Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard model on a 4 x 4 square lattice at U/t = 0.5, 4, and 10, and even number of electrons with Filling factors n ranging from a quarter filling up to half filling. We compare the ground-state energy and density of states at U/t = 0.5 and 4 with second-order perturbation theory in U/t in the paramagnetic phase, and find that while the agreement is reasonable at U/t = 0.5, it becomes worse as the perturbatively determined (i.e., using Stoner s criterion) boundary of the paramagnetic to spin-density-wave instability is approached. In the strong coupling regime (U/t = 10), we find reasonable agreement between the density of states of the Hubbard and the t-J model especially for low doping fractions. In general, we find that at half filling the filled states are separated from the empty states by a gap. At U/t = 10, the density of states shows two bands clearly separated by a Mott-Hubbard gap of order U.
The authors investigate the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine prover9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. prover9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the argument into better focus. Also, the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a 'diagonal argument' and, moreover, one used to establish a positive conclusion rather than a paradox. * This paper is published in the Australasian Journal of Philosophy, 89 (2) (June 2011): 333-350. The authors would like to thank Branden Fitelson for introducing us to otter, prover9, and mace4, and Bill McCune for answering our questions about them.
Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell's reduction of functions to relations over Frege's reduction of relations to functions. There is an interesting system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an oversimplification: one can't assimilate predication to functional application. * This paper is forthcoming in the Journal of Logic and Computation. The authors would like to thank Uri Nodelman for his observations on the first draft of this paper. We'd also like to thank Bernard Linsky for observations on the second draft, which led us to reconceptualize the significance of our results within a more historical context. We'd also like to acknowledge one of the referees of this journal, whose comments led us to clarify and better document the claims in the paper.
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