1,189 research outputs found
Interval-valued algebras and fuzzy logics
In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter
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
Harmonic analysis for real spherical spaces
We give an introduction to basic harmonic analysis and representation theory
for homogeneous spaces attached to a real reductive Lie group . A
special emphasis is made to the case where is real spherical.Comment: Shortened title, typos fixed, more details on dual smooth Frobenius
reciprocity (now Lemma 6.6). 38 pages, lecture notes for the Sanya meeting on
spherical varieties. To appear in Acta Math. Sinic
A temporal semantics for Nilpotent Minimum logic
In [Ban97] a connection among rough sets (in particular, pre-rough algebras)
and three-valued {\L}ukasiewicz logic {\L}3 is pointed out. In this paper we
present a temporal like semantics for Nilpotent Minimum logic NM ([Fod95,
EG01]), in which the logic of every instant is given by {\L}3: a completeness
theorem will be shown. This is the prosecution of the work initiated in [AGM08]
and [ABM09], in which the authors construct a temporal semantics for the
many-valued logics of G\"odel ([G\"od32], [Dum59]) and Basic Logic ([H\'aj98]).Comment: 19 pages, 2 table
Theta functions, fourth moments of eigenforms, and the sup-norm problem I
We give sharp point-wise bounds in the weight-aspect on fourth moments of
modular forms on arithmetic hyperbolic surfaces associated to Eichler orders.
Therefore we strengthen a result of Xia and extend it to co-compact lattices.
We realize this fourth moment by constructing a holomorphic theta kernel on
, for an
indefinite inner-form of over , based on the
Bergman kernel, and considering its -norm in the Weil variable. The
constructed theta kernel further gives rise to new elementary theta series for
integral quadratic forms of signature .Comment: Updated following comment
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