It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitelyintersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph. Let g be an infinite, simple, connected...

Arita and Kobayashi proposed a method for constructing comma-free DNA codes using binary templates, and showed that the separation d of any such binary template of length n satisfies d n/2. Kobayashi, Kondo andArita later produced an infinite family of binary templates with d 11n/30. Here we demonstrate the existence of an infinite family of binary templates with d >n/2 − (18n loge n) 1/2. We a...

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 a countably infinite number of re...

Let u v denote the set of all shuffles of the words u and v . It is shown that for each integer n ≥ 3 there exists a square-free ternary word u of length n such that u u contains a square-free word. This property is then shown to also hold for infinite words, i.e., there exists an infinite square-free word u on three letters such that u can be shuffled with itself to produce an infinite square-...

The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computatio...

We present a theory of Anderson localization on one-dimensional lattice with translation-invariant hopping. find by analytical calculation, the length for arbitrary finite-range hopping in single propagating channel regime. Then examining convergence length, limit infinite range, we revisit problem criteria this model and investigate conditions under which it can be violated. Our results reveal...

We introduce and study a common generalization of 1-error binary perfect codes and perfect single error correcting codes in Lee metric, namely perfect codes on products of paths of length 2 and of infinite length. Both existence and nonexistence results are given.

In this paper we consider chaotic discrete-time systems which generate polyphase sequences. First we give the definition of infinite polyphase sequences with perfect correlation properties. Then a simple design rule for chaotic generators is derived which guarantees that the generated sequences are perfect. Following this rule we construct families of generators of infinite polyphase sequences....

We study transitionless quantum driving in an infinite-range many-body system described by the Lipkin-Meshkov-Glick model. Despite the correlation length being always infinite the closing of the gap at the critical point makes the driving Hamiltonian of increasing complexity also in this case. To this aim we develop a hybrid strategy combining a shortcut to adiabaticity and optimal control that...

In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius ρ = 3 and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length n = 2 − 1 and 2, respectively, where m is even. From these new completely transitive codes, in the usual way, i.e., as coset graphs, new presentations of infinite families o...