Subadditive periodic functions

Abstract

Abstract. Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function f0 : [0, 1) → R is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.

Universal route to thermalization in weakly-nonlinear one-dimensional chains

“…The leading order nonlinearity in the evolution equation for the α-FPUT is cubic; this implies three-wave interactions of the form 2 → 1, 1 → 2 or 3 → 0 (the latter have been already excluded). It is known, see for example [4,31], that the function f k = 2| sin(k/2)| is a subadditive function, i.e.…”

On Subadditive Functions Bounded Above on a “Large” Set

“…Subadditive and sublinear functions play a fundamental role in mathematics and so have attracted the interest of many authors (see e.g. [3], [5], [20], [27], [31], [33], [34], [42]). Examples of subadditive functions include norms, seminorms, and the function R ∋ x → √ |x| ∈ R (see e.g.…”

New characterizations of the Takagi function via functional equations

“…Since f is even, it follows that f (1) = f (−1) ≤ 0. By Lemma 7 we deduce that f is 1-periodic, whence f (x) = f (−x) = f (1 − x) for every x ∈ [0, 1], so f satisfies (6). Now from Corollary 9 we see that f = T .…”