966 research outputs found
Toward a probability theory for product logic: states, integral representation and reasoning
The aim of this paper is to extend probability theory from the classical to
the product t-norm fuzzy logic setting. More precisely, we axiomatize a
generalized notion of finitely additive probability for product logic formulas,
called state, and show that every state is the Lebesgue integral with respect
to a unique regular Borel probability measure. Furthermore, the relation
between states and measures is shown to be one-one. In addition, we study
geometrical properties of the convex set of states and show that extremal
states, i.e., the extremal points of the state space, are the same as the
truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal
logic for probabilistic reasoning on product logic events and prove soundness
and completeness with respect to probabilistic spaces, where the algebra is a
free product algebra and the measure is a state in the above sense.Comment: 27 pages, 1 figur
Rational group ring elements with kernels having irrational dimension
We prove that there are examples of finitely generated groups G together with
group ring elements Q \in \bbQ G for which the von Neumann dimension
\dim_{LG}\ker Q is irrational, so (in conjunction with other known results)
answering a question of Atiyah.Comment: 35 pages, 1 figure; [TDA 3/12/13:] Updated version incorporating
referee's suggestion
Geometric Algebra Techniques for General Relativity
Geometric (Clifford) algebra provides an efficient mathematical language for
describing physical problems. We formulate general relativity in this language.
The resulting formalism combines the efficiency of differential forms with the
straightforwardness of coordinate methods. We focus our attention on
orthonormal frames and the associated connection bivector, using them to find
the Schwarzschild and Kerr solutions, along with a detailed exposition of the
Petrov types for the Weyl tensor.Comment: 34 pages, 0 figures; submitted to Annals of Physic
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