We show that the 2-abelian complexity of the infinite Thue–Morse word is 2-regular, and other properties of the 2-abelian complexity, most notably that it is a concatenation of palindromes of increasing length. We also show sharp bounds for the length of unique extensions of factors of size n, occurring in the Thue–Morse word.

An abelian square is the concatenation of two words that are anagrams of one another. A word of length n can contain Θ(n) distinct factors that are abelian squares. We study infinite words such that the number of abelian square factors of length n grows quadratically with n.

It is shown that various classes of graphs have universal elements. In particular, for each n the class of graphs omitting all paths of length n and the class of graphs omitting all circuits of length at least n possess universal elements in all infinite powers.

We present an infinite series of n-state Eulerian automata whose reset words have length at least (n − 3)/2. This improves the current lower bound on the length of shortest reset words in Eulerian automata. We conjecture that (n − 3)/2 also forms an upper bound for this class and we experimentally verify it for small automata by an exhaustive computation.

The factor complexity function C w (n) of a finite or infinite word w counts the number of distinct factors of w of length n for each n ≥ 0. A finite word w of length |w| is said to be trapezoidal if the graph of its factor complexity C w (n) as a function of n (for 0 ≤ n ≤ |w|) is that of a regular trapezoid (or possibly an isosceles triangle); that is, C w (n) increases by 1 with each n on so...

For every total recursive time bound t, a constant fraction of all compressible (low Kolmogorov complexity) strings is t-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of which every initial segment of length n is compressible to log n yet t-bounded incompressible below 1 4n− log n; and there are countable infinitely many recursiv...

A weakly nonlinear analysis is conducted for localized necking of a hyperelastic solid cylinder under axial stretching based on the exact theory elasticity. The amplitude equation derived shown to be consistent with one-dimensional model recently proposed by Audoly and Hutchinson (J. Mech. Phys. Solids 97, 2016, 68–91). It that results infinite-length approximation are sufficiently accurate eve...