Let A be a given n-by-n complex matrix with eigenvalues λ, λ2, . . . , λn. Suppose there are nonzero vectors x, y ∈ Cn such that Ax = λx, y∗A = λy∗, and y∗x = 1. Let v ∈ Cn be such that v∗x = 1, let c ∈ C, and assume that λ 6= cλj for each j = 2, . . . , n. Define A(c) := cA+(1− c)λxv∗. The eigenvalues of A(c) are λ, cλ2, . . . , cλn. Every left eigenvector of A(c) corresponding to λ is a scala...

In this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with respect to chains defined over the rational unit interval with a special attention to the completeness with respect to the canonical chain, i.e. the...

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the cla...

The study of realizations of response maps is a topic of control and system theory. Realization theory is used in system identification and control synthesis. A minimal rational realization of a given response map p is a rational realization of p such that the dimension of its state space equals the transcendence degree of the observation field of p. We relate minimality of rational realization...

X iv :c s/ 04 12 00 5v 1 [ cs .S C ] 2 D ec 2 00 4 Jordan Normal and Rational Normal Form Algorithms Bernard Parisse, Morgane Vaughan Institut Fourier CNRS-UMR 5582 100 rue des Maths Université de Grenoble I 38402 St Martin d'Hères Cédex Résumé In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula : (λ · I − A) ·B(λ) = P (λ) · I where B(λ) is (λ · I −...

In this note, we study the general form of some rational recursive sequences. By some modification of the methods and ideas, as well as the transformation from the paper [K. The global attractivity of the rational difference equation y n = y n−k +yn−m 1+y n−k yn−m , Appl. Math. Lett. 20 (2007), 54–58], we give a new proof for the conjectures posed therein.

Let S be a surface in P2 × P2 given by the intersection of a (1,1)form and a (2,2)-form. Then S is a K3 surface with two noncommuting involutions σx and σy . In 1991 the second author constructed two height functions ĥ+ and ĥ− which behave canonically with respect to σx and σy , and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of lo...

Curb sets [Basu and Weibull, Econ. Letters 36 (1991), 141-146] are product sets of pure strategies containing all individual best-responses against beliefs restricted to the recommendations to the remaining players. The concept of minimal curb sets is a set-theoretic coarsening of the notion of strict Nash equilibrium. We introduce the concept of minimal strong curb sets which is a set-theoreti...

Bernstein polynomials on a simplex V are considered. The expansion of a given polynomial p into these polynomials provides bounds for the range of p over V . Bounds for the range of a rational function over V can easily obtained from the Bernstein expansions of the numerator and denominator polynomials of this function. In this paper it is shown that these bounds converge monotonically and line...