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Abstract: Let be the number of -words of length . Say that a -word is left doubly extendable (LDE) if both and are . We show that for any positive real number and positive integer such that the proportion of 2’s is greater than in each LDE word of length exceeding , there are positive constants and such that for all positive integers . With the best value known for , and large , this gives [ABSTRACT FROM AUTHOR]

Abstract: Given a sequence of real numbers and a positive integer , the Sum Selection Problem is to find the segment such that the rank of the sum is over all segments. We present a deterministic algorithm for this problem that runs in time. The previously best known result for this problem is a randomized algorithm that runs in expected time. Applying this algorithm we can obtain a deterministic algorithm for the k Maximum Sums Problem, i.e., the problem of enumerating the largest sum segments, that runs in time. The previously best known randomized and deterministic algorithms for the k Maximum Sums Problem run respectively in expected time and in worst case time. [Copyright &y& Elsevier]

Abstract: Let the (subword) complexity of a sequence over a finite set be the function , where denotes the number of distinct blocks of size in . In this paper, we study the complexity of when each , , is a -automatic sequence over a finite set and are pairwise coprime integers. As an application, we answer a question of Allouche and Shallit regarding morphic real numbers. [Copyright &y& Elsevier]

Abstract: Sturmian sequences appear in the work of Markoff on approximations of real numbers and minima of quadratic functions. In particular, Christoffel words, or equivalently pairs of relatively prime nonnegative integers, parametrize the Markoff numbers. It was asked by Frobenius if this parametrization is injective. We answer this conjecture for a particular subclass of these numbers, and show that a special Sturmian sequence of irrational slope determines the order of the Markoff numbers in this subclass. [Copyright &y& Elsevier]

Abstract: To any sequence of real numbers , we can associate another sequence , which Knuth calls its binomial transform. This transform is defined through the rule We study the properties of this transform, obtaining rules for its manipulation and a table of transforms, that allow us to invert many transforms by inspection. We use these methods to perform a detailed analysis of skip lists, a probabilistic data structure introduced by Pugh as an alternative to balanced trees. In particular, we obtain the mean and variance for the cost of searching for the first or the last element in the list (confirming results obtained previously by other methods), and also for the cost of searching for a random element (whose variance was not known). We obtain exact solutions, although not always in closed form. From them we are able to find the corresponding asymptotic expressions. [Copyright &y& Elsevier]