It is shown that every 2, set of reals which contains reals of arbitrarily high Turing degree in the hyperarithmetic hierarchy contains reals of every Turing degree above the degree of Kleene's 0. As an application it is shown that every Turing degree above the Turing degree of Kleene*s 0 is a minimal cover. In this paper we consider a particular verification of the following nebulously stated ...

In this paper, we take a GL-quantale as the truth value table to study a new rough set model—L-valued fuzzy rough sets. The three key components of this model are: an L-fuzzy set A as the universal set, an L-valued relation of A and an L-fuzzy set of A (a fuzzy subset of fuzzy sets). Then L-valued fuzzy rough sets are completely characterized via both constructive and axiomatic approaches. 

We demonstrate a close connection between the decoherent histories (DH) approach to quantum mechanics and the quantum state diffusion (QSD) picture, for open quantum systems described by a master equation of Lindblad form. The (physically unique) set of variables that localize in the QSD picture also define an approximately decoherent set of histories in the DH approach. The degree of localizat...

Let X be a set of points in general position in the plane. General position means that no three points lie on a line and no two points have the same x-coordinate. Y ⊆ X is a cup, resp. cap, if the points of Y lie on the graph of a convex, resp. concave function. Denote the points of Y by p1, p2, . . . , pm according to the increasing x-coordinate. The set Y is open in X if there is no point of ...

It is shown that c1n max{k + 1, log n} ≤ sup 06=p∈Pc n,k ‖p′‖[−1,1] ‖p‖[−1,1] ≤ c2n max{k + 1, log n} with absolute constants c1 > 0 and c2 > 0, where Pc n,k denotes the set of all polynomials of degree at most n with complex coefficients and with at most k (0 ≤ k ≤ n) zeros in the open unit disk. Here ‖ · ‖[−1,1] denotes the supremum norm on [−1, 1]. This result should be compared with the ine...

For polynomial vector fields in R, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a ...

For a connected graph G of order n, a set S ⊆ V (G) is a geodetic set of G if each vertex v ∈ V (G) lies on a x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A set S of vertices of a connected graph G is an open geodetic set of G if for ...

In the present paper a pure exchange, general equilibrium model is considered and the equilibrium set is studied. It is shown for all total endowments and an open and dense set of preferences that if there are l ̧ 2 commodities and m ̧ 2 consumers then there exists a set of distributions of endowments with nonempty interior such that the associated economies have at least l¡1+minfl;mg equilibri...

In three case studies on notions of fuzzy logic and fuzzy set theory (Dubois–Prade’s gradual elements, the entropy of a fuzzy set, and aggregation operators), the paper exemplifies methodological differences between traditional and deductive fuzzy logic. While traditional fuzzy logic admits various interpretations of membership degrees, deductive fuzzy logic always interprets them as degrees of...

We present a direct and fairly simple proof of the following incidence bound: Let P be a set of m points and L a set of n lines in R, for d ≥ 3, which lie in a common algebraic two-dimensional surface of degree D that does not contain any 2-flat, so that no 2-flat contains more than s ≤ D lines of L. Then the number of incidences between P and L is