Let I be a finite interval and r ∈ N. Denote by ∆ s + L q the subset of all functions y ∈ L q such that the s-difference ∆ s τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by ∆ s + W r p , the class of functions x on I with the seminorm x (r) L p ≤ 1, such that ∆ s τ x ≥ 0, τ > 0. For s = 3,. .. , r + 1, we obtain two-sided estimates of the shape preserving widths

It is well established that systems of totally positive blending functions, such as the Bernstein and B-spline bases, preserve monotonicity and convexity and are generally ‘shape preserving’ [9 ]. In this paper we show that total positivity is equivalent to the preservation of all orders of convexity. By a system we understand a sequence of functions (u0, . . . , un) defined on an interval [a, ...

A second-order energy-preserving scheme is studied for the solution of the semilinear Cauchy Problem un uxx -uyy + u =0 (t > 0 ; x, y e R). Smooth data functions of compact support are prescribed at t = 0. On any time interval [0, 7"], second-order convergence (up to logarithmic corrections) to the exact solution is established in both the energy and uniform norms.

We introduce a basis of rational polynomial-like functions P 0 ,. .. , P n−1 for the free module of functions Z/nZ → Z/mZ. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the functions lcm(k) P k where lcm(k) is the least common multiple of 2,. .. , k (viewed in Z/mZ). As a consequence , when n ≥ m, the number of such functions is indep...

Fuzzy relations are mappings from pairs of elements into the interval [0, 1]. As a replacement for the complement operation one can use the mapping that sends x to 1 − x. Together with the concepts of t-norm and t-conorm a weak form of Boolean algebra can be defined. However, to our knowledge so far no notion of domain or codomain has been investigated for fuzzy relations. These might, however,...

1 Some Homework Exercises In introductory automata theory, one can nd a wealth of entertaining automata-theoretic puzzles such as the classical rst halves problem: Show that if A is a regular set, then so is the set of all rst halves of strings in A: FirstHalves(A) = fx j 9y jyj = jxj and xy 2 Ag : Once students master the basic pebbling technique for solving such problems, they can move on to ...