2,666 research outputs found
Implication functions in interval-valued fuzzy set theory
Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory
Applications of self-distributivity to Yang-Baxter operators and their cohomology
Self-distributive (SD) structures form an important class of solutions to the
Yang--Baxter equation, which underlie spectacular knot-theoretic applications
of self-distributivity. It is less known that one go the other way round, and
construct an SD structure out of any left non-degenerate (LND) set-theoretic
YBE solution. This structure captures important properties of the solution:
invertibility, involutivity, biquandle-ness, the associated braid group
actions. Surprisingly, the tools used to study these associated SD structures
also apply to the cohomology of LND solutions, which generalizes SD cohomology.
Namely, they yield an explicit isomorphism between two cohomology theories for
these solutions, which until recently were studied independently. The whole
story leaves numerous open questions. One of them is the relation between the
cohomologies of a YBE solution and its associated SD structure. These and
related questions are covered in the present survey
General relativistic dynamics of compact binaries at the third post-Newtonian order
The general relativistic corrections in the equations of motion and
associated energy of a binary system of point-like masses are derived at the
third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian
expansion of the metric in harmonic coordinates at the 3PN approximation. The
metric is parametrized by appropriate non-linear potentials, which are
evaluated in the case of two point-particles using a Lorentzian version of an
Hadamard regularization which has been defined in previous works.
Distributional forms and distributional derivatives constructed from this
regularization are employed systematically. The equations of motion of the
particles are geodesic-like with respect to the regularized metric. Crucial
contributions to the acceleration are associated with the non-distributivity of
the Hadamard regularization and the violation of the Leibniz rule by the
distributional derivative. The final equations of motion at the 3PN order are
invariant under global Lorentz transformations, and admit a conserved energy
(neglecting the radiation reaction force at the 2.5PN order). However, they are
not fully determined, as they depend on one arbitrary constant, which reflects
probably a physical incompleteness of the point-mass regularization. The
results of this paper should be useful when comparing theory to the
observations of gravitational waves from binary systems in future detectors
VIRGO and LIGO.Comment: 78 pages, submitted to Phys. Rev. D, with minor modification
Cramer's rule applied to flexible systems of linear equations
Abstract. Systems of linear equations, called flexible systems, with coefficients having uncertainties of type o (.) or O (.) are studied. In some cases an exact solution may not exist but a general theorem that guarantees the existence of an admissible solution, in terms of inclusion, is resented.
This admissible solution is produced by Cramer’s Rule; depending on the size of the uncertainties appearing in the matrix of coefficients and in the constant term vector some adaptations may be needed
The Algebraic Intersection Type Unification Problem
The algebraic intersection type unification problem is an important component
in proof search related to several natural decision problems in intersection
type systems. It is unknown and remains open whether the algebraic intersection
type unification problem is decidable. We give the first nontrivial lower bound
for the problem by showing (our main result) that it is exponential time hard.
Furthermore, we show that this holds even under rank 1 solutions (substitutions
whose codomains are restricted to contain rank 1 types). In addition, we
provide a fixed-parameter intractability result for intersection type matching
(one-sided unification), which is known to be NP-complete.
We place the algebraic intersection type unification problem in the context
of unification theory. The equational theory of intersection types can be
presented as an algebraic theory with an ACI (associative, commutative, and
idempotent) operator (intersection type) combined with distributivity
properties with respect to a second operator (function type). Although the
problem is algebraically natural and interesting, it appears to occupy a
hitherto unstudied place in the theory of unification, and our investigation of
the problem suggests that new methods are required to understand the problem.
Thus, for the lower bound proof, we were not able to reduce from known results
in ACI-unification theory and use game-theoretic methods for two-player tiling
games
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