We prove that a family of polynomials encoding a Graph Ordering Principle (GOP(G)) requires refutations in Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) of degree Ω(n). Here n is the number of vertices in the graph G. This is the first linear degree lower bound for PC and PCR refutations for ordering principles. Our result answers to a question raised by Bonet and Gales...

We solve the boson normal ordering problem for ( q(a†)a+ v(a†) )n with arbitrary functions q(x) and v(x) and integer n, where a and a† are boson annihilation and creation operators, satisfying [a, a†] = 1. This consequently provides the solution for the exponential e †)a+v(a†)) generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle...

Cluster density maximization and (maximal) cluster covering have emerged as ordering principles for quasicrystalline structures. The concepts behind these ordering principles are reviewed and illustrated with several examples. For two examples, Gummelt’s aperiodic decagon model and a cluster model for octagonal Mn–Si–Al quasicrystals, these ordering principles can enforce perfectly ordered, qua...

We show that the semi-linear ordering principle for continuous functions implies the determinacy of all Wadge and Lipschitz games. c © 2006 Elsevier B.V. All rights reserved.

The Trichotomy Principle says that a pair of sets A and B either admits a bijection or else precisely one of these sets injects into the other. Hartogs established logical equivalence between the Trichotomy Principle and the Well-Ordering Principle. As ZF suffices to prove the Schröder-Bernstein theorem, the heart of Trichotomy lies in the existence of some injection connecting A and B (in eith...

The Knuth-Bendix ordering is usually preferred over the lexicographic path ordering in successful implementations of resolution and superposition, but it is incompatible with certain requirements of hierarchic superposition calculi. Moreover, it does not allow non-linear definition equations to be oriented in a natural way. We present an extension of the Knuth-Bendix ordering that makes it poss...

We identify four orthogonal dimensions along which information in engineering design organizations can be ordered, namely (i) the principle of ordering (top down vs. bottom up), (ii) the structure of ordering (aggregation-decomposition vs. generalization-specialization), (iii) the breadth of ordering (diversity vs. parsimony) and (iv) the perspective of ordering (horizontal vs. historical). Fou...

There exists a utilitarian tradition à la Sidgwick of treating equal generations equally in the form of anonymity. Diamond showed that no social evaluation ordering over infinite utility streams satisfying the Pareto principle, Sidgwick’s equity principle, and the axiom of continuity exists. We introduce two versions of egalitarianism in the spirit of the Pigou-Dalton transfer principle and the...