On some variations of coloring problems of infinite words

Given a finite coloring (or finite partition) of the free semigroup A(+) over a set A, we consider various types of monochromatic factorizations of right sided infinite words x is an element of A(omega). Some stronger versions of the usual notion of monochromatic factorization are introduced. A factorization is called sequentially monochromatic when concatenations of consecutive blocks are monochromatic. A sequentially monochromatic factorization is called ultra monochromatic if any concatenation of arbitrary permuted blocks of the factorization has the same color of the single blocks. We establish links, and in some cases equivalences, between the existence of these factorizations and fundamental results in Ramsey theory including the infinite Ramsey theorem, Hindman&#39;s finite sums theorem, partition regularity of IF sets and the Milliken Taylor theorem. We prove that for each finite set A and each finite coloring so : A(+) -&gt; C, for almost all words x is an element of A(omega), there exists y in the subshift generated by x admitting a so-ultra monochromatic factorization, where &quot;almost all&quot; refers to the Bernoulli measure on A(omega). (C) 2015 Elsevier Inc. All rights reserved.</p

Recurrence in the dynamical system (X,〈Ts〉s∈S) and ideals of βS

A dynamical system is a pair ( X , 〈 T s 〉 s ∈ S ) , where X is a compact Hausdorff space, S is a semigroup, for each s ∈ S , T s is a continuous function from X to X , and for all s , t ∈ S , T s ∘ T t = T s t . Given a point p ∈ β S , the Stone-Čech compactification of the discrete space S , T p : X → X is defined by, for x ∈ X , T p ( x ) = p − lim s ∈ S T s ( x ) . We let β S have the operation extending the operation of S such that β S is a right topological semigroup and multiplication on the left by any point of S is continuous. Given p , q ∈ β S , T p ∘ T q = T p q , but T p is usually not continuous. Given a dynamical system ( X , 〈 T s 〉 s ∈ S ) , and a point x ∈ X , we let U ( x ) = p ∈ β S : T p ( x ) is uniformly recurrent . We show that each U ( x ) is a left ideal of β S and for any semigroup we can get a dynamical system with respect to which K ( β S ) = ⋂ x ∈ X U ( x ) and c ℓ K ( β S ) = ⋂ U ( x ) : x ∈ X and U ( x ) is closed . And we show that weak cancellation assumptions guarantee that each such U ( x ) properly contains K ( β S ) and has U ( x ) ∖ c ℓ K ( β S ) ≠ ∅

Variations of the Morse-Hedlund Theorem for k-Abelian Equivalence

In this paper we investigate local-to-global phenomena for a new family of complexity functions of infinite words indexed by k >= 0. Two finite words u and v are said to be k-abelian equivalent if for all words x of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equivalence relations, bridging the gap between the usual notion of abelian equivalence (when k = 1) and equality (when k = infinity). Given an infinite word w, we consider the associated complexity function which counts the number of k-abelian equivalence classes of factors of w of length n. As a whole, these complexity functions have a number of common features: Each gives a characterization of periodicity in the context of bi-infinite words, and each can be used to characterize Sturmian words in the framework of aperiodic one-sided infinite words. Nevertheless, they also exhibit a number of striking differences, the study of which is one of the main topics of our paper

Engineering Genetically Encoded Nanosensors for Real-Time In Vivo Measurements of Citrate Concentrations

Citrate is an intermediate in catabolic as well as biosynthetic pathways and is an important regulatory molecule in the control of glycolysis and lipid metabolism. Mass spectrometric and NMR based metabolomics allow measuring citrate concentrations, but only with limited spatial and temporal resolution. Methods are so far lacking to monitor citrate levels in real-time in-vivo. Here, we present a series of genetically encoded citrate sensors based on Förster resonance energy transfer (FRET). We screened databases for citrate-binding proteins and tested three candidates in vitro. The citrate binding domain of the Klebsiella pneumoniae histidine sensor kinase CitA, inserted between the FRET pair Venus/CFP, yielded a sensor highly specific for citrate. We optimized the peptide linkers to achieve maximal FRET change upon citrate binding. By modifying residues in the citrate binding pocket, we were able to construct seven sensors with different affinities spanning a concentration range of three orders of magnitude without losing specificity. In a first in vivo application we show that E. coli maintains the capacity to take up glucose or acetate within seconds even after long-term starvation

Suffix Automata and Standard Sturmian Words

Blumer et al. showed (cf. [3,2]) that the suffix automaton of a word w must have at least |w|+1 states and at most 2|w|-1 states. In this paper we characterize the language L of all binary words w whose minimal suffix automaton S(w) has exactly |w| + 1 states; they are precisely all prefixes of standard Sturmian words. In particular, we give an explicit construction of suffix automaton of words that are palindromic prefixes of standard words. Moreover, we establish a necessary and sufficient condition on S(w) which ensures that if w ∈ L and a ∈ {0, 1} then wa ∈ L. By using such a condition, we show how to construct the automaton S(wa) from S(w). More generally, we provide a simple construction that by starting from an automaton recognizing all suffixes of a word w over a finite alphabet A, allows to obtain an automaton that recognizes the suffixes of wa, a ∈