We study finite delay evolution equation { x′(t) = Ax(t) + F (t, xt), t ≥ 0, x0 = φ ∈ C ([−r, 0] , E) , where linear operator A is non-densely defined and satisfies the Hille-Yosida condition. First we obtain some properties of “integral solutions” in this case, and prove the compactness of an operator determined by integral solutions. This allows us to apply Horn’s fixed point theorem to prove...

Equations of the form X = φ(X) are considered, where the unknown X is a set of natural numbers. The expression φ(X) may contain the operations of set addition, defined as S + T = {m + n | m ∈ S, n ∈ T}, union and intersection, as well as ultimately periodic constants. An equation with a non-periodic solution of exponential growth is constructed. At the same time it is demonstrated that no sets ...

It is known that there are recurrent words with constant abelian complexity three, but not with constant complexity four. We prove that there are recurrent words with ultimately constant complexity c for every c.

Neighborhoods and neighborhood sequences play important roles in several branches of pattern analysis. In earlier papers in Z only certain special (e.g. periodic or octagonal) sequences were investigated. In this paper we study neighborhood sequences which are either ultimately periodic or allow at every neighborhood to do nothing at no cost. We give finite procedures and descriptive theoretica...

Cobham's theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with respect to two multiplicatively independent bases coincide on a set of density one, then they also coincide on a set of density one with a periodic sequence. W...

According to biological strategy for pest control, a mathematical model with periodic releasing virus particles for insect viruses attacking pests is considered. By using Floquet’s theorem, small-amplitude perturbation skills and comparison theorem, we prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pesteradication period...

We study the behavior of Conway’s RATS (reverse-add-then-sort) sequences in base 3. An independent proof is given of Gentges’ result that all RATS sequences in base 3 are ultimately periodic. We also prove Erdős-Kac and Hardy-Ramanujan type results for the periods of such sequences.

Infinite cellular automata have been studied mostly using empirical and sta tistical techniques, with some combinatorial analysis. Here we show how concepts of universal algebra such as subdirect decomposition and chains of varieties can be applied to their study. Cellular automata with ultimately periodic behavior are shown to correspond to varieties of groupoids. Relat ionships between these ...

Much efforts have been made for the realization of hybrid Josephson junctions incorporating various materials for the fundamental studies of exotic physical phenomena as well as the applications to superconducting quantum devices. Nonetheless, the efforts have been hindered by the diffusive nature of the conducting channels and interfaces. To overcome the obstacles, we vertically sandwiched a c...

numeration systems An abstract numeration system is a triple S = (L,Σ, <) where L is an infinite regular language over the totally ordered alphabet (Σ, <) By enumerating words of L in the radix order (induced by <), we define a one-to-one correspondence between N and L repS : N → L : n 7→ (n + 1)th word of L valS = rep −1 S : L → N Abstract numeration systems Example S = (a∗b∗, {a, b}, a < b) L...