multi quadric quasi interpolation method coupled with fdm for the degasperis procesi equation in this paper the degasperis procesi dp equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form u p based on the equivalent formulation of dp equation a new numerical method is presented the multi quadric mq quasi interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order tvd method is used to approximate the time derivative several examples are presented to demonstrate the effectiveness of the proposed method

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

numerical solution of the degasperis procesi equation by the cubic b spline quasi interpolation method abstract in this paper a numerical scheme is presented to solve the non dissipative degasperis procesi equation based on the u p formulation the cubic b spline quasi interpolation coupled with the finite difference method is applied to approximate the spatial derivatives and an optimal third order tvd runge kutta method to estimate the time derivative of the dependent variable the accuracy and effectiveness of the proposed method are validated by six classical problems numerical results indicate that the proposed scheme is simple easy to implement with high accuracy

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

two dimensional pattern matching with rotations the problem of pattern matching with rotation is that of finding all occurrences of a two dimensional pattern in a text in all possible rotations we prove an upper and lower bound on the number of such different possible rotated patterns subsequently given an m m array pattern and an n n array text over some finite alphabet we present a new method yielding an o n2m3 time algorithm for this problem

longest property preserved common factor a new string processing framework abstract we introduce a new family of string processing problems given two or more strings we are asked to compute a factor common to all strings that preserves a specific property and has maximal length we consider three fundamental string properties square free factors periodic factors and palindromic factors under three different settings one per property in the first setting we are given a string x and we are asked to construct a data structure over x answering the following type of online queries given a string y find a longest square free factor common to x and y in the second setting we are given k strings and an integer 1 k k and we are asked to find a longest periodic factor common to at least k strings in the third one we are given two strings and we are asked to find a longest palindromic factor common to the two strings we present linear time solutions for all settings this is a full and extended version of a paper from spire 2018

sequence similarity measures based on bounded hamming distance a growing number of measures of sequence similarity are being based on some underlying notion of relative compressibility within this paradigm similar sequences are expected to share a large number of common substrings or subsequences or more complex patterns or motifs and so on in this paper measures of sequence similarity are introduced and studied in which patterns in a pair are considered similar if they coincide up to a preset number of mismatches that is within a bounded hamming distance it is shown here that for some such measures bounds are achievable that are slightly better than o n 2 preliminary experiments demonstrate the potential applicability to phylogeny and classification of these similarity measures

dissipative conservative galerkin method using discrete partial derivatives for nonlinear evolution equations a new method is proposed for designing galerkin schemes that retain the energy dissipation or conservation properties of nonlinear evolution equations such as the cahn hilliard equation the korteweg de vries equation or the nonlinear schrodinger equation in particular as a special case dissipative or conservative finite element schemes can be derived the key device there is the new concept of discrete partial derivatives as examples of the application of the present method dissipative or conservative galerkin schemes are presented for the three equations with some numerical experiments

a local discontinuous galerkin method for the camassa holm equation in this paper we develop analyze and test a local discontinuous galerkin ldg method for solving the camassa holm equation which contains nonlinear high order derivatives the ldg method has the flexibility for arbitrary h and p adaptivity we prove the l 2 stability for general solutions and give a detailed error estimate for smooth solutions and provide numerical simulation results for different types of solutions of the nonlinear camassa holm equation to illustrate the accuracy and capability of the ldg method

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

k difference matching in amortized linear time for all the words in a text given a text x of length n we study the problem of solving the k difference problem for all the words either with fixed or variable length taken from the text itself the result finds its application in pattern discovery in biosequences where over or under represented words are extracted from the input sequences the proposed algorithm runs in amortized linear time per word this improves the complexity obtained by applying well known algorithms to each of the o n fixed length words or o n 2 variable length words in x by factor of k klogk or mlogm depending on the chosen algorithm the space required is o n if we just count the occurrences or o n 2 if we also store the positions this second scenario can be used as the basis for other applications such as searching gapped factors with mismatches or approximate pattern matching extended to any word

space efficient representation of truncated suffix trees with applications to markov order estimation suffix trees st are useful in many text processing applications for example to determine the number of occurrences of patterns of arbitrary length in an input string x if the length n of x is large the memory required to represent the st may become a practical performance bottleneck this problem can be alleviated in cases where a nontrivial upper bound is known on the lengths of the patterns of interest by using a truncated st tst however conventional tst implementations still require n bits of memory since they store x we describe a new tst representation that avoids this limitation by storing all the information necessary to reconstruct the tst edge labels in a string y that is often much shorter than x we apply tsts to the implementation of markov order estimators where an upper bound k n on the estimated order can be derived or it is imposed for consistency for example the new representation allows for estimator implementations with sublinear space complexity in some cases of interest in other cases we show experimentally that even when the new representation does not have an asymptotic advantage it still achieves very significant memory savings in practice

extracting string motif bases for quorum higher than two bases of generators of motifs consisting of strings in which some positions can be occupied by a don t care provide a useful conceptual tool for their description and a way to reduce the time and space involved in the discovery process in the last few years a few algorithms have been proposed for the extraction of a basis building in large part on combinatorial properties of strings and their autocorrelations currently the most efficient techniques for binary alphabets and quorum q 2 require time quadratic in the length of the host string the present paper explores properties of motif bases for quorum q 2 both with binary and general alphabets by also showing that important results holding for quorum q 2 cannot be extended to this more general case furthermore the extraction of motifs in which a bound is set on the maximum allowed number of don t cares is addressed and suitable algorithms are proposed whose computational complexity depends on the fixed bound

a dispersively accurate compact finite difference method for the degasperis procesi equation in this paper we are aimed to solve the non dissipative degasperis procesi equation based on the u p formulation to resolve the computational difficulty at the wave crest where the first order derivative may diverge and the shockpeakon solution may form the first order spatial derivative term in the two step equations will be approximated in a conservative form the resulting equations will be approximated by the symplecticity preserving time stepping scheme and the spatial discretization scheme that can optimize the numerical wavenumber for the first order spatial derivative term this scheme will be developed in a three point grid stencil with the accuracy order of seventh within the combined compact finite difference framework besides the validation of numerical accuracy we will in particular address the discrete conservation of hamiltonians even when peakon collides with antipeakon and generates as a result a shockpeakon we will also demonstrate the capability of applying the proposed numerical method to sharply resolve some important features of the third order dispersive dp equation

conservative finite difference schemes for the degasperis procesi equation we consider the numerical integration of the degasperis procesi equation which was recently introduced as a completely integrable shallow water equation for the equation we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi hamiltonian form of the equation at the same time we also prove the unique solvability of the schemes and show some numerical examples

motif trie an efficient text index for pattern discovery with don t cares abstract we introduce the motif trie data structure which has applications in pattern matching and discovery in genomic analysis plagiarism detection data mining intrusion detection spam fighting and time series analysis to name a few here the extraction of recurring patterns in sequential and textual data is one of the main computational bottlenecks for this we address the problem of extracting maximal patterns with at most k don t care symbols and at least q occurrences according to a maximality notion we define we apply the motif trie to this problem also showing how to build it efficiently as a result we give the first algorithm that attains a stronger notion of output sensitivity where the cost for an input sequence of n symbols is proportional to the actual number of occurrences of each pattern which is at most n much smaller in practice this avoids the best known cost of o n c per pattern for constant c 1 which is otherwise impractical for massive sequences with large n

the module structure of a group action on a polynomial ring abstract for any representation of a p group g on a vector space of dimension 3 over a finite field k of characteristic p we show how the symmetric algebra regarded as a kg module can be expressed as a direct sum of kg modules each one of which is isomorphic to a summand in low degree it follows that for any group g only a finite number of isomorphism classes of summands can occur

irredundant tandem motifs eliminating the possible redundancy from a set of candidate motifs occurring in an input string is fundamental in many applications the existing techniques proposed to extract irredundant motifs are not suitable when the motifs to search for are structured i e they are made of two or several subwords that co occur in a text string s of length n the main effort of this work is studying and characterizing a compact class of tandem motifs that is pairs of substrings occurring in tandem within a maximum distance of d symbols in s where d is an integer constant given in input to this aim we first introduce the concept of maximality related to four specific conditions that hold only for this class of motifs then we eliminate the remaining redundancy by defining the notion of irredundancy for tandem motifs we prove that the number of non overlapping irredundant tandem motifs is o d 2n which considering d as a constant leads to a linear number of tandems in the length of the input string this is an order of magnitude less than previously developed compact indexes for tandem extraction the notions and bounds provided for tandem motifs are generalized for the case r 2 if r is the number of subwords composing the motifs finally we also provide an algorithm to extract irredundant tandem motifs

an operator splitting method for the degasperis procesi equation an operator splitting method is proposed for the degasperis procesi dp equation by which the dp equation is decomposed into the burgers equation and the benjamin bona mahony bbm equation then a second order tvd scheme is applied for the burgers equation and a linearized implicit finite difference method is used for the bbm equation furthermore the strang splitting approach is used to construct the solution in one time step the numerical solutions of the dp equation agree with exact solutions e g the multipeakon solutions very well the proposed method also captures the formation and propagation of shockpeakon solutions and reveals wave breaking phenomena with good accuracy

a measure of dna sequence dissimilarity based on mahalanobis distance between frequencies of words summary a number of algorithms exist for searching genetic databases for biologically significant similarities in dna sequences past research has shown that word based search tools are computationally efficient and can find similarities or dissimilarities invisible to other algorithms like fasta we characterize a family of word based dissimilarity measures that define distance between two sequences by simultaneously comparing the frequencies of all subsequences of n adjacent letters i e n words in the two sequences applications to real data demonstrate that currently used word based methods that rely on euclidean distance can be significantly improved by using mahalanobis distance which accounts for both variances and covariances between frequencies of n words furthermore in those cases where mahalanobis distance may be too difficult to compute using standardized euclidean distance which only corrects for the variances of frequencies of n words still gives better performance than the euclidean distance also a simple way of combining distances obtained at different n words is considered the goal is to obtain a single measure of dissimilarity between two dna sequences the performance ranking of the preceding three distances still holds for their combined counterparts all results obtained in this paper are applicable to amino acid sequences with minor modifications

structure theorems over polynomial rings given a polynomial ring r over a field k and a finite group g we consider a finitely generated graded rg module s we regard s as a kg module and show that various conditions on s are equivalent such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of d karagueuzian p symonds the module structure of a group action on a polynomial ring a finiteness theorem preprint http www ma umist ac uk pas preprints

suffix tree characterization of maximal motifs in biological sequences finding motifs in biological sequences is one of the most intriguing problems for string algorithm designers due to on the one hand the numerous applications of this problem in molecular biology and on the other hand the challenging aspects of the computational problem indeed when dealing with biological sequences it is necessary to work with approximations that is to identify fragments that are not necessarily identical but just similar according to a given similarity notion and this complicates the problem existing algorithms run in time linear with respect to the input size nevertheless the output size can be very large due to the approximation namely exponential in the approximation degree this often makes the output unreadable as well as slowing down the inference itself a high degree of redundancy has been detected in the set of motifs that satisfy traditional requirements even for exact motifs moreover it has been observed many times that only a subset of these motifs namely the maximal motifs could be enough to provide the information of all of them in this paper we aim at removing such redundancy we extend some notions of maximality already defined for exact motifs to the case of approximate motifs with hamming distance and we give a characterization of maximal motifs on the suffix tree given that this data structure is used by a whole class of motif extraction tools we show how these tools can be modified to include the maximality requirement without changing the asymptotical complexity

group actions on rings and the cech complex abstract we have previously shown that when a finite group g acts on a polynomial ring s in n variables over a finite field k only finitely many isomorphism classes of indecomposable k g modules occur as summands of s we have also shown that the regularity of the invariant subring s g is at most zero which has various consequences for example s g is generated in degrees at most n g 1 provided n g 2 both of these theorems depend on the structure theorem of karagueuzian and the author which is proved by means of a long and complicated calculation the aim of this paper is to prove these results using a more conceptual method

the similarity metric a new class of distances appropriate for measuring similarity relations between sequences say one type of similarity per distance is studied we propose a new normalized information distance based on the noncomputable notion of kolmogorov complexity and show that it is in this class and it minorizes every computable distance in the class that is it is universal in that it discovers all computable similarities we demonstrate that it is a metric and call it the similarity metric this theory forms the foundation for a new practical tool to evidence generality and robustness we give two distinctive applications in widely divergent areas using standard compression programs like gzip and gencompress first we compare whole mitochondrial genomes and infer their evolutionary history this results in a first completely automatic computed whole mitochondrial phylogeny tree secondly we fully automatically compute the language tree of 52 different languages

a three point combined compact difference scheme a new three point combined compact difference ccd scheme is developed for numerical models the major features of the ccd scheme are three point implicit sixth order accuracy and inclusion of boundary values due to its combination of the first and second derivatives the ccd scheme becomes more compact and more accurate than normal compact difference schemes the efficient twin tridiagonal for calculating derivatives and triple tridiagonal for solving partial difference equation with the ccd scheme methods are also presented besides the ccd scheme has sixth order accuracy at periodic boundaries and fifth order accuracy at nonperiodic boundaries the possibility of extending to a three point eighth order scheme is also included

the smallest automaton recognizing the subwords of a text let a partial deterministic finite automaton be a dfa in which each state need not have a transition edge for each letter of the alphabet we demonstrate that the smallest partial dfa for the set of all subwords of a given word w iwl 2 has at most 21w 2 states and 3 wl 4 transition edges independently of the alphabet size we give an algorithm to build this smallest partial dfa from the input w on line in linear time

a fast solver of the shallow water equations on a sphere using a combined compact difference scheme the spectral like three point combined compact difference sp ccd scheme is developed for the shallow water models on a sphere which is originally proposed by chu and fan j comput phys 140 1998 370 a standard test set proposed by williamson et al j comput phys 102 1992 211 is used to evaluate the accuracy of the new method it is shown that the accuracy of the method is comparable to that of the spherical harmonics model as a whole the relative efficiency of the method with respect to the spherical harmonics model appears from the resolution of 512 256 and it becomes significant for the higher resolutions the test results suggest that the method could be extended to numerical atmospheric models

motif patterns in 2d motif patterns consisting of sequences of intermixed solid and don t care characters have been introduced and studied in connection with pattern discovery problems of computational biology and other domains in order to alleviate the exponential growth of such motifs notions of maximal saturation and irredundancy have been formulated whereby more or less compact subsets of the set of all motifs can be extracted that are capable of expressing all others by suitable combinations in this paper we introduce the notion of maximal irredundant motifs in a two dimensional array and develop initial properties and a combinatorial argument that poses a linear bound on the total number of such motifs the remainder of the paper presents approaches to the discovery of irredundant motifs both by offline and incremental algorithms

optimized prefactored compact schemes the numerical simulation of aeroacoustic phenomena requires high order accurate numerical schemes with low dispersion and dissipation errors in this paper we describe a strategy for developing high order accurate prefactored compact schemes requiring very small stencil support these schemes require fewer boundary stencils and offer simpler boundary condition implementation than existing compact schemes the prefactorization strategy splits the central implicit schemes into forward and backward biased operators using fourier analysis we show it is possible to select the coefficients of the biased operators such that their dispersion characteristics match those of the original central compact scheme and their numerical wavenumbers have equal and opposite imaginary components this ensures that when the forward and backward stencils are added the original central compact scheme is recovered to extend the resolution characteristic of the schemes an optimization strategy is employed in which formal order of accuracy is sacrificed in preference to enhanced resolution characteristics across the range of wavenumbers realizable on a given mesh the resulting optimized schemes yield improved dispersion characteristics compared to the standard sixth and eighth order compact schemes making them more suitable for high resolution numerical simulations in gas dynamics and computational aeroacoustics the efficiency accuracy and convergence characteristics of the new optimized prefactored compact schemes are demonstrated by their application to several test problems

truncated suffix trees and their application to data compression the suffix tree is a fundamental data structure in the area of string algorithms and it has been used in many applications including data compression in this paper we propose a data structure called the truncated suffix tree which is a truncated version of the suffix tree we also present two linear time construction algorithms for truncated suffix trees and two algorithms that delete suffixes from truncated suffix trees the truncated suffix tree is particularly a useful data structure for lz77 that compresses using a sliding window of a fixed size our algorithms lead to two implementations of lz77 that maintain sliding windows by truncated suffix trees we also present a technique of finding the longest match in a sliding window which is a crucial step in lz77

galois structure of homogeneous coordinate rings suppose g is a finite group acting on a projective scheme x over a commutative noetherian ring r we study the rg modules h 0 x f l n when n 0 and f and are coherent g sheaves on x such that l is an ample line bundle we show that the classes of these modules in the grothendieck group g 0 rg of all finitely generated rg modules lie in a finitely generated subgroup under various hypotheses we show that there is a finite set of indecomposable rg modules such that each h 0 x f l n is a direct sum of these indecomposables with multiplicities given by generalized hilbert polynomials for n 0