Intersection points of planar curves can be computed

Consider two paths ϕ , ψ : [ 0 ; 1 ] → [ 0 ; 1 ] 2 in the unit square such that ϕ ( 0 ) = ( 0 , 0 ), ϕ ( 1 ) = ( 1 , 1 ), ψ ( 0 ) = ( 0 , 1 ) and ψ ( 1 ) = ( 1 , 0 ). By continuity of ϕ and ψ there is a point of intersection. We prove that from ϕ and ψ we can compute closed intervals S ϕ , S ψ ⊆ [ 0 ; 1 ] such that ϕ ( S ϕ ) = ψ ( S ψ ).

Improved Arithmetic of Complex Fans

Complex fans are sets of complex numbers whose magnitudes and angles range in closed intervals. The fact that the sum of two fans is a disordered shape gives rise to the need for computational methods to find the minimal enclosing fan. Cases where the sum of two fans contains the origin of the complex plane as a boundary point are of special interest. The result of the addition is then enclosed by circles in current methods, but under certain circumstances this turns out to be an overestimate. The focus of this article is the diagnosis and treatment of such cases.

Approximation by modified Jain–Baskakov operators

In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions {e_{0}} and {e_{1}}. It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

Estimates for the Bounds of the Essential Spectrum of a 2 × 2 Operator Matrix

We consider a 2 × 2 operator matrix Aμ, μ > 0, related with the lattice systems describing three particles in interaction, without conservation of the number of particles on a d-dimensional lattice. We obtain an analogue of the Faddeev type integral equation for the eigenfunctions of Aμ. We describe the two- and three-particle branches of the essential spectrum of Aμ via the spectrum of a family of generalized Friedrichs models. It is shown that the essential spectrum of Aμ consists of the union of at most three bounded closed intervals. We estimate the lower and upper bounds of the essential spectrum of Aμ with respect to the dimension d ∈ N of the torus Td and the coupling constant μ > 0.

On the Banzhaf-like Value for Cooperative Games with Interval Payoffs

By using Moore’s subtraction operator and a total order on the set of closed intervals, we introduce a new variation of the Banzhaf value for cooperative interval games called the interval Banzhaf-like value which may accommodate the shortcomings of the interval Banzhaf value. We first reveal the relation between this introduced value and the interval Banzhaf value. Then, we present two sets of properties that may be used to determine whether an interval value is median-indifferent to the interval Banzhaf-like value. Finally, in order to overcome the disadvantages of the interval Banzhaf-like value, we propose the contracted interval Banzhaf-like value and give an axiomatization of this proposed value.

Full Information Optimal Scoring

Ramsay and Wiberg used a new version of item response theory that represents test performance over nonnegative closed intervals such as [0, 100] or [0, n] and demonstrated that optimal scoring of binary test data yielded substantial improvements in point-wise root-mean-squared error and bias over number right or sum scoring. We extend these results by showing that optimal scoring of the full information in option choices produces about as much further improvement in these measures of score performance as was achieved by going from sum scoring to optimal binary scoring.

Normed Interval Space and Its Topological Structure

Based on the natural vector addition and scalar multiplication, the set of all bounded and closed intervals in R cannot form a vector space. This is mainly because the zero element does not exist. In this paper, we endow a norm to the interval space in which the axioms are almost the same as the axioms of conventional norm by involving the concept of null set. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets, which can generate many different topologies.