generalized vector coherent state theory of sp 2n r vector operators and of sp 2n r contains implies u n reduced wigner coefficients sp 2n r contains implies u n reduced wigner coefficients coupling a positive discrete series unitary irreducible representation with the 2n dimensional non unitary irreducible representation characterized by 1 are evaluated by making use of generalized vector coherent state techniques general formulae are given in terms of the k matrix elements of vector coherent state theory and of some u n 6j or 9j type recoupling coefficients explicit analytic expressions are obtained in some important special cases such results should allow a determination of wsp 2n r approximately w n p 2n r and osp p 2n r matrix representations to be completed because both the w n generators and the osp p 2n r odd generators transform under the sp 2n r irrep 1

efficient 2 dimensional approximate matching of half rectangular figures efficient algorithms exist for the approximate two dimensional matching problem for rectangles this is the problem of finding all occurrences of an m m pattern in an n n text with no more than k mismatch insertion and deletion errors in computer vision it is important to generalize this problem to non rectangular figures we make progress towards this goal by defining half rectangular figures of height m and area a the approximate two dimensional matching problem for half rectangular patterns can be solved using a dynamic programming approach in time o an2 we show an o kn2formula formula k2n2 algorithm which combines convolutions with dynamic programming note that our algorithm is superior to previous known solutions for k m13 at the heart of the algorithm are the smaller matching problem and the k aligned ones with location problem these are interesting problems in their own right efficient algorithms to solve both these problems are presented

multistate perceptrons learning rule and perceptron of maximal stability a new perceptron learning rule which works with multilayer neural networks made of multistate units is obtained and the corresponding convergence theorem is proved the definition of a perceptron of maximal stability is enlarged in order to include these new multistate perceptrons and a proof of the existence and uniqueness of such optimal solutions is outlined

optimal off line detection of repetitions in a string an algorithm is presented to detect within optimal time o n log n and space o n off line on a ram all of the distinct repetitions in a given textstring on a finite alphabet the proposed strategy is self contained as it depends more heavily on algorithmic design considerations than on the combinatorial properties of the output it is based on a new data structure the leaf tree which is particularly suited to exploit simple properties of the suffix tree associated with the string to be analyzed

learning grey toned patterns in neural networks the problem of learning multi state patterns in neural networks is investigated an analysis of the space of couplings gardner approach yields the distribution of local fields the critical storage capacity alpha c and the minimum number of errors for an overloaded network for noisy local fields the classification error is minimized if the local fields of the patterns are allowed to lie in intervals of finite width a fast converging adaptive learning algorithm is presented which finds the coupling matrix of optimal stability

the effect of correlations in neural networks the effect of correlations in neural networks is investigated by considering biased input and output patterns statistical mechanics is applied to study training times and internal potentials of the minover and adaline learning algorithms for the latter a direct extension to generalization ability is obtained comparison with computer simulations shows good agreement with theoretical predictions with biased patterns the authors find a decrease in training times and internal potentials for the minover algorithm which however does not lead to faster storage of a given information measure in adaline training characteristic times undergo a transition from order 1 to order n at any finite bias for the learning of patterns as well as for the decay of the generalization error this leads to a rescaling of the gain parameters

a black box for online approximate pattern matching we present a deterministic black box solution for online approximate matching given a pattern of length m and a streaming text of length n that arrives one character at a time the task is to report the distance between the pattern and a sliding window of the text as soon as the new character arrives our solution requires o s j 1 l o g 2 mt n 2 j 1 n time for each input character where t n m is the total running time of the best offline algorithm the types of approximation that are supported include exact matching with wildcards matching under the hamming norm approximating the hamming norm k mismatch and numerical measures such as the l 2 and l 1 norms for these examples the resulting online algorithms take o log 2m o mlogm o log 2m e 2 o klogklogm o log 2m and o mlogm time per character respectively the space overhead is linear in the pattern size which we show is optimal for any deterministic algorithm

symmorphic space groups of finite crystal lattices the symmorphic space group of a finite n dimensional crystal lattice is studied and its factorization is presented this lattice is determined by a direct product of cyclic groups and then its translation point and space groups are defined as modular images of corresponding ones for a lattice of infinite extent a factorization of holohedries of an infinite lattice is used in order to present a symmorphic finite space group as a direct product as an example and as a very special case a space group in the case of the hypercubic lattice is studied and its irreducible representations are investigated the results obtained suggest that for vectors lying on a surface of the first brillouin zone i e for which at least one coordinate is equal to 0 or 1 2 an additional index describing their symmetry properties should be introduced this enables the authors to make a more detailed classification of states energy levels on the other hand finite symmorphic space groups can be used within e g the so called finite lattice approach

quasiperiodic sturmian words and morphisms we characterize all quasiperiodic sturmian words a sturmian word is not quasiperiodic if and only if it is a lyndon word moreover we study links between sturmian morphisms and quasiperiodicity

novel fisher discriminant classifiers at the present several applications need to classify high dimensional points belonging to highly unbalanced classes unfortunately when the training set cardinality is small compared to the data dimensionality small sample size problem the classification performance of several well known classifiers strongly decreases similarly the classification accuracy of several discriminative methods decreases when non linearly separable and unbalanced classes are treated in this paper we firstly survey state of the art methods that employ improved versions of linear discriminant analysis lda to deal with the above mentioned problems secondly we propose a family of classifiers based on the fisher subspace estimation which efficiently deal with the small sample size problem non linearly separable classes and unbalanced classes the promising results obtained by the proposed techniques on benchmark datasets and the comparison with state of the art predictors show the efficacy of the proposed techniques

finding consensus and optimal alignment of circular strings circular strings are different from linear strings in that the last symbol is considered to precede the first symbol even though circular strings are biologically important only a few efforts have been made to solve computational problems on circular strings in this paper we introduce consensus problems for circular strings of length n and present the first non trivial algorithms to find a consensus and an optimal alignment for circular strings by the hamming distance they are o n 2logn time algorithms for three circular strings and an o n 3logn time algorithm for four circular strings our algorithms are o n logn times faster than the naive algorithms directly using the solutions for the linear consensus problems which take o n 3 time for three circular strings and o n 4 time for four circular strings this speedup was achieved by reducing the problems into correlations and by formulating and solving systems of linear equations moreover our algorithms use only o n space

a dilution algorithm for neural networks a dilution algorithm to enlarge the storage capacity per synapse alpha eff of neural networks is proposed the algorithm is a hybrid method where hebb s rule is used to select a fraction of couplings to be removed afterwards the perceptron of optimal stability for the remaining couplings is learned the authors present an analytical calculation and the results of the numerical simulations in comparison with the fully connected or the randomly diluted perceptron the effective storage capacity alpha eff is remarkably enlarged

comparing large scale graphs based on quantum probability theory abstract in this paper a new measurement to compare two large scale graphs based on the theory of quantum probability is proposed an explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions it is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues but also about the corresponding eigenvectors moreover we prove that the proposed distance is graph invariant and sub structure invariant examples with various graphs are given and distances between graphs with few vertices are checked computational results for real large scale graphs show that its accuracy is better than any existing methods and time cost is extensively cheap

on solutions of the yang baxter equations without additivity the relationship is described between solutions found by ge and xue 1991 and by the present author 1987

a generalization of the cosine sine functional equation on groups abstract the functional equation f xy f x g y g x f y h x h y is solved where f g h are complex functions defined on a group

distributions and size scalings for strength in a one dimensional random lattice with load redistribution to nearest and next nearest neighbors lattice and network models with elements that have random strength are useful tools in explaining various statistical features of failure in heterogeneous materials including the evolution of failure clusters and overall strength distributions and size effects models have included random fuse and spring networks where monte carlo simulation coupled to scaling analysis from percolation theory has been a common approach unfortunately severe computational demands have limited the network sizes that can be treated to gain insight at large size scales interest has returned to idealized fiber bundle models in one dimension many models can be solved exactly or asymptotically in increasing size n but at the expense of major simplification of the local stress redistribution mechanism models have typically assumed either equal load sharing among nonfailed elements or nearest neighbor local load sharing lls where a failed element redistributes its load onto its two nearest flanking survivors the present work considers a one dimensional fiber bundle model under tapered load sharing tls which assumes load redistribution to both the nearest and next nearest neighbors in a two to one ratio this rule reflects features found in a discrete mechanics model for load transfer in two dimensional fiber composites and planar lattices we assume that elements have strength 1 or 0 with probability p and q 1 ensuremath p respectively we determine the structure and probabilities for critical configurations of broken fibers which lead to bundle failure under a given load we obtain rigorous asymptotic results for the strength distribution and size effect as stackrel ensuremath rightarrow n ensuremath infty with precisely determined constants and exponents the results are a nontrivial extension of those under lls in that failure clusters are combinatorially much more complicated and contain many bridging fibers consequently certain probabilities are eigenvalues from recursive equations arising from the structure of tls next nearest neighbor effects weaken the material beyond what is predicted under lls keeping only nearest neighbor overloads our results question the validity of scaling relationships that are based largely on monte carlo simulations on networks of limited size since some failure configurations appear only in extremely large bundles the dilemma has much in common with the petersburg paradox

period recovery of strings over the hamming and edit distances a string t of length m is periodic in p of length p if p is a substring of t and t i t i p t i t i p for all 0 i m p 1 0 i m p 1 and m 2p m 2 p the shortest such prefix p is called the period of t i e p t 0 p 1 p t 0 p 1 in this paper we investigate the period recovery problem given a string s of length n find the primitive period s p such that the distance between s and a string t that is periodic in p is below a threshold we consider the period recovery problem over both the hamming distance and the edit distance for the hamming distance case we present an o nlog n o n log n time algorithm where is given as n 2 p for 0 0 for the edit distance case n 3 75 p and 0 0 we provide an o n 4 3 o n 4 3 time algorithm

the subsequence composition of a string words that appear as constrained subsequences in a text string are considered as possible indicators of the host string structure hence also as a possible means of sequence comparison and classification the constraint consists of imposing a bound on the number w of positions in the text that may intervene between any two consecutive characters of a subsequence a subset of such w sequences is then characterized that consists in intuitive terms of sequences that could not be enriched with more characters without losing some occurrence in the text a compact spatial representation is then proposed for these representative sequences within which a number of parameters can be defined and measured in the final part of the paper such parameters are empirically analyzed on a small collection of text strings endowed with various degrees of structure

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

covering problems for partial words and for indeterminate strings indeterminate strings are a subclass of non standard words having non deterministic nature in a classic string every position contains exactly one symbol we say it is a solid symbol while in an indeterminate string a position may contain a set of symbols possible at this position such sets are called non solid symbols the most important subclass of indeterminate strings are partial words where each non solid symbol is the whole alphabet in this case non solid symbols are also called don t care symbols we consider the problem of finding a shortest cover of an indeterminate string i e finding a shortest solid string whose occurrences cover the whole indeterminate string we show that this classical problem becomes np complete for indeterminate strings and even for partial words the proof of this fact is one of the main results of this paper our other main results focus on design of algorithms efficient with respect to certain parameters of the input so called fpt algorithms for the shortest cover problem for the indeterminate string covering problem we obtain an o nk2 2kk3 time algorithm where k is the number of non solid symbols while for the partial word covering problem we obtain a running time of o nk2 2o klog k additionally we prove that unless the exponential time hypothesis is false no 2o k no 1 time solution exists for either problem which shows that our algorithm for partial words is close to optimal we also present an algorithm for both problems parameterized both by k and the alphabet size with a simple implementation a preliminary version of this article was presented at the 25th international symposium on algorithms and computation isaac 2014 lncs vol 8889 pp 220 232 springer 2014 12

distribution of the activities in a diluted neural network the dynamics of asymmetrically diluted neural networks can be solved exactly in the present work the distribution of the neural activities is calculated analytically for zero temperature parallel dynamics this distribution depends on the number of stored patterns and is a continuous function in the good retrieval phase the continuous part of the distribution of activities is due to the asymmetry of the synapses since it is known that networks with symmetric interactions always have a distribution of activities which is a sum of a few delta functions the expression for the distribution of activities is also given for a mixture of two patterns which have a non zero overlap

sign constrained synapses and biased patterns in neural networks the storage of biased patterns is examined in neural networks with sign constrained synapses every neuron has outgoing synapses which are either all inhibitory or all excitatory for random patterns stored in such networks it is known that the presence of a discrete gauge symmetry makes the maximal storage capacity independent of the proportion of excitatory neurons to inhibitory neurons when the stored patterns are biased however this discrete gauge symmetry is broken with the result that the maximal capacity depends on the proportion of excitatory neurons to inhibitory ones the dependence of the capacity on the fraction of excitatory neurons in the network f is calculated using the space of interactions approach it is found that the storage capacity is maximal at f 0 5 this result is true regardless of the particular value of the bias in the stored patterns the significance of this result in the neurophysiological context is discussed

quasiperiodicity and string covering in this paper we study word regularities and in particular extensions of the notion of the word period quasiperiodicity covers and seeds we present overviews of algorithms for computing the quasiperiodicity the covers and the seeds of a given word we also present an overview of an algorithm that finds maximal word factors with the above regularities finally we show how fine and wilf s theorem fails if we try to extend it directly to quasiperiodicity as well as a new property on concatenation of periodic words

generalized string matching given a pattern string of length n and an object string of length m the string matching problem asks for the positions of all occurrences of the pattern in the object string this paper investigates a generalization of string matching in which the pattern is a sequence of pattern elements each compatible with a set of symbols the alphabet of symbols is infinite with its members encoded in a finite alphabet in contrast to standard string matching which can be solved in simultaneous linear time and constant space it is shown that generalized string matching requires a time space product of omega n 2 log n on a powerful model of computation when the alphabet is restricted to n symbols our proof uses a method of borodin the obvious algorithm for generalized string matching requires time o nm where n is the length of the encoding of the pattern and m is that of the object string we describe an algorithm which solves generalized string matching in time o n m mn 1 2 o

screening and vertex operators for u n screening and vertex operators for u n are given in terms of generators of the group left action on the flag manifold u n 1 su n their matrix elements in a basis of holomorphic sections of a vector bundle are computed

learning and generalization in multineuron interacting feed forward neural networks we consider learning and generalization in the multi interacting feed forward network model recently proposed by h o carmesin 1994 with an a priori definition of the net architecture based on symmetries presented by the function to be learnt we define a generalized hebb rule extend the maximum stability learning algorithm to multi interactions and obtain training and generalization curves for rules where different orders of synapses are not correlated the results obtained for the simple perceptron concerning the hebb rule and through replica calculations in the space of couplings may be straightforwardly adapted to multi interactions through a simple renormalization of the total number of independent couplings analytical and numerical simulation results are compared and show excellent agreement

extending alignments with k mismatches and l gaps recently the problem of extending an alignment with k mismatches and a single gap for pairwise sequence alignment was introduced flouri et al 2011 the authors considered the problem of extending an alignment under the hamming distance model by also allowing the insertion of a single gap and presented a q email protected time algorithm to solve it where m is the length of the shortest sequence to be extended and b is the maximum allowed length of the single gap very recently it was shown flouri et al 2012 that this problem is strongly and directly motivated by the next generation re sequencing application aligning tens of millions of short dna sequences against a reference genome in this article we consider an extension of this problem extending an alignment with k mismatches and two gaps and present a q email protected time algorithm to solve it this extension is proved to be fundamental in the next generation re sequencing application alachiotis et al 2012 in addition we present a generalisation of our solution to solve the problem of extending an alignment with k mismatches and gaps in time q email protected the presented solutions work provided that all gaps in the alignment must occur in one of the two sequences

training of optimal cluster separation networks finding the optimal separation of two clusters of normalized vectors corresponds to training thresholds and weights in a neural network of maximum stability in order to achieve this two local iterative algorithms are presented which treat threshold and weights all in one avoiding the need to calculate any intermediate test quantities convergence is proved and the separation stability obtained is shown to match theoretical predictions and to be superior to existing algorithms

optimal storage of invariant sets of patterns in neural network memories the authors investigate optimal conditions for the storage of invariant sets of patterns in neural network memories the sets of patterns that they consider are highly correlated non random and invariant with respect to some symmetry transformations they generalize gardner s method to study the fractional volume in the space of neural interactions that allow for storage of invariant patterns they demonstrate that optimal storage conditions correspond to non trivial relations between the number of stored patterns their stability and other symmetry specific characteristics of patterns

learning the unlearnable three neural network training algorithms are presented which are robust to nonlearnable problems the first algorithm converges to the gardner stability limit if the learning problem is linearly separable and otherwise finds a locally maximally stable solution the second algorithm is a robust version of rosenblatt s perceptron learning algorithm which will converge to a solution of the learning problem if one exists and otherwise will converge locally to a solution with a certain fraction of wrongly mapped patterns the third algorithm is suited most favourably to unlearnable problems it will always find a solution if the problem is learnable and otherwise it locally maximizes the number of patterns which are stored correctly the error rate of this algorithm and other known algorithms for unlearnable problems are compared for two benchmark problems proofs of the existence of solutions are given convergence is proven as well to be global in the case of learnable and local in unlearnable cases

pattern matching image compression algorithmic and empirical results we propose a non transform image compression scheme based on approximate 1d pattern matching called pattern matching image compression pmic the main idea behind it is a lossy extension of the lempel ziv data compression scheme in which one searches for the longest prefix of an uncompressed image that approximately occurs in the already processed image this main algorithm is enhanced with several new features such as searching for reverse approximate matching recognizing sub strings in images that are additively shifted versions of each other introducing a variable and adaptive maximum distortion level d and so forth these enhancements are crucial to the overall quality of our scheme and their efficient implementation leads to algorithmic issues of interest in their own right both algorithmic and experimental results are presented our scheme turns out to be competitive with jpeg and wavelet compression for good quality graphical images we also review related theoretical results

dilution in boolean perceptrons that learn from noisy examples we investigate the effect of dilution after learning on the generalization ability of single layer boolean perceptrons that learn from noisy examples we present a thorough comparison between the relative performances of several well known learning rules in particular we show that the effect of dilution is always deleterious and that the bayes algorithm always gives the best generalization performance

two parametric solution to graded yang baxter equation and two parametric uuvgl 1 mod 1 algebra as a hopf algebra the authors discuss a two parametric solution of the graded yang baxter equation ybe and perform the yang baxterization to obtain the solution to the quantum ybe in the formalism developed previously they present the two parametric quantized superalgebra uuvgl 1 mod 1 and prove that this algebra is a hopf algebra with the hopf operations explicitly provided

time dependent fiber bundles with local load sharing fiber bundle models where fibers have random lifetimes depending on their load histories are useful tools in explaining time dependent failure in heterogeneous materials such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings various asymptotic and approximate theories have been developed for bundles with various geometries and fiber load sharing mechanisms but numerical verification has been hampered by severe computational demands in larger bundles to gain insight at large size scales interest has returned to idealized fiber bundle models in 1d such simplified models typically assume either equal load sharing els among survivors or local load sharing lls where a failed fiber redistributes its load onto its two nearest flanking survivors such models can often be solved exactly or asymptotically in increasing bundle size n yet still capture the essence of failure in real materials the present work focuses on 1d bundles under lls as in previous works a fiber has failure rate following a power law in its load level with breakdown exponent ensuremath rho surviving fibers under fixed loads have remaining lifetimes that are independent and exponentially distributed we develop both new asymptotic theories and new computational algorithms that greatly increase the bundle sizes that can be treated in large replications e g one million fibers in thousands of realizations in particular we develop an algorithm that adapts several concepts and methods that are well known among computer scientists but relatively unknown among physicists to dramatically increase the computational speed with no attendant loss of accuracy we consider various regimes of ensuremath rho that yield drastically different behavior as n increases for 1 2l ensuremath rho l 1 els and lls have remarkably similar behavior they have identical lifetime distributions at ensuremath rho 1 with approximate gaussian bundle lifetime statistics and a finite limiting mean for ensuremath rho g1 this gaussian behavior also applies to els whereas lls behavior diverges sharply showing brittle weakest volume behavior in terms of characteristic elements derived from critical cluster formation for 0l ensuremath rho l1 2 els and lls again behave similarly but the bundle lifetimes are dominated by a few long lived fibers and show characteristics of strongest link extreme value distributions which apply exactly to ensuremath rho 0

the generalization probability of a perceptron using the delta rule compares the generalization probability of a perceptron using the delta rule with other learning algorithms the numerical results indicate that the delta rule gives results which are superior to hebb s rule and comparable with the maximal stability perceptron rule minover both for randomly generated linear separable example sets and for optimal queries

a work time optimal algorithm for computing all string covers abstract in recent study of repetitive structures of strings generalized notions of periods have been introduced a typical regularity the period u of a given string x grasps the repetitiveness of x since x is a prefix of a string constructed by concatenations of u a substring w of x is called a cover of x if x can be constructed by concatenations and superpositions of w the notion cover is a generalization of periods in the sense that superpositions as well as concatenations are considered to define it whereas only concatenations are considered for periods we consider the all covers problem i e that of computing all the covers of a given string of length n we present an optimal o log log n time crcw pram algorithm for the all covers problem since there is an log log n lower bound on the time complexity of the all covers problem our algorithm is work time optimal

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

indeterminate strings prefix arrays undirected graphs an integer array y y 1 n is said to be feasible if and only if y 1 n and for every i 2 n i i y i n 1 a string is said to be indeterminate if and only if at least one of its elements is a subset of cardinality greater than one of a given alphabet otherwise it is said to be regular a feasible array y is said to be regular if and only if it is the prefix array of some regular string we show using a graph model that every feasible array of integers is a prefix array of some indeterminate or regular string and for regular strings corresponding to y we use the model to provide a lower bound on the alphabet size we show further that there is a 1 1 correspondence between labelled simple graphs and indeterminate strings and we show how to determine the minimum alphabet size of an indeterminate string x based on its associated graph g x thus in this sense indeterminate strings are a more natural object of combinatorial interest than the strings on elements of that have traditionally been studied

the set of parameterized k covers problem the problem of the set of k covers is a distance measure for strings another well studied string comparison measure is that of parameterized matching we consider the problem of the set of parameterized k covers k spc which combines k cover measure with parameterized matching we prove that k spc is np complete we describe an approach to solve k spc this approach is based on constructing a logical model for k spc

faster image template matching in the sum of the absolute value of differences measure given an m spl times m image i and a smaller n spl times n image p the computation of an m n 1 spl times m n 1 matrix c where c i j is of the form c i j spl sigma sub k 0 sup n 1 spl sigma sub k 0 sup n 1 f i i k j k p k k 0 spl les i j spl les m n for some function f is often used in template matching frequent choices for the function f are f x y x y sup 2 and f x y m y for the case when f x y x y sup 2 it is well known that c is computable in o m sup 2 log n time for the case f x y y on the other hand the brute force o m n 1 sup 2 n sup 2 time algorithm for computing c seems to be the best known this paper gives an asymptotically faster algorithm for computing c when f x y x y one that runs in time o min s n spl radic log n m sup 2 log n time where s is the size of the alphabet i e the number of distinct symbols that appear in i and p this is achieved by combining two algorithms one of which runs in o sm sup 2 log n time the other in o m sup 2 n spl radic log n time we also give a simple monte carlo algorithm that runs in o m sup 2 log n time and gives unbiased estimates of c

structure and representations of the hyperoctahedral group in this paper the main properties of the symmetry group of the n dimensional cube are reviewed and formulated with respect to possible applications in lattice theories the connection between the hyperoctahedral group wn and the orthogonal group o n is investigated by means of the canonical representation

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

efficient computations of l 1 and l rearrangement distances recently a new pattern matching paradigm was proposed pattern matching with address errors in this paradigm approximate string matching problems are studied where the content is unaltered and only the locations of the different entries may change specifically a broad class of problems was defined the class of rearrangement errors in this type of error the pattern is transformed through a sequence of rearrangement operations each with an associated cost the natural 1 and 2 rearrangement systems were considered the best algorithm presented for general patterns that may have repeating symbols is o nm in this paper we show that the problem can be approximated in linear time for general patterns another natural rearrangement system is considered in this paper the rearrangement distance for this new rearrangement system efficient exact solutions for different variants of the problem are provided as well as a faster approximation

solutions of the nonadditive ybe by spectral parameter factorization and re yang baxterization r matrix solutions for the nonadditive yang baxter equation are obtained in arbitrary dimension using a factorization ansatz in spectral parameters a symmetry transformation determined for the particle conserving r matrix is used to formulate a re yang baxterization scheme with multicomponent spectral parameters extending the notion of conventional yang baxterization the possible scope for such a transformation which has a striking resemblance to reshetikhin s deformation of r matrices but extended now to include spectral parameters is discussed

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

collage of two dimensional words we consider a new operation on one dimensional resp two dimensional word languages obtained by piling up one on top of the other words of a given recognizable language resp two dimensional recognizable language on a previously empty one dimensional resp two dimensional array the resulting language is the set of words seen from above a position in the array is labeled by the topmost letter we show that in the one dimensional case the language is always recognizable this is no longer true in the two dimensional case which is shown by a counter example and we investigate in which particular cases the result may still hold

a disorder solution for a generalised mixed spin model on anisotropic kagome lattices a disorder solution is given for a generalised mixed spin model on an anisotropic kagome lattice because of its lower rotational symmetry the solution is obtained by decimation along four different directions in the case of a spin 1 2 and spin 1 mixed model the intra row correlations in both horizontal and vertical directions are zero when a relation imposed on the interactions is satisfied

non linear conductivity of granular superconductors a novel breakdown problem the authors study a percolating network of highly hysteretic josephson junctions in which each junction is modelled as a piecewise ohmic hysteretic device at the percolation threshold the average voltage drop across a network on a large l l grid is assumed to fall to zero as v approximately i ic l eta as the applied current is reduced to the critical current ic l the finite size monte carlo study gives the value eta 2 0 or 0 1 this value is remarkably close to the result eta 2 obtained by a scaling analysis of a continuum version of the model

matrix elements of the labelling operators for su 4 su 2 su 2 a matrix representation of the labelling operators of moshinsky and nagel 1963 and of partensky and maguin 1978 in the non orthogonal draayer basis of su 4 is derived this allows one to solve explicitly the missing label problem for the spin isospin multiplets in the arbitrary su 4 supermultiplets the simplification and degeneration of these representations are considered for two parametric irreducible representations of su 4 as well as for some special cases of irreps typical of nuclear theory the expansion of the linearly dependent states of the draayer basis is discussed

measures of distance between probability distributions abstract in statistical estimation problems measures between probability distributions play significant roles hellinger coefficient jeffreys distance chernoff coefficient directed divergence and its symmetrization j divergence are examples of such measures here these and like measures are characterized through a composition law and the sum form they possess the functional equations f pr qs f ps qr r s f p q p q f r s and f pr qs f ps qr f p q f r s are instrumental in their deduction

a perceptron with a skeletal weight space a perceptron whose space of interactions can interpolate between spherically constrained and binary valued synapses is introduced and investigated as an associative neural network memory for maximally stable storage and where the weight space remains connected the critical storage capacity alpha c is found to be reduced by a factor determined solely by the geometry of the weight space and is shown to interpolate within the replica symmetric approximation between alphac 2 in the gardner model limit and alphac 4 pi various comparisons of the synaptic weights with those of the binary perceptron show that such differences as remain between this weight space and that of the true binary perceptron are crucial to obtaining alphac or 4 pi although these differences limit the use of such models in realizing optimal binary networks they may yet provide worthwhile binary systems by simple weight clipping simulation results are presented in support of the theoretical analyses

repetitive perhaps but certainly not boring in this paper some of the work done on repetitions in strings is surveyed especially that of an algorithmic nature several open problems are described and conjectures formulated about some of them

efficient detection of quasiperiodicities in strings a string z is quasiperiodic if there is a second string w z such that the occurrences of w in z cover z entirely i e every position of z falls within some occurrence of w in z it is shown here that all maximal quasiperiodic substrings of a string x of n symbols can be detected in time o n log2 n

statistical mechanics of unsupervised structure recognition a model of unsupervised learning is studied where the environment provides n dimensional input examples that are drawn from two overlapping gaussian clouds we consider the optimization of two different objective functions the search for the direction of the largest variance in the data and the largest separating gap stability between clusters of examples respectively by means of a statistical mechanics analysis we investigate how well the underlying structure is inferred from a set of examples the performances of the learning algorithms depend crucially on the actual shape of the input distribution a generic result is the existence of a critical number of examples needed for successful learning the learning strategies are compared with methods different in spirit such as the estimation of parameters in a model distribution and an information theoretical approach

linear dimensionality reduction via a heteroscedastic extension of lda the chernoff criterion we propose an eigenvector based heteroscedastic linear dimension reduction ldr technique for multiclass data the technique is based on a heteroscedastic two class technique which utilizes the so called chernoff criterion and successfully extends the well known linear discriminant analysis lda the latter which is based on the fisher criterion is incapable of dealing with heteroscedastic data in a proper way for the two class case the between class scatter is generalized so to capture differences in co variances it is shown that the classical notion of between class scatter can be associated with euclidean distances between class means from this viewpoint the between class scatter is generalized by employing the chernoff distance measure leading to our proposed heteroscedastic measure finally using the results from the two class case a multiclass extension of the chernoff criterion is proposed this criterion combines separation information present in the class mean as well as the class covariance matrices extensive experiments and a comparison with similar dimension reduction techniques are presented

quasiperiodic and lyndon episturmian words recently the second two authors characterized quasiperiodic sturmian words proving that a sturmian word is non quasiperiodic if and only if it is an infinite lyndon word here we extend this study to episturmian words a natural generalization of sturmian words by describing all the quasiperiods of an episturmian word which yields a characterization of quasiperiodic episturmian words in terms of their directive words even further we establish a complete characterization of all episturmian words that are lyndon words our main results show that unlike the sturmian case there is a much wider class of episturmian words that are non quasiperiodic besides those that are infinite lyndon words our key tools are morphisms and directive words in particular normalized directive words which we introduced in an earlier paper also of importance is the use of return words to characterize quasiperiodic episturmian words since such a method could be useful in other contexts

usefulness of the karp miller rosenberg algorithm in parallel computations on strings and arrays abstract the karp miller rosenberg 1972 algorithm was one of the first efficient almost linear sequential algorithms for finding repeated patterns and for string matching in the area of efficient sequential computations on strings it was soon superseded by more efficient and more sophisticated algorithms we show that the karp miller rosenberg algorithm kmr must be considered as a basic technique in parallel computations for many problems variations of kmr give the known most efficient parallel algorithms the representation of the set of basic factors subarrays of a string array produced by the algorithm is an extremely useful data structure in parallel algorithms on strings and arrays this gives also a general unifying framework for a large variety of problems we show that the following problems for strings and arrays can be solved by almost optimal parallel algorithms pattern matching longest repeated factor subarray longest common factor subarray maximal symmetric factor subarray also the following problems for strings can be solved within the same complexity bounds finding squares testing even palstars and compositions of k palindromes for k 2 3 4 computing lyndon factorization and building minimal pattern matching automata in the model without concurrent writes the parallel time is o log n 2 with n processors and in the model with concurrent writes the time for most of the problems is o log n with n processors for two problems related to the one dimensional case longest repeated factor and longest common factor there were designed parallel algorithms using suffix trees apostolico et al 1988 however our data structure is simpler and furthermore for the two dimensional case suffix trees do not work the complexity of our algorithms does not depend on the size of the alphabet except for the computation of pattern matching automata

normal forms of quasiperiodic strings here we consider the problem of computing normal forms of quasiperiodic strings a string x is quasiperiodic if it can be constructed by concatenation and superpositions of one of its proper factor cover the notion of quasiperiodicity is a generalization of periodicity in the sense that superpositions as well as concatenations are allowed to define it it is shown here that given a quasiperiodic string x there exists a unique factorization of x into roots of its shortest cover and how we can efficiently build such a factorization in linear time these forms can be used for example to test whether or not a string v covers xk for some integer k where v covers x

density of states and dynamic scaling on the vicsek snowflake fractal real space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the heisenberg ferromagnet and antiferromagnet on the infinite vicsek snowflake fractal the authors have calculated the associated dynamic exponents zf and za by linearising about the respective fixed points of the ferromagnetic antiferromagnetic recursion relations and find them to satisfy the relation za zf 2 in addition using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho f omega and rho a omega have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho alpha omega approximately bz alpha df rho lambda alpha omega near their respective fixed points alpha a f df denoting the fractal dimension finally the amplitudes galpha omega appearing in the solution to the scaling equation rho alpha omega approximately galpha omega omega dfz alpha 1 for the density of states rho alpha omega were found to be periodic functions of ln omega with period ln lambda alpha where lambda alpha is the eigenvalue associated with ferromagnetic alpha f or antiferromagnetic alpha a fixed point

optimal correspondence of string subsequences the definition of optimal correspondent subsequence ocs which extends the finite alphabet editing error minimization matching to infinite alphabet penalty minimization matching is given the authors prove that the string distance derived from ocs is a metric an algorithm to compute the string to string ocs is given the computational complexity of ocs is analyzed ocs is more efficient than relaxation and elastic matching for 1d problems an algorithm combining syntactic information in template matching is given to show the ease of integrating regular grammar into the ocs technique since in different applications different penalty functions may be required two of them are discussed one pointwise and the other piecewise the pointwise application consists of a stereo epipolar line matching problem solved by using string to string ocs the feasibility of applying ocs to upc bar code recognition is investigated showing the elegance of string to regular expression ocs compared to the relaxation and elastic matching techniques

nature of the largest cluster size distribution at the percolation threshold two distinct distribution functions psp m and pns m of the scaled largest cluster sizes m are obtained at the percolation threshold by numerical simulations depending on the condition whether the lattice is actually spanned or not with r pc the spanning probability the total distribution of the largest cluster is given by ptot m r pc psp m 1 r pc pns m the three distributions apparently have similar forms in three and four dimensions while in two dimensions ptot m does not exhibit a familiar form by studying the first and second cumulants of the distribution functions the different behaviour of ptot m in different dimensions may be quantified

minimal critical exponent of quasiperiodic words abstract a quasiperiodic word is a finite or infinite word that can be obtained by concatenations or overlaps of a given finite word we proved that the least critical exponent of a quasiperiodic binary word is 7 3 for larger alphabets given any 0 we provide a construction of a ternary quasiperiodic word with critical exponent 2

the statistical mechanics of constructive algorithms the storage capacity of multilayer networks with overlapping receptive fields is studied for constructive algorithms using boolean perceptrons as their basic building block which have been investigated within a replica framework the assumption of weak coupling between subsequently constructed perceptrons is verified within a replica symmetric rs ansatz and shown to be negligible in most cases in comparison with correction due to replica symmetry breaking rsb in individual perceptrons the capacities of a tiling like and variants of the upstart algorithm are then calculated within rs and one step rsb with the quenched average taken over the individual units separately for networks with up to k 4000 and k 600 units respectively within this treatment the storage capacity seems to exhibit a power law behaviour in with an exponent n that may depend on the algorithm and the stability however due to finite size effects in k reliable estimates of n could not be extracted nevertheless the results strongly indicate that n should be strictly smaller than 1 within one step rsb whereas within rs the mitchison durbin bound is violated for finite k and n 1 may hold asymptotically

stochastic renormalization group in percolation i fluctuations and crossover abstract a generalization of the renormalization group which describes order parameter fluctuations in finite systems is developed in the specific context of percolation this stochastic renormalization group srg expresses statistical self similarity through a non stationary branching process the srg provides a theoretical basis for analytical or numerical approximations both at and away from criticality whenever the correlation length is much larger than the lattice spacing regardless of the system size for example the srg predicts order parameter distributions and finite size scaling functions for the complete crossover between phases for percolation the simplest srg describes structural quantities conditional on spanning such as the total cluster mass or the minimum chemical distance between two boundaries in these cases the central limit theorem for independent random variables holds at the stable off critical fixed points while a fractal central limit theorem describing long range correlations holds at the unstable critical fixed point this first part of a series of articles explains these basic concepts and a general theory of crossover subsequent parts will focus on limit theorems and comparisons of small cell srg approximations with simulation results

coupling coefficients of so n and integrals involving jacobi and gegenbauer polynomials the expressions for the coupling coefficients 3j symbols for most degenerate symmetric representations of orthogonal groups so n in a canonical basis with so n restricted to so n 1 and different semicanonical or tree bases with so n restricted to so n so n n n n are considered respectively in context of integrals involving triplets of the gegenbauer and the jacobi polynomials since the directly derived triple hypergeometric series do not reveal the apparent triangle conditions of the 3j symbols they are rearranged using their relation with semistretched isofactors of the second kind for the complementary chain sp 4 su 2 su 2 and analogy with the stretched 9j coefficients of su 2 into formulae with more rich limits for summation intervals and obvious triangle conditions the isofactors of class one representations of orthogonal groups or class two representations of unitary groups and of course the related integrals involving triplets of the gegenbauer and the jacobi polynomials turn into double sums in the cases of canonical so n so n 1 or u n u n 1 and semicanonical so n so n 2 so 2 chains as well as into the 4f3 1 series under more specific conditions some ambiguities of the phase choice of the complementary group approach are adjusted as well as problems with an alternating sign parameter of so 2 representations in the so 3 so 2 and so n so n 2 so 2 chains

mean field analysis of hierarchical associative networks with magnetisation a hopfield type neural network that can store ultrametrically organised patterns with a finite magnetisation or bias is studied both the patterns and their ancestors are remembered in one network a homogeneous field or threshold is included in the model in zero field the replica symmetric mean field equations map onto those of the hopfield model implying that transition temperatures storage capacity etc are the same by varying appropriately the field and or a constant added to all the synaptic strengths it is possible to climb up and down the hierarchical tree or to focus on a certain level in the hierarchy and thereby increase the capacity slightly

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

optimality of some algorithms to detect quasiperiodicities improving a 1993 algorithm of apostolico and ehrenfeucht independently iliopoulos and mouchard in 1999 and brodal and pedersen in 2000 provided o nlog n algorithms to determine all maximal quasiperiodicities of a word of length n we show here the optimality of this bound providing an infinite family of words w containing o w log w maximal quasiperiodicities we also show that this bound is not reached for the celebrated family of fibonacci words

approximate pattern matching on elastic degenerate text an elastic degenerate string is a sequence of n sets of strings of total length n it has been introduced to represent a multiple alignment of several closely related sequences e g pan genome compactly in this representation substrings of these sequences that match exactly are collapsed while in positions where the sequences differ all possible variants observed at that location are listed the natural problem that arises is finding all matches of a deterministic pattern of length m in an elastic degenerate text there exists a non combinatorial o nm1 381 n time algorithm to solve this problem on line 1 in this paper we study the same problem under the edit distance model and present an o k2mg kn time and o m space algorithm where g is the total number of strings in the elastic degenerate text and k is the maximum edit distance allowed we also present a simple o kmg kn time and o m space algorithm for solving the problem under hamming distance

vector coherent state theory of the semidirect sum lie algebras wsp 2n r positive discrete series irreducible representations of the semidirect sum lie algebras wsp 2n r w n sp 2n r are given in wsp 2n r contains implies sp 2n r contains implies u n bases the branching rule for the reduction of a wsp 2n r irreducible representation into a direct sum of multiplicity free sp 2n r irreducible representations is accounted for by the explicit construction of a wsp 2n r contains implies sp 2n r contains implies u n basis of vector valued holomorphic functions

the isoscalar factors of o6 o6 using the tensor basis and utilising the wigner eckart theorem the authors obtain the matrix representations 00 and 10 of o6 st o3 s o3 t the isoscalar factors of o6 st o6 st o6 st o3 s o3 t are also calculated

random access to grammar compressed strings and trees grammar based compression where one replaces a long string by a small context free grammar that generates the string is a simple and powerful paradigm that captures sometimes with slight reduction in efficiency many of the popular compression schemes including the lempel ziv family run length encoding byte pair encoding sequitur and re pair in this paper we present a novel grammar representation that allows efficient random access to any character or substring without decompressing the string let s be a string of length n compressed into a context free grammar mathcal s of size n we present two representations of mathcal s achieving o log n random access time and either o n cdot alpha k n construction time and space on the pointer machine model or o n construction time and space on the ram here alpha k n is the inverse of the k th row of ackermann s function our representations also efficiently support decompression of any substring in s we can decompres

approximate periods of strings the study of approximately periodic strings is relevant to diverse applications such as molecular biology data compression and computer assisted music analysis here we study different forms of approximate periodicity under a variety of distance functions we consider three related problems for two of which we derive polynomial time algorithms we then show that the third problem is np complete

scattering operators on fock space ii representations of the scattering operator generated by sl 2 r for pt i see ibid vol 18 p 3141 1985 a representation for unitary scattering operators acting on a symmetric fock space and invariant under an so n internal symmetry group is constructed a group of transformations commuting with so n is seen to be isomorphic to sl 2 r the representations of sl 2 r acting on the fock space are shown to come from the discrete series of representations of sl 2 r these representations are used to label the equivalent irreducible representations of so n and the partial wave amplitudes of the scattering operators are shown to be matrix elements of the discrete series of representations of sl 2 r the example of isospin internal symmetry and the pion triplet is briefly discussed

efficient seed computation revisited the notion of the cover is a generalization of a period of a string and there are linear time algorithms for finding the shortest cover the seed is a more complicated generalization of periodicity it is a cover of a superstring of a given string and the shortest seed problem is of much higher algorithmic difficulty the problem is not well understood no linear time algorithm is known in the paper we give linear time algorithms for some of its versions computing shortest left seed array longest left seed array and checking for seeds of a given length the algorithm for the last problem is used to compute the seed array of a string i e the shortest seeds for all the prefixes of the string in o n 2 time we describe also a simpler alternative algorithm computing efficiently the shortest seeds as a by product we obtain an o nlog n m time algorithm checking if the shortest seed has length at least m and finding the corresponding seed we also correct some important details missing in the previously known shortest seed algorithm iliopoulos et al 1996 14

enhanced string covering a factor u of a string y is a cover of y if every letter of y lies within some occurrence of u in y thus every cover u is also a border both prefix and suffix of y if u is a cover of a superstring of y then u is a seed of y covers and seeds are two formalisations of quasiperiodicity and there exist linear time algorithms for computing all the covers and seeds of y a string y covered by u thus generalises the idea of a repetition that is a string composed of exact concatenations of u even though a string is coverable somewhat more frequently than it is a repetition still a string that can be covered by a single u is rare as a result seeking to find a more generally applicable and descriptive notion of cover many articles were written on the computation of a minimum k cover of y that is the minimum cardinality set of strings of length k that collectively cover y unfortunately this computation turns out to be np hard therefore in this article we propose new simple easily computed and widely applicable notions of string covering that provide an intuitive and useful characterisation of a string the enhanced cover the enhanced left cover and the enhanced left seed

string matching under a general matching relation in standard string matching each symbol matches only itself in other string matching problems e g the string matching with don t cares problem a symbol may match several symbols in general an arbitrary many to many matching relation might hold between symbols we consider a general string matching problem in which such a matching relation is specified and those positions in a text t of length n are sought at which the pattern p of length m matches under this relation depending upon the existence of a simple and easily recognizable property in the given matching relation we show that string matching either requires linear i e o n m time or is at least as hard as boolean convolution as an application we show that the matching relations of several independently studied string matching problems do indeed fall into the latter hard category we also give a generic string matching algorithm that works far any matching relation and has complexity o nm except for very large matching relations

efficient algorithms for shortest partial seeds in words abstract a factor u of a word w is a cover of w if every position in w lies within some occurrence of u in w a factor u is a seed of w if it is a cover of a superstring of w covers and seeds extend the classical notions of periodicity we introduce a new notion of partial seed that is a factor covering as a seed at least positions in a given word we use the cover suffix tree recently introduced in the context of partial covers kociumaka et al 2015 13 an o n log n time algorithm constructing such a tree is known however it appears that partial seeds are more complicated than partial covers our algorithms require algebraic manipulations of special functions related to edges of the modified cover suffix tree and the border array we present a procedure for computing shortest partial seeds that works in o n time if the cover suffix tree is already given this is a full version which includes all the proofs of a paper that appeared at cpm 2014 1