Next Level Odd-One-Out Puzzles

A commonly occurring task in intelligence tests or recreational riddles is to “find the odd one out”, that is, to determine a unique element of a set of objects that is somehow special. It is somewhat arbitrary what exactly the relevant feature is that makes one object different. But once that is settled, the answer becomes obvious. Not so with a puzzle popularized by Tanya Khovanova to express her dislike for this type of puzzle. Here, it is a more complicated relation between the objects and the features that determines the odd object, because there is only one object that does not have a unique feature expression. This puzzle inspired me to look for even more complicated relations between objects, features and feature expressions that appear to be even more symmetric, but actually still single out a “special object”. This paper provides useful definitions, a theoretical basis, solution algorithms, and several examples for this kind of puzzle.

A pFFT Accelerated BEM Linear Strength Potential Flow Solver

In this paper the development of a linear shape function, Galerkin Boundary Element Method (BEM) for solving the direct potential flow integral equation around arbitrary 3-Dimensional bodies is described. The solution of the potential flow for both constant and linear shape functions over a triangulated body surface is examined. In order to facilitate a larger and more practical number of panels, an iterative GMRES [1] matrix solution method is coupled with a precorrected Fast Fourier Transform (pFFT) approximate matrix vector product (MVP)[2]. The pFFT algorithm is described and the differences in attaining MVP’s for linear and constant strength panel distributions are highlighted. A simple flat sheet wake model is included to solve the lifting body problem. The pFFT is shown to reduce the solution time to O(nlog(n)) operations (n is the number of panels). The results from flat panel surface representations of the body show that the convergence rate of the solution is at best O(n) for both linear and constant basis function representations of the solution. When the constant basis solution is sampled at the centroid of the panel, the error converges at a similar rate to the linear basis solution error, namely (O(n)); however, when the solution is sampled at surface points other than the centroid, the constant basis representation will converge at a slower rate O(n1/2), while the linear basis solution converges at a rate of O(n) for all points on the body.

Completeness of Root Functions for Thermal Conduction in a Strip with Piecewise Continuous Coefficients

We consider the boundary layer problem associated with the steady thermal conduction problem in a thin laminated plate. Two cases of boundary conditions, Dirichlet and Neumann, are treated in the paper. Transmission conditions across the interfaces should be added since the plate is laminated. The study of the structure of the solution in the matching region of the layer with the basis solution in the plate leads to consideration of an eigenvalue problem for a second-order operator pencil with piecewise continuous coefficients and the corresponding boundary and transmission conditions. Twofold completeness of root functions of the latter problem is proved. The boundary layer term can then be expressed as a combination of these functions.