Showing papers by "Turku Centre for Computer Science published in 2020"

All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words.

Abstract: We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function f : N → R, we construct an infinite binary word whose abelian exponents have limit superior growth rate f . As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word. 2012 ACM Subject Classification Mathematics of computing → Combinatorics on words

On $k$-abelian equivalence and generalized Lagrange spectra

Abstract: We study the set of $k$-abelian critical exponents of all Sturmian words. It has been proven that in the case $k = 1$ this set coincides with the Lagrange spectrum. Thus the sets obtained when $k > 1$ can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when $k > 1$ the spectrum is a dense non-closed set. This is in contrast with the case $k = 1$, where the spectrum is a closed set containing a discrete part and a half-line. We describe explicitly the least accumulation points of the generalized spectra. Our geometric approach allows the study of $k$-abelian powers in Sturmian words by means of continued fractions.

Avoiding abelian powers cyclically

Abstract: We study a new notion of cyclic avoidance of abelian powers. A finite word $w$ avoids abelian $N$-powers cyclically if for each abelian $N$-power of period $m$ occurring in the infinite word $w^\omega$, we have $m \geq |w|$. Let $\mathcal{A}(k)$ be the least integer $N$ such that for all $n$ there exists a word of length $n$ over a $k$-letter alphabet that avoids abelian $N$-powers cyclically. Let $\mathcal{A}_\infty(k)$ be the least integer $N$ such that there exist arbitrarily long words over a $k$-letter alphabet that avoid abelian $N$-powers cyclically. We prove that $5 \leq \mathcal{A}(2) \leq 8$, $3 \leq \mathcal{A}(3) \leq 4$, $2 \leq \mathcal{A}(4) \leq 3$, and $\mathcal{A}(k) = 2$ for $k \geq 5$. Moreover, we show that $\mathcal{A}_\infty(2) = 4$, $\mathcal{A}_\infty(3) = 3$, and $\mathcal{A}_\infty(4) = 2$.

Identification of Drug Targets in Breast Cancer Metabolic Network.

Abstract: Genome-scale metabolic models have been proven to be valuable for defining cancer or to indicate the severity of cancer. However, identifying effective metabolic drug target (DT) of the active small-molecule compound is difficult to unravel and needs to be investigated. In this study, we identify effective DT for breast cancer using proposed network analysis of enzyme-centric network in the metabolic model. Our network-based analysis revealed that high degree nodes (HDNs) of enzymes are key to progression/development of cancer. These HDNs show high interconnections inside the network. It has been found that these HDNs are crucial driver nodes for effectively targeting in breast cancer metabolic network. Furthermore, based on the correlation and principal component analysis, we have shown that certain proteins play a significant role in the network and can be used as an effective DT in cancer therapeutics. In addition, these proteins stimulate the active site of enzymes to activate the target metabolites. Overall, we have shown that a better understanding of the metabolic networks using statistical model could be valuable in DT identification for developing effective therapeutic approaches and personalized medicine.

On prefix palindromic length of automatic words

Abstract: The prefix palindromic length $\mathrm{PPL}_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. Since 2013, it is still unknown if $\mathrm{PPL}_{\mathbf{u}}(n)$ is unbounded for every aperiodic infinite word $\mathbf{u}$, even though this has been proven for almost all aperiodic words. At the same time, the only well-known nontrivial infinite word for which the function $\mathrm{PPL}_{\mathbf{u}}(n)$ has been precisely computed is the Thue-Morse word $\mathbf{t}$. This word is $2$-automatic and, predictably, its function $\mathrm{PPL}_{\mathbf{t}}(n)$ is $2$-regular, but is this the case for all automatic words? In this paper, we prove that this function is $k$-regular for every $k$-automatic word containing only a finite number of palindromes. For two such words, namely the paperfolding word and the Rudin-Shapiro word, we derive a formula for this function. Our computational experiments suggest that generally this is not true: for the period-doubling word, the prefix palindromic length does not look $2$-regular, and for the Fibonacci word, it does not look Fibonacci-regular. If proven, these results would give rare (if not first) examples of a natural function of an automatic word which is not regular.

Inner symmetries of the spatially singular part of the solutions of the Burgers equation and their Lie representations

Abstract: We describe two new discrete symmetries of the inviscid Burgers (or Riemann–Hopf) equation u t + uu x = 0 . We derived both of them using a local, formal approach of Hopf algebraic renormalization, a tool recently used in algorithmic computations. We prove that one of them is a Lie point transformation. Symmetries generate new exact solutions from the known solutions and provide useful frames of reference in the study of shock wave formation.