73 research outputs found

    Quasiperiodicity and non-computability in tilings

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    We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the fixed point construction; we improve this general technique and make it enforce the property of local regularity of tilings needed for quasiperiodicity. We prove also a stronger result: any effectively closed set can be recursively transformed into a tile set so that the Turing degrees of the resulted tilings consists exactly of the upper cone based on the Turing degrees of the later.Comment: v3: the version accepted to MFCS 201

    Fingerprints in Compressed Strings

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    The Karp-Rabin fingerprint of a string is a type of hash value that due to its strong properties has been used in many string algorithms. In this paper we show how to construct a data structure for a string S of size N compressed by a context-free grammar of size n that answers fingerprint queries. That is, given indices i and j, the answer to a query is the fingerprint of the substring S[i,j]. We present the first O(n) space data structures that answer fingerprint queries without decompressing any characters. For Straight Line Programs (SLP) we get O(logN) query time, and for Linear SLPs (an SLP derivative that captures LZ78 compression and its variations) we get O(log log N) query time. Hence, our data structures has the same time and space complexity as for random access in SLPs. We utilize the fingerprint data structures to solve the longest common extension problem in query time O(log N log l) and O(log l log log l + log log N) for SLPs and Linear SLPs, respectively. Here, l denotes the length of the LCE

    A new upper bound for the cross number of finite Abelian groups

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    In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite Abelian groups. Given a finite Abelian group, this upper bound appears to depend only on the rank and on the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite Abelian groups holds asymptotically in at least two different directions.Comment: 21 pages, to appear in Israel Journal of Mathematic

    Could the 2017 ILAE and the four-dimensional epilepsy classifications be merged to a new "Integrated Epilepsy Classification"?

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    Over the last few decades the ILAE classifications for seizures and epilepsies (ILAE-EC) have been updated repeatedly to reflect the substantial progress that has been made in diagnosis and understanding of the etiology of epilepsies and seizures and to correct some of the shortcomings of the terminology used by the original taxonomy from the 1980s. However, these proposals have not been universally accepted or used in routine clinical practice. During the same period, a separate classification known as the "Four-dimensional epilepsy classification" (4D-EC) was developed which includes a seizure classification based exclusively on ictal symptomatology, which has been tested and adapted over the years. The extensive arguments for and against these two classification systems made in the past have mainly focused on the shortcomings of each system, presuming that they are incompatible. As a further more detailed discussion of the differences seemed relatively unproductive, we here review and assess the concordance between these two approaches that has evolved over time, to consider whether a classification incorporating the best aspects of the two approaches is feasible. To facilitate further discussion in this direction we outline a concrete proposal showing how such a compromise could be accomplished, the "Integrated Epilepsy Classification". This consists of five categories derived to different degrees from both of the classification systems: 1) a "Headline" summarizing localization and etiology for the less specialized users, 2) "Seizure type(s)", 3) "Epilepsy type" (focal, generalized or unknown allowing to add the epilepsy syndrome if available), 4) "Etiology", and 5) "Comorbidities & patient preferences"

    Symbolic computation in RL/1

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