176 research outputs found
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
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
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
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Some basic mathematical tools such as convex sets, polytopes and
combinatorial topology, are used quite heavily in applied fields such as
geometric modeling, meshing, computer vision, medical imaging and robotics.
This report may be viewed as a tutorial and a set of notes on convex sets,
polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay
Triangulations. It is intended for a broad audience of mathematically inclined
readers. I have included a rather thorough treatment of the equivalence of
V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and
H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin
elimination method (a version of Gaussian elimination for inequalities) is
discussed in some detail. I also included some material on projective spaces,
projective maps and polar duality w.r.t. a nondegenerate quadric in order to
define a suitable notion of ``projective polyhedron'' based on cones. To the
best of our knowledge, this notion of projective polyhedron is new. We also
believe that some of our proofs establishing the equivalence of V-polyhedra and
H-polyhedra are new.Comment: 183 page
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