HIDDEN PATTERN STATISTICS

Abstract. Two fundamental problems in combinatorics on words and string manipulation are string matching and sequence comparison. In string matching one searches for all occurrences of a given string, understood as a sequence of consecutive symbols, in a text. In sequence comparison a subsequence rather than a string is searched in a text The string matching problem has been extensively studied in literature from algorithmic and probabilistic points of view. The sequence comparison problem, also known as hidden pattern problem, is harder and it has been much less investigated. In this paper we study the number of occurrences of a given pattern of length as a subsequence in a random text of length generated by a memoryless source. In particular, we consider two versions of this problem, namely the unconstrained one in which the subsequence can appear anywhere in the text, and the constrained one that puts bounds on the distances between symbols of the word. We determine the mean and the variance of the number of occurrences, and establish a Gaussian limit law. These results are obtained via combinatorics on words, formal languages, and methods of analytic combinatorics based on generating functions and moment methods. The motivation to study this problem comes from an attempt at finding a reliable threshold for intrusion detections, from textual data processing applications, and from molecular biology. 1

Magnetic Flux Emergence, Activity, Eruptions and Magnetic Clouds: Following Magnetic Field from the Sun to the Heliosphere

We present an overview of how the principal physical properties of magnetic flux which emerges from the toroidal fields in the tachocline through the turbulent convection zone to the solar surface are linked to solar activity events, emphasizing the effects of magnetic field evolution and interaction with other magnetic structures on the latter. We compare the results of different approaches using various magnetic observables to evaluate the probability of flare and coronal mass ejection (CME) activity and forecast eruptive activity on the short term (i.e. days). Then, after a brief overview of the observed properties of CMEs and their theoretical models, we discuss the ejecta properties and describe some typical magnetic and composition characteristics of magnetic clouds (MCs) and interplanetary CMEs (ICMEs). We review some individual examples to clarify the link between eruptions from the Sun and the properties of the resulting ejecta. The importance of a synthetic approach to solar and interplanetary magnetic fields and activity is emphasized. © 2008 Springer Science+Business Media B.V