sums of functions of nearest neighbor distances moment bounds limit theorems and a goodness of fit test abstract the limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances are both established as a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

on the least number of palindromes contained in an infinite word we investigate the least number of palindromic factors in an infinite word we first consider general alphabets and give answers to this problem for periodic and non periodic words closed or not under reversal of factors we then investigate the same problem when the alphabet has size two

rich square free words abstract a word w is rich if it has w 1 distinct palindromic factors including the empty word a word is square free if it does not have a factor uu where u is a non empty word pelantova and starosta 2013 16 proved that every infinite rich word contains a square we will give another proof for that result pelantova and starosta marked with r n the length of a longest rich square free word on an alphabet of size n the exact value of r n was left as an open question we will give an upper and a lower bound for r n the lower bound is conjectured to be exact but it is not explicit we will also generalize the notion of repetition threshold for a limited class of infinite words the repetition thresholds for episturmian and rich words are left as an open question

a new characteristic property of rich words originally introduced and studied by the third and fourth authors together with j justin and s widmer 2008 rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes several characterizations of rich words have already been established a particularly nice characteristic property is that all complete returns to palindromes are palindromes in this note we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix

extensions of rich words a word w is rich if it has w 1 many distinct palindromic factors including the empty word this article contains several results about rich words particularly related to extending them a word w can be eventually extended richly in n ways if there exists a finite word u and n distinct letters a alph w such that wua is rich we will prove that every non unary rich word can be eventually extended richly in at least two different ways but not always in three or more ways we will also prove that every rich word can be extended to both periodic and aperiodic infinite rich words the defect of a finite word w is defined by d w w 1 pal w this concept has been studied in various papers here we will define a new concept infinite defect for a finite word w the definition is d w min d z zis an infinite word which has factorw we will show that the infinite defect of a finite word is always finite and give some upper bounds for it the difference between defect and infinite defect is also investigated we will also give an upper and a lower bound for the number of rich words a new class of words two dimensional rich words is also introduced

burrows wheeler transform and palindromic richness the investigation of the extremal case of the burrows wheeler transform leads to study the words w over an ordered alphabet a a 1 a 2 a k with a 1

goodness of fit testing in mathbb r m based on the weighted empirical distribution of certain nearest neighbor statistics let x1 x be a random sample in r from an unknown density f x recently bickel and breiman have introduced a goodness of fit test for this situation based on the empirical distribution function of the variables wi expt ng x1 v ri i 1 n where g x is the hypothesized density and v r represents the volume of the nearest neighbor sphere centered at xi under the null hypothesis h f x g x the empirical process is asymptotically independent of g in this paper a weighted version of the empirical process is shown to produce tests which are still essentially distribution free under h but in addition have asymptotic power against sequences of alternatives contiguous to g the optimal weight function is obtained as a function of the particular sequence of alternatives chosen and consistency behavior against fixed alternatives is determined monte carlo results illustrate the power performance of the tests for various densities and weight functions

palindrome complexity we study the palindrome complexity of infinite sequences on finite alphabets i e the number of palindromic factors blocks of given length occurring in a given sequence we survey the known results and obtain new results for some sequences in particular for rote sequences and for fixed points of primitive morphisms of constant length belonging to class p of hof knill simon we also give an upper bound for the palindrome complexity of a sequence in terms of its block complexity

on theta palindromic richness in this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism q it generalizes the notion of a palindrome into a q palindrome a word invariant under q for languages closed under q we give the relation email protected complexity and factor complexity we generalize the notion of richness email protected and we prove analogous characterizations of words that are q rich especially in the case of set of factors invariant under q a criterion for q richness of q episturmian words is given together with other examples of q rich words

palindromic richness for languages invariant under more symmetries for a given finite group g consisting of morphisms and antimorphisms of a free monoid a we study infinite words with language closed under the group g we focus on the notion of g richness which describes words rich in generalized palindromic factors i e in factors w satisfying q w w for some antimorphism email protected g we give several equivalent descriptions which are generalizations of known characterizations of rich words in the terms of classical palindromes and show two examples of g rich words

rich sturmian and trapezoidal words in this paper we explore various interconnections between rich words sturmian words and trapezoidal words rich words first introduced by the second and third authors together with j justin and s widmer constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors every finite sturmian word is rich but not conversely trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite sturmian words unfortunately this property does not characterize finite sturmian words in this note we show that the only trapezoidal palindromes are sturmian more generally we show that sturmian palindromes can be characterized either in terms of their subword complexity the trapezoidal property or in terms of their palindromic complexity we also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity

the radial curvature of an end that makes eigenvalues vanish in the essential spectrum i the concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the laplacian in particular this paper discusses the question of whether there is an eigenvalue of the laplacian embedded in the essential spectrum or not the borderline behavior of the radial curvatures for this problem will be determined we will assume that the radial curvature k rad of an end converges to a constant 1 at infinity with the decay order k rad 1 o r 1 and prove the absence of eigenvalues embedded in the essential spectrum furthermore in order to show that this decay order k rad 1 o r 1 is sharp we will construct a manifold with the radial curvature decay k rad 1 o r 1 and with an eigenvalue frac n 1 2 4 1 embedded in the essential spectrum frac n 1 2 4 infty of the laplacian

balancing and clustering of words in the burrows wheeler transform compression algorithms based on burrows wheeler transform bwt take advantage of the fact that the word output of bwt shows a local similarity and then turns out to be highly compressible the aim of the present paper is to study such clustering effect by using notions and methods from combinatorics on words the notion of balance of a word plays a central role in our investigation empirical observations suggest that balance is actually the combinatorial property of input word that ensure optimal bwt compression moreover it is reasonable to assume that the more balanced the input word is the more local similarity we have after bwt and therefore the better the compression is this hypothesis is here corroborated by experiments on real text by using local entropy as a measure of the degree of balance of a word in the setting of combinatorics on words a sound confirmation of previous hypothesis is given by a result of mantaci et al 2003 27 which states that in the case of a binary alphabet there is an equivalence between circularly balanced words words having a clusterized bwt and the conjugates of standard words in the case of alphabets of size greater than two there is no more equivalence the last section of the present paper is devoted to investigate the relationships between these notions and other related ones as for instance palindromic richness in the case of a general alphabet

convexity and absolute continuity of the laplace beltrami operator on montre que l existence de bonnes fonctions convexes implique l absolue continuite par rapport a la mesure de lebesgue des mesures spectrales de l extension auto adjointe du laplacien c 0 m dans l 2 m

density and spectrum of minimal submanifolds in space forms let varphi m m rightarrow n n be a minimal proper immersion in an ambient space suitably close to a space form mathbb n n k of curvature k le 0 in this paper we are interested in the relation between the density function theta r of m and the spectrum of its laplace beltrami operator in particular we prove that if theta r has subexponential growth when k 0 or sub polynomial growth k 0 along a sequence then the spectrum of m m is the same as that of the space form mathbb n m k notably the result applies to anderson s smooth solutions of plateau s problem at infinity on the hyperbolic space independently of their boundary regularity we also give a simple condition on the second fundamental form that ensures m to have finite density in particular we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density

on k abelian palindromes abstract a word is called a palindrome if it is equal to its reversal in the paper we consider a k abelian modification of this notion two words are called k abelian equivalent if they contain the same number of occurrences of each factor of length at most k we say that a word is a k abelian palindrome if it is k abelian equivalent to its reversal a question we deal with is the following how many distinct palindromes can a word contain it is well known that a word of length n can contain at most n 1 distinct palindromes as its factors such words are called rich on the other hand there exist infinite words containing only finitely many distinct palindromes as their factors such words are called poor we show that in the k abelian case there exist infinite words containing finitely many distinct k abelian palindromic factors for rich words we show that there exist finite words of length n containing n 2 distinct k abelian palindromes as their factors