamometry provide an adequate measure of anisotropy and heterogeneity during orthogonal cutting.