Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible

This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting -1 as an eigenvalue and then to all orthogonal matrices. We review a method due to Hermann Weyl and another method involving multiplication by a diagonal matrix whose entries are +1 or -1. The second remark has to do with ways of flipping the signs of the entries of a diagonal matrix, C, with nonzero diagonal entries, obtaining a new matrix, E, so that E + A is invertible, where A is any given matrix (invertible or not).Comment: 7 page

The Completeness of Propositional Resolution: A Simple and Constructive<br> Proof

It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a resolution refutation. As a consequence, none of these proofs actually gives an algorithm for producing a resolution refutation from an unsatisfiable set of clauses. In this note, we give a simple and constructive proof of the completeness of propositional resolution which consists of an algorithm together with a proof of its correctness.Comment: 7 pages, submitted to LMC

Fast and Simple Methods For Computing Control Points

The purpose of this paper is to present simple and fast methods for computing control points for polynomial curves and polynomial surfaces given explicitly in terms of polynomials (written as sums of monomials). We give recurrence formulae w.r.t. arbitrary affine frames. As a corollary, it is amusing that we can also give closed-form expressions in the case of the frame (r, s) for curves, and the frame ((1, 0, 0), (0, 1, 0), (0, 0, 1) for surfaces. Our methods have the same low polynomial (time and space) complexity as the other best known algorithms, and are very easy to implement.Comment: 15 page

Democracy and compliance in public goods games

I investigate if, how, and why the effect of a contribution rule in a public goods game depends on how it is implemented: endogenously chosen or externally imposed. The rule prescribes full contributions to the public good backed by a nondeterrent sanction for those who do not comply. My experimental design allows me to disentangle to what extent the effect of the contribution rule under democracy is driven by self-selection of treatments, information transmitted via the outcome of the referendum, and democracy per se. In case treatments are endogenously chosen via a democratic decision-making process, the contribution rule significantly increases contributions to the public good. However, democratic participation does not affect participants’ contribution behavior directly, after controlling for self-selection of treatments and the information transmitted by voting

On the Choice and the Effects of Rule-Based Contribution Schemes in Public Good Games

In this experiment, we endogenize the choice of which contribution scheme is implemented in a public good game. We investigate three rule-based contribution schemes and a voluntary contribution mechanism (VCM). The game is implemented either as a Single- or a Multi-Phase Game. In the Single-Phase Game, the contribution schemes are exogenously implemented. In the Multi-Phase Game, we let subjects vote on the rule-based contribution schemes. If a scheme gets a sufficient majority it is implemented. In case a sufficient majority is failed, subjects have to make their contributions to the public good using the VCM. We find that the endogenous choice of a contribution scheme has an impact on the level of contributions. In case of a rule-based contribution scheme which equalizes payoffs, contributions are higher if subjects chose the scheme for themselves than in case the scheme is implemented exogenously. The contrary holds for the VCM. Contributions are higher if the VCM is implemented exogenously than in case a sufficient majority is failed and, therefore, subjects have to play the VCM

ORDER-SORTED RIGID E-UNIFICATION

Rigid E-Unification is a special type of unification which arises naturally when extending Andrew's method of matings to logic with equality. We study the rigid E-Unification problem in the presence of subsorts. We present an order sorted method for the computation of order sorted rigid-E-unifiers. The method is based on an unsorted one which we refine and extend to handle sort information. Our approach is to incorporate the sort information within the method so as to leverage it. We show via examples how the order sorted method is able to detect failures due to sort conflicts at an early stage in the construction of potential rigid E Unifiers. The algorithm presented here is NP-complete, as is the unsorted one. This is significant, specially due to the complications presented by the sort information.Information Systems Working Papers Serie