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this article describes one of options of the main objective of external ballistics under the interval non-determination of parameters. It is supposed that initial data has been set inaccurately, i.e. their possible values belong to some interval. The algorithm of the solution of an objective within the interval analysis has been developed. The evaluation of the interval decision width has been obtained.

SUMMARY First, this paper derives the prefix sum-of-products expression (PreSOP) and the number of products in a PreSOP for an interval function. Second, it derives Ψ(n ,τ p), the number of n-variable interval functions that can be represented with τp products. Finally, it shows that more than 99.9% of the n-variable interval functions can be represented with � 3n − 1� products, when n is sufficiently large. These results are use

We can consider the Riemann-Stieltjes integral $$ \int\limits_a^b f $$ dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral $$ \int\limits_a^b f $$ dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.

Let G be a (finite) connected graph. Intervals and steps in G are objects that depend on the distance function d of G. If u,v∈V(G), then by the u–v interval in G we mean the set{x∈V(G);d(u,x)+d(x,v)=d(u,v)}.By the interval function of G we mean the mapping I of V(G)×V(G) into the power set of V(G) such that I(u,v) is the u–v interval of G. By a step in G we mean an ordered triple (u,v,w) where u,v,w∈V(G), d(u,v)=1 and d(v,w)=d(u,w)−1. A characterization of the interval function of G and a characterization of the set of all steps in G were published by this author in 1994 and 1997, respectively.This paper is a review of author's results on intervals and steps in a connected graph. Some small results or short proofs are new.

In this paper, we analyze the problem of the optimal (narrowest) approximation (enclosure) of a quadratic interval function \(y(x_1 ,...,x_n ) = [y(x_1 ,...,x_n ) \bar y(x_1 ,...x_n )]\) (i.e., an interval function for which both endpoint functions \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} (x_1 ,...,x_n ) {\text{and}} \bar y(x_1 ,...x_n )\), ..., xn) are quadratic) by a linear interval function. show that in general, this problem is computationally intractable (NP-hard). For a practically important 1D case (n = 1), we present an efficient algorithm.

SUMMARY This paper shows a method to represent interval functions by using head-tail expressions. The head-tail expressions represent greaterthanGT(X : A )f unctions,less-than LT(X : B )f unctions, and interval functions IN0(X : A, B )m ore efficiently than sum-of-products expressions. Let n be the number of bits to represent the largest value in the interval (A, B). This paper proves that a head-tail expression (HT) represents an interval function with at most n words in a ternary content addressable memory (TCAM) realization. It also shows the average numbers of factors to represent interval functions by HTs for up to n = 16, which were obtained by a computer simulation. It also conjectures that, for sufficiently large n, the average number of factors to represent n-variable interval functions by HTs is at most 2 n − 5 .E xperimental results also show that, forn ≥ 10

Let X be a Banach space, and let J be a nonempty interval in R, which may be bounded or unbounded, and open or closed at either end. Recall that an interval is called nondegenerate if it has nonempty interior or equivalently contains more than one point.

The aim of this paper is to introduce the notion of interval additive generators of interval t-norms as interval representations of additive generators of t-norms, considering both the correctness and the optimality criteria, in order to provide a more systematic methodology for the selection of interval t-norms in the various applications. We prove that interval additive generators satisfy the main properties of punctual additive generators discussed in the literature.

Another problem where interval computations axe used is a problem of approximating functions with simpler ones. In this chapter, we show that already the problem of optimal (narrowest) approximation of a quadratic interval function f (x1,..., xn) by a linear one is NP-hard. For a practically important 1D case (n = 1), an efficient approximation is possible.

This paper describes an improvement of the solution by Krawczyk, Moore and Jones (KMJ algorithm) for nonlinear equations based on the interval analysis. When the KMJ algorithm is applied to practical problems such as the determination of the operating point of a multistable electronic circuit, a large computation time is required. The reason for the large computation time is as follows: (i) The existence of the solution is determined using the direct-replacement type interval extensions of functions, which enlarges the search region for the solutions; (ii) the region partitioning is repeated until a region X is obtained such that K(X) ⊆ X, where K(X) is Krawczyk's interval function and X is the interval region. The situation is demonstrated by an example. As a solution for these problems, the following method is proposed. (1) The given initial region is partitioned into subregions, and some subregions are excluded from consideration if the solution is judged as nonexistent by the interval extensions of functions; (2) when K(X) ⊆ X, does not apply, the region X is partitioned into XC δ X - (K(X) ∩ X) and XK ∩ K(X) ∩ X, and the KMJ algorithm is applied to XC and XK to examine the existence of the solutions. Finally, as a computation example, the KMJ algorithm with the above two elaborations is applied to the nonlinear circuit equations with two to four variables. It is demonstrated that the computation time can be reduced considerably with the original KMJ algorithm.

An interval function as «universal primitive» in the sense of Marcinkiewicz is constructed for measurable functions defined in a compact interval of R n .

An interval function Y assigns an interval $Y(x) = (y(x),\bar y(x)]$ in the extended real number system to each x in its interval $X = [a,b]$ of definition. The integral of Y over $[a,b]$ is taken to be the interval $\int_a^b {Y(x)dx = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\int } _a^b \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} (x)dx,\smallint _a^b \bar y(x)]} $, where $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\int } _a^b \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} (x)$ is the lower Darboux integral of the lower endpoint function $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} $, and $\bar \int _a^b \bar y(x)dx$ is the upper Darboux integral of the upper endpoint function $\bar y$. Since these Darboux integrals always exist in the extended real number system, it follows that all interval functions are integrable, no matter how nasty the endpoint functions $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y...