languages with mismatches in this paper we study some combinatorial properties of a class of languages that represent sets of words occurring in a text s up to some errors more precisely we consider sets of words that occur in a text s with k mismatches in any window of size r the study of this class of languages mainly focuses both on a parameter called repetition index and on the set of the minimal forbidden words of the language of factors of s with errors the repetition index of a string s is defined as the smallest integer such that all strings of this length occur at most in a unique position of the text s up to errors we prove that there is a strong relation between the repetition index of s and the maximal length of the minimal forbidden words of the language of factors of s with errors moreover the repetition index plays an important role in the construction of an indexing data structure more precisely given a text s over a fixed alphabet we build a data structure for approximate string matching having average size o s log k 1 s and answering queries in time o x occ x for any word x where occ is the list of all occurrences of x in s up to errors

power of the scan statistic in detecting a changed segment in a bernoulli sequence summary fu curnow 1990 derive recursive equations to find the level of significance and power of a likelihood ratio test for a changed segment of specified length based on the scan statistic the maximum number of successes within the specified length their method is computationally feasible for segment lengths of 20 or less we present and evaluate highly accurate approximations as well as bounds for the power function of this test that are computationally feasible even for very large segment lengths we also evaluate power when the duration of the increased length used in the test statistic does not correspond to the actual length scientists dealing with data sequences modelled by independent bernoulli trials frequently seek criteria to detect changes in the underlying process certain criteria have been developed to be sensitive to situations in which the probability of success on an individual trial changes at a point in the sequence fu curnow 1990 consider a version of a two change point problem with known distance m between the change points and with the trial probability of success at the two ends being equal they give recurrence relations for the probability of type i error and the power computational difficulties arise on even large computers when the number of trials is large in the present paper we derive highly accurate approximations that allow computation of power for all segment lengths consider a sequence of n independent 0 1 random variables x1 x2 xn let 1 denote success with pr x 1 i t we wish to test the null hypothesis of a constant probability of success ho i t i t 1 n against the alternative that some contiguous sequence of m events starting at an unknown trial z has a higher probability of success fu curnow 1990 note that the generalized likelihood ratio test for this alternative rejects ho for large values of the scan statistic

bifurcations of travelling wave solutions for 2 1 dimensional boussinesq type equation variants of the two dimensional boussinesq type equations with positive and negative exponents are studied the bifurcation theory of dynamic systems is fruitfully used to carry out the analysis the dynamical behavior of different physical structure solitary patterns solitons kink breaking and periodic wave solutions is obtained the quantitative change in the physical structure of the solutions is shown to depend on the systemic parameters under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given

contours and dimple for the gneiting class of space time correlation functions we offer a dual view of the dimple problem related to space time correlation functions in terms of their contours we find that the dimple property kent et al 2011 in the gneiting class of correlations is in one to one correspondence with nonmonotonicity of the parametric curve describing the associated contour lines further we show that given such a nonmonotonic parametric curve associated with a given level set all the other parametric curves at smaller levels inherit the nonmonotonicity we propose a modified gneiting class of correlations having monotonically decreasing parametric curves and no dimple along the temporal axis

string matching bounds via coding it is known that the length l x 1 n of the longest block appearing at least twice in a randomly chosen sample path of length n drawn from an i i d process is asymptotically almost surely equal to c log n where the constant c depends on the process a simple coding argument will be used to show that for a class of processes called the finite energy processes l x 1 n is almost surely upper bounded by c log n where c is a constant while the coding technique does not yield the exact constant c it does show clearly what is needed to obtain log n bounds

approximate p values for local sequence alignments assume that two sequences from a finite alphabet are optimally aligned according to a scoring system that rewards similarities according to a general scoring scheme and penalizes gaps insertions and deletions under the assumption that the letters in each sequence are independent and identically distributed and the two sequences are also independent approximate p values are obtained for the optimal local alignment when either i there are at most a fixed number of gaps or ii the gap initiation cost is sufficiently large in the latter case the approximation can be written in the same form as the well known case of ungapped alignments

an extreme value theory for long head runs for an infinite sequence of independent coin tosses withp heads p 0 1 the longest run of consecutive heads in the firstn tosses is a natural object of study we show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn 1 p i i d exponential random variables these results are extended to the case of the longest head run interrupted byk tails the mean length of this run is shown to be log n klog n k 1 log 1 p log k k 1 2 r1 n o 1 where log log1 p 0 577 is the euler mascheroni constant ln 1 p andr1 n is small the variance is 2 6 2 1 12 r2 n o 1 wherer2 n is again small upper and lower class results for these run lengths are also obtained and extensions discussed

the hausdorff dimension of level sets described by erdos renyi average abstract let 0 1 and be an integer function defined on n 0 satisfying 1 n n define the level set e r x 0 1 lim n a n n x where a n n x is the n n erdos renyi average of x 0 1 in this paper we will give descriptions for the hausdorff dimension of e r under the assumption n as n which complement simultaneously an early classic result of besicovitch and the new strong law of large number established by p erdos and a renyi moreover for the case n m ultimately where m 1 is an integer the hausdorff dimension of e r is also determined by us in the last section

the dimple in gneiting s spatial temporal covariance model gneiting 2002 proposed a nonseparable covariance model for spatial temporal data in present paper we show that in certain circumstances his model possesses a counterintuitive dimple which detracts from its modelling appeal

nonseparable stationary covariance functions for space time data geostatistical approaches to spatiotemporal prediction in environmental science climatology meteorology and related fields rely on appropriate covariance models this article proposes general classes of nonseparable stationary covariance functions for spatiotemporal random processes the constructions are directly in the space time domain and do not depend on closed form fourier inversions the model parameters can be associated with the data s spatial and temporal structures respectively and a covariance model with a readily interpretable space time interaction parameter is fitted to wind data from ireland

hitting probabilities and large deviations let fyngn2z be a sequence of random variables in r d and let a r d then pfyn 2 a for some ng is the hitting probability of the set a by the sequence fyng we consider the asymptotic behavior as m 1 o fpfyn 2 ma some ng pfhitting mag whenever 1 the probability law of yn n satises the large deviation principle and 2 the central tendency of yn n is directed away from the given set a for a particular function i w e showpfyn 2 ma some n ge m i a

critical phenomena in sequence matching we give a generalization of the result of erdos and renyi on the length r n of the longest head run in the first n tosses of a coin consider two independent sequences x 1 x 2 cdots x m and y 1y 2 cdots y n suppose that x 1 x 2 cdots are i i d mu and y 1 y 2 cdots are i i d nu where mu and nu are possibly different distributions on a common finite alphabet s let p equiv p x 1 y 1 in 0 1 the length of the longest matching consecutive subsequence is m m n equiv max k x i r y j r for r 1 to k for some 0 leq i leq m k 0 leq j leq n k for m and n rightarrow infty with log m log mn rightarrow lambda in 0 1 our result is that there is a constant k equiv k mu nu lambda in 0 1 rbrack such that p lim m m n log 1 p mn k 1 the proof uses large deviation methods the constant k is determined from a variational formula involving the kullback liebler distance or relative entropy a simple necessary and sufficient condition for k 1 is given for the case m n lambda 1 2 and mu nu k 1 the set of mu nu lambda for which k 1 has nonempty interior the boundary of this set is the location of a phase transition the results generalize to more than two sequences and to markov chains a strong law of large numbers is given for the proportion of letters within the longest matching word the limiting proportion exhibits critical behavior similar to that of k

speeding up dynamic programming algorithms for finding optimal lattice paths many problems can be reduced to finding an optimal lattice path for example minimum path integrals can be computed by discretizing to a lattice graph and then using the optimal lattice path to approximate the minimum continuous path finding an optimal alignment between two sequences can also be reduced to finding an optimal lattice path dynamic programming algorithms are generally well suited to such problems but can be slow and require too much storage if the lattice is too large for example if the lattice dimension is too high faster algorithms requiring less computer storage can often be constructed by restricting calculations to a computational volume known to contain the optimal path upper and lower bounds on path distances from the problem domain can often help to produce small computational volumes

asymptotic properties of data compression and suffix trees recently wyner and ziv see ibid vol 35 p 1250 8 1989 have proved that the typical length of a repeated subword found within the first n positions of a stationary ergodic sequence is 1 h log n in probability where h is the entropy of the alphabet this finding was used to obtain several insights into certain universal data compression schemes most notably the lempel ziv data compression algorithm wyner and ziv have also conjectured that their result can be extended to a stronger almost sure convergence in this paper we settle this conjecture in the negative in the so called right domain asymptotic that is during a dynamic phase of expanding the data base we prove under an additional assumption involving mixing conditions that the length of a typical repeated subword oscillates almost surely a s between 1 h sub 1 log n and 1 h sub 2 log n where d

bifurcation studies on travelling wave solutions for nonlinear intensity klein gordon equation abstract in this paper we study the travelling waves for the n 1 dimensional klein gordon equation with nonlinear intensity by using the bifurcation theory and methods of planar dynamic systems with the aid of maple we obtain all bifurcations and phase portraits in the parametric space moreover we explain the reason for appearance of nonsmooth travelling waves and for persistence of smoothness of smooth travelling waves and show the existence of smooth and nonsmooth solitary and periodic waves under different regions of parametric space various sufficient conditions to guarantee the existence of the above solutions are given

longest runs in a sequence of m dependent random variables let x i i 1 be a sequence of random 0 s and 1 s we define the lengthr n of the longest head run as follows r n max left k leqq n mathop max limits 0 leqq i leqq n k 1 x i 1 cdots x i k 1 1 right the limit law for the distribution ofr n asn was found in 1943 the rate of convergence in that limit theorem remained unknown until recently even for bernoulli independent trials supposing that x i i 1 is a stationary sequence ofm dependent r v s we given an estimate on the rate of convergence in that limit theorem this estimate provides the unimprovable raten 1 lnn for the independent case a result of lim inf type is obtained too

matching rectangles in d dimensions algorithms and laws of large numbers abstract for each point of the integer lattice zd let x and y be independent identically distributed random variables with p x y p 0 1 let s n be the volume of the largest d dimensional cube in 1 n d with the property that x y at every point of the cube r n is similarly defined to be the maximum volume of perfectly matching rectangles it is proved that if all possible shifts of the x lattice relative to the y lattice are allowed p lim n s n log n lim n r n log n 2d 1 where log is to base 1 p the corresponding limit without shifts is d algorithms to find largest squares and rectangles with and without shifts are also given

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

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

a note on the run length function for intermittent maps abstract we study the run length function for intermittent maps in particular we show that the longest consecutive zero digits resp one digits having a time window of polynomial resp logarithmic length our proof is relatively elementary in the sense that it only relies on the classical borel cantelli lemma and the polynomial decay of intermittent maps our results are compensational to the erd s renyi law obtained by denker and nicol in 8

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

on the shortest distance between orbits and the longest common substring problem abstract in this paper we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions the decay of the shortest distance depends on the correlation dimension for irrational rotations we prove a different behaviour depending on the irrational exponent of the angle of the rotation for random processes this problem corresponds to the longest common substring problem we extend the result of 6 on sequence matching to mixing processes with exponential decay

critical phenomena for sequence matching with scoring consider two independent sequences x 1 x n and y 1 y n suppose that x 1 x n are i i d x and y 1 y n are i i d y where x and y are distributions on finite alphabets x and y respectively a score f x x y r is assigned to each pair x i y j and the maximal nonaligned segment score is m n max 0 i j n 0 l 1 f x i l y j l our result is that m n log n x y a s with determined by a tractable variational formula moreover the pair empirical measure of x i l y j l during the segment where m n is achieved converges to a probability measure which is accessible by the same formula these results generalize to x i y j taking values in any polish space to intrasequence scores under shifts to long quality segments and to more than two sequences

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

tail estimates for one dimensional random walk in random environment suppose that the integers are assigned i i d random variables x taking values in the unit interval which serve as an environment this environment defines a random walk x k called a rwre which when atx moves one step to the right with probability x and one step to the left with probability 1 x solomon 1975 determined the almost sure asymptotic speed rate of escape of a rwre for certain environment distributions where the drifts 2 x 1 can take both positive and negative values we show that the chance of the rwre deviating below this speed has a polynomial rate of decay and determine the exponent in this power law for environments which allow only positive and zero drifts we show that these large deviation probabilities decay like exp cn1 3 this differs sharply from the rates derived by greven and den hollander 1994 for large deviation probabilities conditioned on the environment as a by product we also provide precise tail and moment estimates for the total population size in a branching process with random environment

maximal length of common words among random letter sequences on considere des suites de lettres aleatoires basees sur un alphabet fini genere par des processus stationnaires uniformement melangeants on etudie les proprietes des distributions asymptotiques de la longueur du plus grand mot commun

2d pattern matching image and video compression theory algorithms and experiments we propose a lossy data compression framework based on an approximate two dimensional 2d pattern matching 2d pmc extension of the lempel ziv 1977 1978 lossless scheme this framework forms the basis upon which higher level schemes relying on differential coding frequency domain techniques prediction and other methods can be built we apply our pattern matching framework to image and video compression and report on theoretical and experimental results theoretically we show that the fixed database model used for video compression leads to suboptimal but computationally efficient performance the compression ratio of this model is shown to tend to the generalized entropy for image compression we use a growing database model for which we provide an approximate analysis the implementation of 2d pmc is a challenging problem from the algorithmic point of view we use a range of techniques and data structures such as k d trees generalized run length coding adaptive arithmetic coding and variable and adaptive maximum distortion level to achieve good compression ratios at high compression speeds we demonstrate bit rates in the range of 0 25 0 5 bpp for high quality images and data rates in the range of 0 15 0 5 mbps for a baseline video compression scheme that does not use any prediction or interpolation we also demonstrate that this asymmetric compression scheme is capable of extremely fast decompression making it particularly suitable for networked multimedia applications

space time covariance functions this work considers a number of properties of space time covariance functions and how these relate to the spatial temporal interactions of the process first it examines how the smoothness away from the origin of a space time covariance function affects for example temporal correlations of spatial differences models that are not smoother away from the origin than they are at the origin such as separable models have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments these results are used to obtain a parametric class of spectral densities whose corresponding space time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time second this work considers models that

optimal data compression algorithm abstract a practical source coding scheme based on approximate string matching is proposed it is an approximate fixed length string matching data compression combined with a block coder based on the empirical distribution a lemma on approximate string matching which is an extension of the kac lemma is proved it is shown based on the lemma that the deterministic algorithm converts the stationary and ergodic source u into an output process v and under the assumption that v is a stationary process after the scheme has run for an infinite time the optimal compression ratio r d is achieved this reduces the problem of the universal lossy coder to the proof of stationarity of the output process v in the proposed algorithm the main advantages of the proposed method are the asymptotic sequential behavior of the encoder and the simplicity of the decoder

an extreme value theory for sequence matching on considere des suites finies x 1 x 2 x m et y 1 y 2 y m ou les lettres x i et y i sont choisies i i d sur un alphabet denombrable avec p p x 1 y 1 0 1 on etudie la distribution de la plus longue sequence contigue de couples entre les x et les y avec au plus k mauvais couplages

approximations for distributions of scan statistics abstract certain statistical applications deal with the extremal distributions of the number of points in a moving interval or window of fixed length this article gives an approximation that is highly accurate for several of these distributions applications include the maximum cluster of points on a line or circle multiple coverage by subintervals or subarcs of fixed size the length of the longest success run in bernoulli trials and the generalized birthday problem

distribution of the length of the longest significance run on a bernoulli net and its applications we consider the length of the longest significance run in a two dimensional bernoulli net and derive its asymptotic limit distribution our results can be considered as generalizations of known theorems in significance runs we give three types of theoretical results 1 reliability style lower and upper bounds 2 erdos renyi law and 3 the asymptotic limit distribution to understand the rate of convergence to the asymptotic distributions we carry out numerical simulations the convergence rates in a variety of situations are presented to understand the relation between the length of the longest significance run s and the success probability p we propose a dynamic programming algorithm to implement simultaneous simulations insights from numerical studies are important for choosing the values of design parameters in a particular application which motivates this article the distribution of the length of the longest significance run in a bernoulli net is critical in applying a multiscale metho

flexible syntactic matching of curves and its application to automatic hierarchical classification of silhouettes curve matching is one instance of the fundamental correspondence problem our flexible algorithm is designed to match curves under substantial deformations and arbitrary large scaling and rigid transformations a syntactic representation is constructed for both curves and an edit transformation which maps one curve to the other is found using dynamic programming we present extensive experiments where we apply the algorithm to silhouette matching in these experiments we examine partial occlusion viewpoint variation articulation and class matching where silhouettes of similar objects are matched based on the qualitative syntactic matching we define a dissimilarity measure and we compute it for every pair of images in a database of 121 images we use this experiment to objectively evaluate our algorithm first we compare our results to those reported by others second we use the dissimilarity values in order to organize the image database into shape categories the veridical hierarchical organization stands as evidence to the quality of our matching and similarity estimation

scale free duplication dynamics a model for ultraduplication empirical studies of the genome wide length distribution of duplicated sequences have revealed an algebraic tail common to nearly all clades the decay of the tail is often well approximated by a single exponent that takes values within a limited range we propose and study here scale free duplication dynamics a class of model for genome sequence evolution that generates the observed shapes of this distribution a transition between self similar and non self similar regimes is exhibited our model accounts plausibly for the observed form of the algebraic tail which is not produced by standard models for generating long range sequence correlations

an erdos renyi law with shifts abstract motivated by the comparison of dna sequences a generalization is given of the result of erdos and renyi on the length r n of the longest run of heads in the first n tosses of a coin consider two sequences x 1 x 2 x n and y 1 y 2 y n the length of the longest matching consecutive subsequence allowing shifts is m n max m x i k y j k for k 1 to m for some 0 i j n m suppose that all the letters are independent and identically distributed the length of the longest match without shifts has the same distribution as r n the length of the longest head run for a biased coin with p p x i y i described by the erdos renyi law p lim n r n log 1 p n 1 1 for matching with shifts our result is p lim n m n log 1 p n 2 1 loosely speaking allowing shifts doubles the length of the longest match the case of markov chains is also handled

cylindrical solitary pulses in a two dimensional stabilized kuramoto sivashinsky system abstract by linearly coupling a generalized zakharov kuznetsov equation alias the two dimensional 2d benney equation to an extra linear dissipative equation a 2d extension of a recently proposed stabilized kuramoto sivashinsky system is developed the model applies to the description of surface waves on 2d liquid layers in various physical settings the extra equation provides for the stability of the zero state in the system thus paving a way to the existence of stable 2d localized solitary pulses sps a perturbation theory based on a family of cylindrical solitons existing in the conservative counterpart of the system is developed by treating dissipation and gain in the model as small perturbations it is shown that the system may select two steady state solitons from the continuous family provided by the conservative counterpart of which the one with larger amplitude is expected to be stable numerical simulations support the analytical predictions quite well additionally it is found that a shallow quasi one dimensional 1d trough is attached to the stable sp if the integration domain is not very large and an explanation to this feature is proposed stable double humped bound states of two pulses are found too

a suboptimal lossy data compression based on approximate pattern matching a practical suboptimal variable source coding algorithm for lossy data compression is presented this scheme is based on approximate string matching and it naturally extends the lossless lempel ziv 1977 data compression scheme among others we consider the typical length of an approximately repeated pattern within the first n positions of a stationary mixing sequence where d percent of mismatches is allowed we prove that there exists a constant r sub 0 d such that the length of such an approximately repeated pattern converges in probability to 1 r sub 0 d log n pr but it almost surely oscillates between 1 r sub spl infin d log n and 2 r sub 1 d log n where r sub spl infin d r sub 0 d r sub 1 d 2 are some constants these constants are natural generalizations of renyi entropies to the lossy environment more importantly we show that the compression ratio of a lossy data compression scheme based on such an approximate pattern matching is asymptotically equal to r sub 0 d we also establish the asymptotic behavior of the so called approximate waiting time n sub l which is defined as the time until a pattern of length c repeats approximately for the first time we prove that log n sub l l spl rarr r sub 0 d pr as l spl rarr spl infin in general r sub 0 d r d where r d is the rate distortion function thus for stationary mixing sequences we settle in the negative the problem investigated by steinberg and gutman by showing that a lossy extension of the wyner ziv 1989 scheme cannot be optimal

a comparison of scores of two protein structures with foldings let x i i 1 y i i 1 u u i i 1 and v v i i 1 be four i i d sequences of random variables suppose u and v are uniformly distributed on 0 1 3 for each realization of u j 1 j n x i p 1 p n is constructed as a certain permutation of x p 1 p n for any 1 i n also y j p 1 p n 1 j n are constructed the same way based on y j and v j for a score function f we show that formula math has an asymptotic extreme distribution with the same parameters as in the one dimensional case this model is constructed for a comparison of scores of protein structures with foldings

all traveling wave exact solutions of the variant boussinesq equations in this article we employ the complex method to obtain all meromorphic solutions of complex variant boussinesq equations 1 then find out related traveling wave exact solutions of system vb the idea introduced in this paper can be applied to other nonlinear evolution equations our results show that all rational and simply periodic solutions w r 1 k x t w r 2 k x t w s 1 k x t and w s 2 k x t of system vb are solitary wave solutions and there exist some rational solutions wr 2 z and simply periodic solutions ws 2 z which are not only new but also not degenerated successively by the elliptic function solutions we believe that this method should play an important role for finding exact solutions in the mathematical physics we also give some computer simulations to illustrate our main results

some monotonicity properties of schur powers of matrices and related inequalities abstract a series of inequalities are developed relating the spectral radius a b of the schur product a b of two nonnegative matrices a and b with those of a a and b b yielding a b a a b b 1 2 as a corollary it is proved that the spectral radius of the schur powers r a r a r a a a r factors satisfies 1 r log r is decreasing while 1 r 1 log r is increasing the latter provided a is a stochastic matrix the entropy of a finite stationary markov chain is identified with d r dr r 1 a number of majorization comparisons for the spectral radius of schur powers is given

objective method for estimating asymptotic parameters with an application to sequence alignment sequence alignment is an indispensable computational tool in modern molecular biology the model underlying biological sequence alignment is of interest to physicists because it approximates the statistical mechanics of dna and protein annealing while bearing an intimate relationship to models of directed polymers in random media recent methods for determining the statistics of random sequence alignments have reduced the computation time to less than 1 s opening up some interesting possibilities for online computation with biological search engines before implementation however the methods required an objective technique for computing regression coefficients pertinent to an asymptotic regime typically physicists estimate parameters pertinent to an asymptotic regime subjectively they eyeball their data estimate the asymptotic regime where the regression model holds with reasonable accuracy and then regress data only within the estimated asymptotic regime our publicly available computer program arrp replaces the subjective assessment of the asymptotic regime with an objective change point detection method increasing confidence in the scientific objectivity of the parameter estimates asymptotic regression has potential applications across most of physics

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

maximal repetition and zero entropy rate maximal repetition of a string is the maximal length of a repeated substring this paper investigates maximal repetition of strings drawn from stochastic processes strengthening previous results two new bounds for the almost sure growth rate of maximal repetition are identified an upper bound in terms of conditional renyi entropy of order gamma 1 given a sufficiently long past and a lower bound in terms of unconditional shannon entropy gamma 1 both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time we also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity in particular it is shown that a power law logarithmic growth of maximal repetition with respect to the string length recently observed for texts in natural language may hold only if the conditional renyi entropy rate given a sufficiently long past equals zero according to this observation natural language cannot be faithfully modeled by a typical hidden markov process which is a class of basic language models used in computational linguistics

erdős renyi laws for gibbs measures can one detect a phase transition from a single large sample of a gibbs measure what information does one get on the other gibbs distributions with the same potential we approach these questions via erd s renyi laws in particular we prove almost sure limit theorems for sets of empirical distributions of sub samples of the given one for suitable sub samples size this set converges to the set of stationary gibbs measures moreover we formulate erd s renyi laws for general families of random variables with suitable large deviation principles

analytic solutions of the 2 1 dimensional nonlinear evolution equations using the sine cosine method in this paper we establish exact solutions for 2 1 dimensional nonlinear evolution equations the sine cosine method is used to construct exact periodic and soliton solutions of 2 1 dimensional nonlinear evolution equations many new families of exact traveling wave solutions of the 2 1 dimensional boussinesq breaking soliton and bkp equations are successfully obtained these solutions may be important of significance for the explanation of some practical physical problems it is shown that the sine cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics