correction of numerov s eigenvalue estimates the error in the estimate of thekth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh lengthh iso k6h4 we show that a simple correction technique of paine de hoog and anderssen reduces the error to one ofo k3h4 numerical examples demonstrate the usefulness of this correction even for low values ofk

a four russians algorithm for regular expression pattern matching given a regular expression r of length p and a word a of length n the membership problem is to determine if a is in the language denoted by r an o pn lg n time algorithm is presented that is based on a lg n speedup of the standard o pn time simulation of r s nonderministic finite automaton on a using a combination of the node listing and four russians paradigms this result places a new worst case upper bound on regular expression pattern matching moreover in practice the method provides an implementation that is faster than existing software for small regular expressions

the chebyshev collocation path following method for solving sixth order sturm liouville problems in this paper we implement a chebyshev collocation method to approximate the eigenvalues of nonsingular sixth order sturm liouville problem this method transforms the sturm liouville problem to a sparse singular linear system which is solved by the path following technique numerical results demonstrate the accuracy and efficiency of the present algorithm

spaces of rna secondary structures abstract we prove two topological theorems in physical chemistry namely we introduce a hybrid of transverse and tangential measures on train tracks to prove sphericity of various simplicial complexes which arise from certain idealized models of physical chemistry these complexes are at once identified with thurston s space of projective geodesic laminations on an ideal polygon and with the analogue of a compactification described elsewhere of the moduli space of a punctured riemann surface the physical structures we study are various sub collections of the set of all possible planar chemical bonds among the sites of a linear macromolecule each such collection we consider has a natural partial ordering and the geometric realizations of appropriate posets are shown to be topological spheres such a topological statement encodes a wealth of combinatorial data as we briefly discuss in fact our primary motivation here is to study secondary structures on rna this imposes the further restriction that there can be at most one base pair supported at a given site of underlying linear macromolecule and imposing this restriction leads to the class of binary macromolecules our main results here assert the sphericity of certain topological spaces of both arbitrary and binary macromolecules and it is the latter which we hope may have applications to rna our techniques are largely elaborations of elementary topological techniques from techmuller theory and the theory of train tracks

matching a set of strings with variable length don t cares given an alphabet a a pattern p is a word 1 m where i a and a is a distinguished symbol called a variable length don t care symbol pattern p matches a text t a if t u0 1u1 um 1 mum for some u0 um a we address the following problem given a set p of patterns and a text t test whether one of the patterns of p matches t we describe an algorithm that solves the problem in time o t p log p in contrast to most of the existing string matching algorithms such as that of aho corasick our algorithm is not composed of two successive stages preprocessing the pattern resp the text and reading through the text resp the pattern but has these two stages essentially interleaved our approach is based on using the dawg directed acyclic word graph a data structure studied by a blumer j blumer haussler ehrenfeucht crochemore chen seiferas

the affix array data structure and its applications to rna secondary structure analysis efficient string processing in large data sets like complete genomes is strongly connected to the suffix tree and similar index data structures with respect to complex structural string analysis like the search for rna secondary structure patterns unidirectional suffix tree algorithms are inferior to bidirectional algorithms based on the affix tree data structure the affix tree incorporates the suffix tree and the suffix tree of the reverse text in one tree structure but suffers from its large memory requirements in this paper i present a new data structure denoted affix array which is equivalent to the affix tree with respect to its algorithmic functionality but with smaller memory requirements and improved performance i will show a linear time construction of the affix array without making use of the linear time construction of the affix tree i will also show how bidirectional affix tree traversals can be transferred to the affix array and present the impressive results of large scale rna secondary structure analysis based on the new data structure

approximate solution of periodic sturm liouville problems with chebyshev collocation method abstract this study investigates the eigenvalues of regular sturm liouville problems with chebyshev collocation method for the approximate computation of periodic boundary conditions by a truncated chebyshev series corrected eigenvalues with the chebyshev collocation method was obtained by using asymptotic correction technique the results were encouraging

computing similarity between rna structures the primary structure of a ribonucleic acid rna molecule is a sequence of nucleotides bases over the four letter alphabet a c g u the secondary or tertiary structure of an rna is a set of base pairs nucleotide pairs which forms bonds between au and cg for secondary structures these bonds have been traditionally assumed to be one to one and non crossing this paper considers a notion of similarity between two rna molecule structures taking into account the primary the secondary and the tertiary structures we show that for tertiary structures it is max snp hard for both minimization and maximization versions we show a stronger result for the maximization version where it cannot be approximated within ratio 2logn in polynomial time unless npdtime 2polylogn we then present an algorithm that can be used for practical application our algorithm will produce an optimal solution for the case where at least one of the rna involved is of a secondary structure we also show an approximation algorithm

complete controllability of finite level quantum systems complete controllability is a fundamental issue in the field of control of quantum systems not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables in this paper we investigate the question of complete controllability for finite level quantum systems subject to a single control field for which the interaction is of dipole form sufficient criteria for complete controllability of a wide range of finite level quantum systems are established and the question of limits of complete controllability is addressed finally the results are applied to give a classification of complete controllability for four level systems

fast and compact regular expression matching we study 4 problems in string matching namely regular expression matching approximate regular expression matching string edit distance and subsequence indexing on a standard word ram model of computation that allows logarithmic sized words to be manipulated in constant time we show how to improve the space and or remove a dependency on the alphabet size for each problem using either an improved tabulation technique of an existing algorithm or by combining known algorithms in a new way

compressed parameterized pattern matching pattern matching between traditional strings is well defined for both uncompressed and compressed sequences prior to this work parameterized pattern matching p matching was defined predominately by the matching between uncompressed parameterized strings p strings from the constant alphabet and the parameter alphabet in this work we define the compressed parameterized pattern matching compressed p matching problem to find all of the p matches between a pattern p and text t using only p and the compressed text t c initially we present parameterized compression p compression as a new way to losslessly compress data experimentally we show that p compression is competitive with various other standard compression schemes subsequently we provide the compression and decompression algorithms next two different approaches are developed to address the compressed p matching problem 1 using the recently proposed parameterized arithmetic codes pac and 2 using the parameterized border array p border our general solution is independent of the underlying compression scheme the results are further examined for catenate tunstall codes huffman codes and lzss

optimal control of two level quantum systems we study the manipulation of two level quantum systems this research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two level quantum system a quantum bit we consider the problem of driving the evolution operator to a desired state while minimizing an energy type cost mathematically this problem translates into an optimal control problem for systems varying on the lie group of special unitary matrices of dimension two with cost that is quadratic in the control we develop a comprehensive theory of optimal control for two level quantum systems this includes in particular a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions the impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed

suffix arrays a new method for on line string searches a new and conceptually simple data structure called a suffix array for on line string searches is introduced in this paper constructing and querying suffix arrays is reduced to a sort and search paradigm that employs novel algorithms the main advantage of suffix arrays over suffix trees is that in practice they use three to five times less space from a complexity standpoint suffix arrays permit on line string searches of the type is w a substring of a to be answered in time o p log n where p is the length of w and n is the length of a which is competitive with and in some cases slightly better than suffix trees the only drawback is that in those instances where the underlying alphabet is finite and small suffix trees can be constructed in o n time in the worst case versus o n log n time for suffix arrays however an augmented algorithm is given that regardless of the alphabet size constructs suffix arrays in o n expected time albeit with lesser space efficiency it is

mapped barycentric chebyshev differentiation matrix method for the solution of regular sturm liouville problems abstract in this paper using spectral differentiation matrix and an elimination treatment of boundary conditions sturm liouville problems slps are discretized into standard matrix eigenvalue problems the eigenvalues of the original sturm liouville operator are approximated by the eigenvalues of the corresponding chebyshev differentiation matrix cdm this greatly improves the efficiency of the classical chebyshev collocation method for slps where a determinant or a generalized matrix eigenvalue problem has to be computed furthermore the state of the art spectral method which incorporates the barycentric rational interpolation with a conformal map is used to solve regular slps a much more accurate mapped barycentric chebyshev differentiation matrix mbcdm is obtained to approximate the sturm liouville operator compared with many other existing methods the mbcdm method achieves higher accuracy and efficiency i e it produces fewer outliers when a large number of eigenvalues need to be computed the mbcdm method is very competitive hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer

algorithms for pattern matching and discovery in rna secondary structure text indexing structures provide significant advantages in the solution of many problems related to string analysis and comparison and are nowadays widely used in the analysis of biological sequences in this paper we present some applications of affix trees to problems of exact and approximate pattern matching and discovery in rna sequences by allowing bidirectional search for symmetric patterns in the sequences affix trees permit to discover and locate in the sequences patterns describing not only sequence regions but also containing information about the secondary structure that a given region could form with improvements in terms of theoretical and practical efficiency over the existing methods the search can be either exact or approximate where the approximation can be defined simultaneously both for the sequence and the structure of patterns the approach presented in this paper could provide significant help in the analysis of rna sequences where the functional motifs often involve not only sequence but also the structural constraints

identification of dynamical lie algebras for finite level quantum control systems the problem of identifying the dynamical lie algebras of finite level quantum systems subject to external control is considered with special emphasis on systems that are not completely controllable in particular it is shown that the dynamical lie algebra for an n level system with symmetrically coupled transitions such as a system with equally spaced energy levels and uniform transition dipole moments is a subalgebra of so n if n 2 1 and a subalgebra of sp if n 2 general criteria for obtaining either so 2 1 or sp are established

string matching with variable length gaps we consider string matching with variable length gaps given a string t and a pattern p consisting of strings separated by variable length gaps arbitrary strings of length in a specified range the problem is to find all ending positions of substrings in t that match p this problem is a basic primitive in computational biology applications let m and n be the lengths of p and t respectively and let k be the number of strings in p we present a new algorithm achieving time o email protected and space o m a where a is the sum of the lower bounds of the lengths of the gaps in p and a is the total number of occurrences of the strings in p within t compared to the previous results this bound essentially achieves the best known time and space complexities simultaneously consequently our algorithm obtains the best known bounds for almost all combinations of m n k a and a our algorithm is surprisingly simple and straightforward to implement we also present algorithms for finding and encoding the positions of all strings in p for every match of the pattern

a note on the height of suffix trees consider a random word in which the individual symbols are drawn from a finite or infinite alphabet with symbol probabilities p i and let h n be the height of the suffix tree constructed from the first n suffixes of this word it is shown that h n is asymptotically close to 2 log n log 1 sum i p i 2 in many respects the difference is o log log n in probability and the ratio tends to one almost surely and in the mean

approximate solution of the sturm liouville problems with legendre galerkin chebyshev collocation method abstract in this paper we use the legendre galerkin chebyshev collocation method lgcc to approximate the eigenvalues of a regular sturm liouville problem with three kinds of boundary conditions unlike the classical chebyshev collocation method this method preserves the symmetry of the problem numerical results show that the method is of high accuracy

approximate computation of eigenvalues with chebyshev collocation method in this study chebyshev collocation method is investigated for the approximate computation of higher sturm liouville eigenvalues by a truncated chebyshev series using the chebyshev collocation points this method transform the sturm liouville problems and given boundary conditions to matrix equation by solving the algebraic equation system the approximate eigenvalues can be computed hence by using asymptotic correction technique corrected eigenvalues can be obtained

optimal control of one qubit gates we consider the problem of carrying an initial bloch vector to a final bloch vector in a specified amount of time under the action of three control fields a vector control field we show that this control problem is solvable and therefore it is possible to optimize the control we choose the physically motivated criteria of minimum energy expended in the control minimum magnitude of the rate of change of the control and a combination of both we find exact analytical solutions determine the fields for a general one qubit gate and use the y gate as an example we argue that in the case of less than three controls only the physical intuition does not provide a straight reasonable solution and solve the problem in the case of a unique control minimizing the energy consumption

an improvement for chebyshev collocation method in solving certain sturm liouville problems abstract in this paper the chebyshev collocation method with an improved step for computing the approximate eigenvalues of regular sturm liouville problems is proposed the method is investigated for applying to the sturm liouville problems with two points and semi periodic boundary conditions the new step simplifies the computation by solving a generalized matrix eigenvalue problem instead of solving a high order algebraic equation required by the original method and thus more accurate results can be achieved the numerical results are encouraging

parameterized longest previous factor given a string w the longest previous factor lpf problem is to determine the maximum length of a previously occurring factor for each suffix occurring in w the lpf problem is defined for traditional strings exclusively from the constant alphabet s a parameterized string p string is a string composed of symbols from a constant alphabet s and a parameter alphabet p we formulate the lpf problem in terms of p strings by defining the parameterized longest previous factor plpf problem subsequently we present an expected linear time solution to construct the parameterized longest previous factor plpf array given our plpf solution we show how to construct the plcp parameterized longest common prefix array with the same general algorithm we exploit the properties of the plpf data structure to also construct the standard lpf longest previous factor and lcp longest common prefix arrays all in linear time further we provide insight into the practicality of our construction algorithms

efficient pattern matching for rna secondary structures we propose efficient methods to address key pattern matching problems in rna secondary structures using the notion of structural strings a structural string s string is composed of constant symbols and parameter symbols from the alphabets and respectively an individual symbol in the alphabet may be considered a complement of another unique symbol in the notion of matching constants parameters and complements is referred to as the structural matching s match problem which is helpful in matching rna and previously was solved by the structural suffix tree sst other approaches to rna matching that do not openly consider the s match include the use of affix data structures in this paper we provide new data structures and algorithms to address the s match problem specifically we introduce the structural suffix array and structural longest common prefix array and then identify how to s match with these data structures our new s matching solution is then used as the framework to answer various combinatorial queries encountered in matching rna secondary structures

exact matching of rna secondary structure patterns many rna structures are assembled from a collection of rna motifs which appear repeatedly and in various combinations identification of rna structural motifs will enhance our understanding of rna structures and functions searching for secondary structural patterns in sequence databases is the basic technique and fundamental problem for extracting and identifying such motifs a number of algorithms and programs have been developed for this purpose in this paper we adopt a representation of secondary structure called secondary expressions and present two algorithms for finding all exact matches of a given secondary expression