Transition Property for α\alpha-Power Free Languages with α≥2\alpha\geq 2 and k≥3k\geq 3 Letters

In 1985, Restivo and Salemi presented a list of five problems concerning power free languages. Problem $4$ states: Given $\alpha$-power-free words $u$ and $v$, decide whether there is a transition from $u$ to $v$. Problem $5$ states: Given $\alpha$-power-free words $u$ and $v$, find a transition word $w$, if it exists. Let $\Sigma_k$ denote an alphabet with $k$ letters. Let $L_{k,\alpha}$ denote the $\alpha$-power free language over the alphabet $\Sigma_k$, where $\alpha$ is a rational number or a rational "number with $+$". If $\alpha$ is a "number with $+$" then suppose $k\geq 3$ and $\alpha\geq 2$. If $\alpha$ is "only" a number then suppose $k=3$ and $\alpha>2$ or $k>3$ and $\alpha\geq 2$. We show that: If $u\in L_{k,\alpha}$ is a right extendable word in $L_{k,\alpha}$ and $v\in L_{k,\alpha}$ is a left extendable word in $L_{k,\alpha}$ then there is a (transition) word $w$ such that $uwv\in L_{k,\alpha}$. We also show a construction of the word $w$

Transition Property For Cube-Free Words

We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair $(u,v)$ of $d$-ary cube-free words, if $u$ can be infinitely extended to the right and $v$ can be infinitely extended to the left respecting the cube-freeness property, then there exists a "transition" word $w$ over the same alphabet such that $uwv$ is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper. The obtained "transition property", together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity.Comment: 14 pages, 5 figure

Computing the kk-binomial complexity of the Thue--Morse word

Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue--Morse word is aperiodic, its $k$-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form $\Psi^\ell(w)$ for an arbitrary morphism $\Psi$. We also thoroughly describe the factors of the Thue--Morse word by introducing a relevant new equivalence relation

Overshoot mechanism in transient excitation of THz and Gunn oscillations in wide-bandgap semiconductors

A detailed study of high-field transient and direct-current (DC) transport in GaN-based Gunn diode oscillators is carried out using the commercial simulator Sentaurus Device. Applicability of drift-diffusion (DD) and hydrodynamic (HD) models to high-speed, highfrequency devices is discussed in depth, and the results of the simulations from these models are compared. It is shown, for a highly homogeneous device based on a short (2 μm) supercritically doped (1017 cm-3) GaN specimen, that the DD model is unable to correctly take into account some essential physical effects which determine the operation mode of the device. At the same time, the HD model is ideally suited to solve such problems due to its ability to incorporate non-local effects. We show that the velocity overshoot near the device contacts and space charge injection and extraction play a crucial role in defining the operation mode of highly homogeneous short diodes in both the transient regime and the voltagecontrolled oscillation regime. The transient conduction current responses are fundamentally different in the DD and HD models. The DD current simply repeats the velocity-field (v-F) characteristics, and the sample remains in a completely homogeneous state. In the HD model, the transient current pulse with a full width at half maximum of approximately 0.2 ps is increased about twofold due to the carrier injection (extraction) into (from) the active region and the velocity overshoot. The electron gas is characterized by highly inhomogeneous distributions of the carrier density, the electric field and the electron temperature. The simulation of the DC steady states of the diodes also shows very different results for the two models. The HD model shows the trapped stable anodic domain in the device, while the DD model completely retains all features of the v-F characteristics in a homogeneous gas. Simulation of the voltage-controlled oscillator shows that it operates in the accumulation layer mode generating microwave signals at 0.3 to 0.7 THz. In spite of the fact that the known criterion of a Gunn domain mode n0L > (n0L)0 was satisfied, no Gunn domains were observed. The explanation of this phenomenon is given. © 2012 Momox et al

Dramatic Co-Activation of WWOX/WOX1 with CREB and NF-κB in Delayed Loss of Small Dorsal Root Ganglion Neurons upon Sciatic Nerve Transection in Rats

BACKGROUND:Tumor suppressor WOX1 (also named WWOX or FOR) is known to participate in neuronal apoptosis in vivo. Here, we investigated the functional role of WOX1 and transcription factors in the delayed loss of axotomized neurons in dorsal root ganglia (DRG) in rats. METHODOLOGY/PRINCIPAL FINDINGS:Sciatic nerve transection in rats rapidly induced JNK1 activation and upregulation of mRNA and protein expression of WOX1 in the injured DRG neurons in 30 min. Accumulation of p-WOX1, p-JNK1, p-CREB, p-c-Jun, NF-kappaB and ATF3 in the nuclei of injured neurons took place within hours or the first week of injury. At the second month, dramatic nuclear accumulation of WOX1 with CREB (>65% neurons) and NF-kappaB (40-65%) occurred essentially in small DRG neurons, followed by apoptosis at later months. WOX1 physically interacted with CREB most strongly in the nuclei as determined by FRET analysis. Immunoelectron microscopy revealed the complex formation of p-WOX1 with p-CREB and p-c-Jun in vivo. WOX1 blocked the prosurvival CREB-, CRE-, and AP-1-mediated promoter activation in vitro. In contrast, WOX1 enhanced promoter activation governed by c-Jun, Elk-1 and NF-kappaB. WOX1 directly activated NF-kappaB-regulated promoter via its WW domains. Smad4 and p53 were not involved in the delayed loss of small DRG neurons. CONCLUSIONS/SIGNIFICANCE:Rapid activation of JNK1 and WOX1 during the acute phase of injury is critical in determining neuronal survival or death, as both proteins functionally antagonize. In the chronic phase, concurrent activation of WOX1, CREB, and NF-kappaB occurs in small neurons just prior to apoptosis. Likely in vivo interactions are: 1) WOX1 inhibits the neuroprotective CREB, which leads to eventual neuronal death, and 2) WOX1 enhances NF-kappaB promoter activation (which turns to be proapoptotic). Evidently, WOX1 is the potential target for drug intervention in mitigating symptoms associated with neuronal injury

Genomic Targets of Brachyury (T) in Differentiating Mouse Embryonic Stem Cells

The T-box transcription factor Brachyury (T) is essential for formation of the posterior mesoderm and the notochord in vertebrate embryos. Work in the frog and the zebrafish has identified some direct genomic targets of Brachyury, but little is known about Brachyury targets in the mouse.Here we use chromatin immunoprecipitation and mouse promoter microarrays to identify targets of Brachyury in embryoid bodies formed from differentiating mouse ES cells. The targets we identify are enriched for sequence-specific DNA binding proteins and include components of signal transduction pathways that direct cell fate in the primitive streak and tailbud of the early embryo. Expression of some of these targets, such as Axin2, Fgf8 and Wnt3a, is down regulated in Brachyury mutant embryos and we demonstrate that they are also Brachyury targets in the human. Surprisingly, we do not observe enrichment of the canonical T-domain DNA binding sequence 5'-TCACACCT-3' in the vicinity of most Brachyury target genes. Rather, we have identified an (AC)(n) repeat sequence, which is conserved in the rat but not in human, zebrafish or Xenopus. We do not understand the significance of this sequence, but speculate that it enhances transcription factor binding in the regulatory regions of Brachyury target genes in rodents.Our work identifies the genomic targets of a key regulator of mesoderm formation in the early mouse embryo, thereby providing insights into the Brachyury-driven genetic regulatory network and allowing us to compare the function of Brachyury in different species

Computing the k-binomial complexity of the Thue–Morse word

Two words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both the abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word x maps the integer n to the number of classes in the quotient, by this k-binomial equivalence relation, of the set of factors of length n occurring in x. This complexity measure has not been investigation very much. In this paper, we characterize the k-binomial complexity of the Thue–Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue–Morse word is aperiodic, its k-binomial complexity eventually takes only two values. In this paper, we first express the number of occurrences of subwords appearing in iterates of the form Ψ^l(w) for an arbitrary morphism Ψ. We also thoroughly describe the factors of the Thue–Morse word by introducing a relevant new equivalence relation