the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

a new linear time algorithm to compute the genomic distance via the double cut and join distance the genomic distance problem in the hannenhalli pevzner hp theory is the following given two genomes whose chromosomes are linear calculate the minimum number of translocations fusions fissions and inversions that transform one genome into the other this paper presents a new distance formula based on a simple tree structure that captures all the delicate features of this problem in a unifying way and a linear time algorithm for computing this distance

optimal algorithms of gram schmidt type abstract three algorithms of gram schmidt type are given that produce orthogonal decompositions of finite d dimensional symmetric alternating or hermitian forms over division rings the first uses d 3 3 o d 2 products in next that algorithm is adapted in two new directions one is a nearly optimal sequential algorithm whose complexity matches the complexity of matrix multiplication the other is a parallel algorithm

analysis and computation of the elastic wave equation with random coefficients we consider the stochastic initial boundary value problem for the elastic wave equation with random coefficients and deterministic data we propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest we study the convergence rate of the error in the stochastic collocation method in particular we show that the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest we demonstrate that a fast rate of convergence is possible in two cases for the elastic wave solutions with high regular data and for some high regular quantities of interest even in the presence of low regular data we perform numerical examples including a simplified earthquake which confirm the analysis and show that the collocation method is a valid alternative to the more traditional monte carlo sampling method for approximating quantities with high stochastic regularity

some problems on cayley graphs this survey paper presents the historical development of some problems on cayley graphs which are interesting to graph and group theorists such as hamiltonicity or diameter problems to computer scientists and molecular biologists such as pancake problem or sorting by reversals to coding theorists such as the vertex reconstruction problem related to error correcting codes but not related to ulam s problem

bayesian phylogenetic inference from animal mitochondrial genome arrangements summary the determination of evolutionary relationships is a fundamental problem in evolutionary biology genome arrangement data are potentially more informative than deoxyribonucleic acid sequence data for inferring evolutionary relationships between distantly related taxa we describe a bayesian framework for phylogenetic inference from mitochondrial genome arrangement data using markov chain monte carlo methods we apply the method to assess evolutionary relationships between eight animal phyla

sequential vs parallel complexity in simple gene assembly we investigate some differences between the general intramolecular model for gene assembly and its restricted simple model although both models satisfactorily sort all current experimental data we show that the general model offers assembly strategies for a given string that vary in both assembly length and the operations used while the simple model will always use the same number of each type of operation to sort a gene when simple operations are applied in parallel this is given a new twist we prove that for any n 1 there exists a string having maximally parallel assemblies of any length between n and 2n

on the exact controllability of hyperbolic magnetic schrodinger equations abstract in this paper we address the exact controllability problem for the hyperbolic magnetic schrodinger equation which plays an important role in the research of electromagnetics typical techniques such as hamiltonian induced hilbert spaces and pseudodifferential operators are introduced by choosing an appropriate multiplier we proved the observability inequality with sharp constants in particular a genuine compactness uniqueness argument is applied to obtain the minimal time in the final analysis a suitable boundary control is constructed by the systematic hilbert uniqueness method introduced by j l lions compared with the micro local discussion in bardos et al 1992 we do not require the coefficients belong to c actually c 1 is already sufficient for the vector potential of the hyperbolic electromagnetic equation

signed genome rearrangement by reversals and transpositions models and approximations an important problem in computational biology is the genome rearrangement using reversals and transpositions analysis of genome evolving by reversals and transpositions leads to a combinatorial optimization problem of sorting by reversals and transpositions i e sorting of a permutation using reversals and transpositions of arbitrary fragments the reversal operation works on a single segment of the genome by reversing the selected segment two kinds of transpositions have been studied in the literature the first kind of transposition operations delete a segment of the genome and insert it into another position in the genome the second kind of transposition operations delete a segment of the genome and insert its inverse into another position in the genome both transposition operations can be viewed as operations working on two consecutive segments in this paper we introduce a third transposition operation which works on two consecutive segments and study sorting of a signed permutation by reversals and transpositions by allowing reversals and the first kind of transpositions or reversals and the first two kinds of transpositions or reversals and all three kinds of transpositions we have three problem models after establishing a common lower bound on the number of operations needed we present a unified 2 approximation algorithm for all these problems finally we present a better 1 75 approximation for the third problem

on the strong unique continuation of electromagnetic schrodinger equations with random coefficients abstract stochastic partial differential equations spdes with random coefficients are useful mathematical models for the quantification of perturbations in lots of complex engineering and physical applications this paper mainly addresses the strong unique continuation property for the electromagnetic schrodinger operator with random coefficients by the method proposed by n garofalo and f h lin 10 11 appropriate multipliers with physical backgrounds have been introduced to prove a priori estimates moreover its application in an exact controllability problem has been shown in which case the boundary value determines the interior value completely

a two level stochastic collocation method for semilinear elliptic equations with random coefficients in this work we propose a novel two level discretization for solving semilinear elliptic equations with random coefficients motivated by the two grid method for deterministic partial differential equations pdes introduced by xu 1994 our two level stochastic collocation method utilizes a two grid finite element discretization in the physical space and a two level collocation method in the random domain in particular we solve semilinear equations on a coarse mesh t h with a low level stochastic collocation corresponding to the polynomial space p p and solve linearized equations on a fine mesh t h using high level stochastic collocation corresponding to the polynomial space p p we prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with t h and p p the two level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients numerical experiments are provided to verify the theoretical results

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

probabilistic models for stochastic elliptic partial differential equations mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively yet additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities referred to as physical requirements have not been examined systematically it is shown that current models for random coefficients constructed solely by mathematical considerations can violate physical constraints and consequently be of limited practical use we develop alternative models for the random coefficients of stochastic differential equations that satisfy both mathematical and physical constraints theoretical arguments are presented to show potential limitations of current models and establish properties of the models developed in this study numerical examples are used to illustrate the construction of the proposed models assess the performance of these models and demonstrate the sensitivity of the solutions of stochastic differential equations to probabilistic characteristics of their random coefficients

sorting permutations and binary strings by length weighted rearrangements abstract comparative genomics is a line of study which aims at determining dis similarities between genomes one way of doing this is to infer the large scale mutations also called genome rearrangements that are responsible for transforming one genome into another under the parsimony principle numerous efforts have been made over the past years regarding the traditional approach in which all genome rearrangements have the same unit cost however since long rearrangements are more likely to disturb the organism some works considered the length weighted approach in which the cost of each rearrangement is a function of its length the problem is thus modeled as follows given two genomes g 1 and g 2 and a set of allowed genome rearrangements each with a given cost determine their distance i e the total cost of a minimum cost sequence of rearrangements from that transforms g 1 into g 2 in this paper we study length weighted reversals and transpositions mainly considering their prefix variants in which the leftmost part of the genome must always be included we present approximation algorithms and bounds on the distances and diameters i e maximum possible distance over any two genomes of same length for eight problems when genomes are represented as permutations and another eight problems when genomes are represented as binary strings in each case considering two different cost functions

sorting by reversals and the theory of 4 regular graphs abstract we show that the theory of sorting by reversals fits into the well established theory of circuit partitions of 4 regular multigraphs which also involves the combinatorial structures of circle graphs and delta matroids in this way we expose strong connections between the two theories that have not been fully appreciated before we also discuss a generalization of sorting by reversals involving the double cut and join dcj operation finally we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates

quick greedy computation for minimum common string partition in the minimum common string partition problem one is given two strings s and t with the same character statistics and one seeks for the smallest partition of s into substrings so that t can also be partitioned into the same substring multiset the problem is fundamental in several variants of edit distance with block operations e g signed reversal distance with duplicates and edit distance with moves the minimum common string partition problem is known to be np complete and the best approximation algorithm known has an approximation factor of o lognlog n since the minimum common string partition problem is of utmost practical importance one seeks a heuristic that will 1 usually have a low approximation factor and 2 will run fast a simple greedy algorithm is known which iteratively choose non overlapping longest common substrings of the input strings this algorithm has been well studied from an approximation point of view and it has been shown to have a bad worst case approximation factor however all the bad approximation factors presented so far stem from complicated recursive construction in practice the greedy algorithm seems to have small approximation factors however the best current implementation of greedy runs in quadratic time we propose a novel method to implement greedy in linear time

approximation algorithms for sorting by length weighted prefix and suffix operations we addressed sorting permutations by prefix and suffix reversals and transpositions we initiated the study of these rearrangements in a length weighted context we presented approximation algorithms for 10 such problems the traditional approach for the problems of sorting permutations by rearrangements is to consider that all operations have the same unitary cost in this case the goal is to find the minimum number of allowed rearrangements that are needed to sort a given permutation and numerous efforts have been made over the past years regarding these problems on the other hand a long rearrangement which is in fact a mutation is more likely to disturb the organism therefore weights based on the length of the segment involved may have an important role in the evolutionary process in this paper we present the first results regarding problems of sorting permutations by length weighted operations that consider rearrangement models with prefix and suffix variations of reversals and transpositions which are the two most common types of genome rearrangements our main results are o lg 2 n approximation algorithms for 10 such problems

a 2 approximation algorithm for genome rearrangements by reversals and transpositions recently a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of dna sequences was proposed sankoff et al 1992 the approach is based on the global rearrangements e g inversions and transpositions of fragments analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions i e sorting of a permutation using reversals and transpositions of arbitrary fragments we study sorting of signed permutations by reversals and transpositions a problem which adequately models genome rearrangements as the genes in dna are oriented we establish a lower bound and give a 2 approximation algorithm for the problem

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

a phase transition in the random transposition random walk our work is motivated by bourque pevzner s simulation study of the effectiveness of the parsimony method in studying genome rearrangement and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on n elements starting from the identity let d t be the minimum number of transpositions needed to go back to the identity element from the location at time t d t undergoes a phase transition for 0 1 d cn 2 u c n where u is an explicit function satisfying u x x 2 moreover we describe the fluctuations of d cn 2 about its mean at each of the three stages subcritical critical and supercritical the techniques used involve viewing the cycles in the random permutation as a coagulation fragmentation process and relating the behavior to the erd s renyi random graph model

genome rearrangements and sorting by reversals sequence comparison in molecular biology is in the beginning of a major paradigm shift a shift from gene comparison based on local mutations i e insertions deletions and substitutions of nucleotides to chromosome comparison based on global rearrangements i e inversions and transpositions of fragments the classical methods of sequence comparison do not work for global rearrangements and little is known in computer science about the edit distance between sequences if global rearrangements are allowed in the simplest form the problem of gene rearrangements corresponds to sorting by reversals i e sorting of an array using reversals of arbitrary fragments recently kececioglu and sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements one of these problems is gollan s conjecture on the reversal diameter of the symmetric group this paper proves the conjecture further the problem of expected reversal distance between two random permutations is investigated the reversal distance between two random permutations is shown to be very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies the gene rearrangement problem forces us to consider reversals of signed permutations as the genes in dna could be positively or negatively oriented an approximation algorithm for signed permutation is presented which provides a performance guarantee of 3 over 2 finally using the signed permutations approach an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of 7 over 4

regularity of hyperbolic magnetic schrodinger equation with oscillating coefficients this paper mainly discusses the regularity of hyperbolic magnetic schrodinger equation with singular coefficients near the origin we apply the techniques from microlocal analysis to explore the upper bound of loss of regularity furthermore in order to demonstrate the optimality of the result a delicate counterexample with periodic coefficients will be constructed to show the lower bound of loss of regularity by the application of harmonic analysis and instability arguments

testing isomorphism of modules abstract we present a new deterministic algorithm to test constructively for isomorphism between two given finite dimensional modules of a finitely generated algebra the algorithm uses only basic field operations for arbitrary fields this is not possible with the existing methodology furthermore the number of field operations used by the algorithm is bounded by a polynomial in the length of the input the algorithm has been implemented in the computer algebra system magma and we report on its performance our approach has applications to other problems concerning decompositions of modules

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

a simpler and faster 1 5 approximation algorithm for sorting by transpositions an important problem in genome rearrangements is sorting permutations by transpositions the complexity of the problem is still open and two rather complicated 1 5 approximation algorithms for sorting linear permutations are known bafna and pevzner 98 and christie 99 the fastest known algorithm is the quadratic algorithm of bafna and pevzner in this paper we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions hence all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations our main result is a new o n3 2 log n 1 5 approximation algorithm which is considerably simpler than the previous ones and whose analysis is significantly less involved

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

successor rules for flipping pancakes and burnt pancakes a stack of n pancakes can be rearranged in all n ways by a sequence of n 1 flips and a stack of n burnt pancakes can be rearranged in all 2 n n ways by a sequence of 2 n n 1 flips in both cases a computer program can efficiently generate suitable solutions we approach these tasks instead from a human perspective how can we determine the next flip directly from the current stack how can we flip the minimum or maximum number of burnt pancakes overall what if we are only allowed to flip the top n 2 n 1 or n burnt pancakes we answer the first question with simple successor rules that take worst case o n time and amortized o 1 time then we answer the second question exactly for minimization and provide conjectures for maximization for the third question we prove that solutions almost certainly exist for pancakes and burnt pancakes using only these three flips more broadly we discuss how efficiency and optimality can shape iterative solutions to hamilton cycle problems in highly symmetric graphs

discussion on the meeting on statistical modelling and analysis of genetic data david j balding imperial college school of medicine london i extendmy apologies to the authors that an unavoidable commitment arising unexpectedly in the 18 hours before the meeting robbed me of my final preparation so that my comments at the meeting were poorly presented i shall try to do a better job in this written version and to leave enough space i shall omit my introductory comments about the role of statisticians in bioinformatics

pivots determinants and perfect matchings of graphs we consider sequences of local and edge complementations on graphs with loops we allow local complementation only on looped vertices we recall that these operations together form the matrix operation of principal pivot transform or pivot restricted to graphs this fact is not well known and as a consequence various special cases of the properties of pivot have been rediscovered multiple times in this paper we give a gentle overview of various properties of pivot for local and edge complementations on graphs moreover we relate the pivot operation to perfect matchings to obtain a purely graph theoretical characterization of the effect of sequences of pivot operations finally we show that two of the three operations that make up a formal graph model of the biological process of gene assembly in ciliates together form the matrix operation of schur complement restricted to graphs

computing automorphism groups and testing isomorphisms for modular group algebras abstract we describe a new algorithm for computing automorphism groups and checking isomorphism of nilpotent finite dimensional associative algebras over a finite field the algorithm can also be applied to modular group algebras and thus yields a new approach for checking the modular isomorphism problem we report on its application to the groups of order dividing 2 8 and 3 6

computing the automorphism group of a solvable lie algebra we describe an effective algorithm for computing the automorphism group of a finite dimensional solvable lie algebra over a finite field we show that a similar approach can be used to determine a canonical table for such a lie algebra that is a description for the lie algebra which is invariant under isomorphisms hence we also obtain an effective isomorphism test for finite dimensional solvable lie algebras over finite fields

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

an efficient algorithm for one sided block ordering problem under block interchange distance abstract in this work we study the one sided block ordering problem under block interchange distance given two signed permutations and of size n where represents a partially assembled genome consisting of several blocks i e contigs and represents a completely assembled genome the one sided block ordering problem under block interchange distance is to order i e assemble the blocks of such that the block interchange distance between the assembly of and is minimized the one sided block ordering problem is useful in genome resequencing because its algorithms can be used to assemble the contigs of partially assembled resequencing genomes based on their completely assembled genomes by using permutation groups in algebra we design an efficient algorithm to solve the one sided block ordering problem under block interchange distance in o n log n time moreover we show that the assembly of can be done in o n time and its block interchange distance from can also be calculated in advance in o n time

whole genome duplications and contracted breakpoint graphs the genome halving problem motivated by the whole genome duplication events in molecular evolution was solved by el mabrouk and sankoff in the pioneering paper siam j comput 32 2003 pp 754 792 the el mabrouk sankoff algorithm is rather complex inspiring a quest for a simpler solution an alternative approach to the genome halving problem based on the notion of the contracted breakpoint graph was recently proposed in m a alekseyev and p a pevzner ieee acm trans comput biol bioinformatics 4 2007 pp 98 107 this new technique reveals that while the el mabrouk sankoff result is correct in most cases it does not hold in the case of unichromosomal genomes this raises a problem of correcting a flaw in the el mabrouk sankoff analysis and devising an algorithm that deals adequately with all genomes in this paper we efficiently classify all genomes into two classes and show that while the el mabrouk sankoff theorem holds for the first class it is incorrect for the second class the crux of our analysis is a new combinatorial invariant defined on duplicated permutations using this invariant we were able to come up with a full proof of the genome halving theorem and a polynomial algorithm for the genome halving problem

a faster and simpler algorithm for sorting signed permutations by reversals we give a quadratic time algorithm for finding the minimum number of reversals needed to sort a signed permutation our algorithm is faster than the previous algorithm of hannenhalli and pevzner and its faster implementation by berman and hannenhalli the algorithm is conceptually simple and does not require special data structures our study also considerably simplifies the combinatorial structures used by the analysis

on the complexity of unsigned translocation distance translocation is one of the basic operations for genome rearrangement translocation distance is the minimum number of translocations required to transform one genome into the other in this paper we show that computing the translocation distance for unsigned genomes is np hard moreover we show that approximating the translocation distance for unsigned genomes within ratio 1 00017 is np hard

1 ɛ approximation of sorting by reversals and transpositions gu et al gave a 2 approximation for computing the minimal number of inversions and transpositions needed to sort a permutation there is evidence that from the point of view of computational molecular biology a more adequate objective function is obtained if transpositions are given double weight we present a 1 e approximation for this problem based on the exact algorithm of hannenhalli and pevzner for sorting by reversals only

the evolution of the random reversal graph genomes are typically represented as signed permutations where each integer corresponds to a single gene and its sign indicates the orientation of the gene in this paper we study genome rearrangements via reversals sankoff and blanchette 1999 20 flipping entire segments of genes and changing their orientations this abstraction leads to the notion of the reversal graph in which two signed permutations are neighbors if they differ by one reversal we identify the evolution of multiple genomes within the random reversal graph i e the probability space consisting of subgraphs over signed permutations obtained by selecting edges reversals with independent probability n which is an important framework for the analysis of genome rearrangements in the sense of relating the reversal rate n with the global evolution of genomes our main result shows that the structure of the random reversal graph changes dramatically at n 1 n 1 2 for n 1 n 1 2 the random graph consists of components of size at most o n ln n a s and for 1 n 1 2 there emerges a unique largest component of size 2 n n a s where is the survival probability of a certain branching process this giant component is furthermore dense in the reversal graph

polynomial time sortable stacks of burnt pancakes pancake flipping a famous open problem in computer science can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible in that context a prefix reversal of length k reverses the order of the first k elements of the permutation the burnt variant of pancake flipping involves permutations of signed integers and reversals in that case not only reverse the order of elements but also invert their signs although three decades have now passed since the first works on these problems neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known in this work we prove a new lower bound for sorting burnt pancakes and show that an important class of permutations known as simple permutations can be optimally sorted in polynomial time