Related papers: A characterization of eventually periodicity
For a complexity function $C$, the lower and upper $C$-complexity rates of an infinite word $\mathbf{x}$ are \[ \underline{C}(\mathbf x)=\liminf_{n\to\infty} \frac{C(\mathbf{x}\upharpoonright n)}n,\quad \overline{C}(\mathbf…
In this paper we develop combinatorial techniques for the case of string algebras with the aim to give a characterization of string complexes with infinite minimal projective resolution. These complexes will be called \textit{periodic…
We show that if the Sch\"{u}tzenberger graph of every positive word, that contains an $R$-word only once as it's subword, is finite over an Adain presentation $\langle X|u=v\rangle$, then the Sch\"{u}tzenberger graph of every positive word…
We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation \begin{equation*} u'' + cu' + \bigr{(} \lambda a^{+}(x) - \mu a^{-}(x) \bigr{)} g(u) = 0, \end{equation*}…
The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given…
A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many…
We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…
We investigate finite effect algebras and their classification. We show that an effect algebra with $n$ elements has at least $n-2$ and at most $(n-1)(n-2)/2$ nontrivial defined sums. We characterize finite effect algebras with these…
We consider the three conjectures stated in a 2003 paper of Wu, concerning the asymptotics of particular sums of products of binomials, powers and logarithms. These sums relate to the form of the regularised integrals used in loop…
We prove the "End Curve Theorem," which states that a normal surface singularity $(X,o)$ with rational homology sphere link $\Sigma$ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good…
In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all…
We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and…
Let A be a finite subset of the natural numbers containing 0, and let f(n) denote the number of ways to write n in the form $\sum e_j2^j$, where $\e_j \in A$. We show that there exists a computable T = T(A) so that the sequence (f(n) mod 2)…
In this paper we introduce and study a family of complexity functions of infinite words indexed by $k \in \ints ^+ \cup {+\infty}.$ Let $k \in \ints ^+ \cup {+\infty}$ and $A$ be a finite non-empty set. Two finite words $u$ and $v$ in $A^*$…
We present a new algorithm for general reinforcement learning where the true environment is known to belong to a finite class of N arbitrary models. The algorithm is shown to be near-optimal for all but O(N log^2 N) time-steps with high…
We argue that the finiteness of quantum gravity amplitudes in fully compactified theories (at least in supersymmetric cases) leads to a bottom-up prediction for the existence of non-trivial dualities. In particular, finiteness requires the…
We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more…
We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length $k$ is a constant, depending only on $k$ and…
For any positive integer $q$, the sequence of the Euler up/down numbers reduced modulo $q$ was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of $q$ precise conjectures for…
We introduce the concept of an extremely negatively dependent (END) sequence of random variables with a given common marginal distribution. The END structure, as a new benchmark for negative dependence, is comparable to comonotonicity and…