asymptotic normality of a nonparametric estimator of sample coverage this paper establishes a necessary and sufficient condition for the asymptotic normality of the nonparametric estimator of sample coverage proposed by good biometrica 40 1953 237 264 this new necessary and sufficient condition extends the validity of the asymptotic normality beyond the previously proven cases

representation type of hecke algebras of type a in this paper we provide a complete classification of the representation type for the blocks for the hecke algebra of type a stated in terms of combinatorical data the computation of the complexity of young modules is a key component in the proof of this classification result

the stability of the kronecker product of schur functions in the late 1930 s murnaghan discovered the existence of a stabilization phenomenon for the kronecker product of schur functions for n large enough the values of the kronecker coefficients appearing in the product of two schur functions of degree n do not depend on the first part of the indexing partitions but only on the values of their remaining parts we compute the exact value of n when this stable expansion is reached we also compute two new bounds for the stabilization of a particular coefficient of such a product given partitions alpha and beta we give bounds for all the parts of any partition gamma such that the corresponding kronecker coefficient is nonzero finally we also show that the reduced kronecker coefficients are structure coefficients for the heisenberg product introduced by aguiar ferrer and moreira

exponentials of skew symmetric matrices and logarithms of orthogonal matrices two widely used methods for computing matrix exponentials and matrix logarithms are respectively the scaling and squaring and the inverse scaling and squaring both methods become effective when combined with pade approximation this paper deals with the computation of exponentials of skew symmetric matrices and logarithms of orthogonal matrices our main goal is to improve these two methods by exploiting the special structure of skew symmetric and orthogonal matrices geometric features of the matrix exponential and logarithm and extensions to the special euclidean group of rigid motions are also addressed

the global dimension of schur algebras for gl2 and gl3 abstract we first define the notion of good filtration dimension and weyl filtration dimension in a quasi hereditary algebra we calculate these dimensions explicitly for all irreducible modules in sl 2 and sl 3 we use these to show that the global dimension of a schur algebra for gl 2 and gl 3 is twice the good filtration dimension to do this for sl 3 we give an explicit filtration of the modules by modules of the form f l where is a dominant weight and is p restricted

injective schur modules abstract we determine the partitions for which the corresponding induced module or schur module in the language of buchsbaum et al 1 is injective in the category of polynomial modules for a general linear group over an infinite field equivalently which weyl modules are projective polynomial modules since the problem is essentially no more difficult in the quantised case we address it at this level of generality expressing our results in terms of the representation theory of hecke algebras at the parameter q we determine the partitions for which the corresponding specht module is a young module when 1 q 0 in the classical case this problem was addressed by d hemmer 10 the nature of the set of partitions appearing in our solution gives a new formulation of carter s condition on regular partitions on the other hand we note in remark 2 22 that the result on irreducible weyl modules for the quantised schur algebra s q n n 15 theorem 5 39 given in terms of carter partitions may be also used to obtain the main result presented here

tensor products of young modules abstract we apply the brauer construction to tensor products of trivial source modules if u and v are trivial source kg modules with vertices p and q respectively we describe a family x of subgroups of g depending only on p and q such that for every subgroup h in x there exists a direct summand m of u v with vertex h and a correspondence for such summands we apply this to the special case of young modules for symmetric groups in particular deriving similar results concerning young vertices and some reduction theorems for multiplicities of young modules in a tensor product of young modules some new lower bounds on certain cartan numbers for schur algebras are obtained as an application of the results on young modules

an action of the hecke monoid on rational modules for the borel subgroup of a quantised general linear group abstract we construct an action of the hecke monoid on the category of rational modules for the quantum negative borel subgroup of the quantum general linear group we also show that this action restricts to the category of polynomial modules for this quantum subgroup and induces an action on the category of modules for the quantised borel schur algebra s n r

blocks of affine quantum schur algebras abstract the affine quantum schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine gl n finite dimensional irreducible modules for the affine quantum schur algebra s n r v were classified in 15 where v c is not a root of unity we will classify blocks of the affine quantum schur algebra s n r v in this paper

homological properties of quantised borel schur algebras and resolutions of quantised weyl modules abstract we continue the development of the homological theory of quantum general linear groups previously considered by the first author the development is used to transfer information to the representation theory of quantised schur algebras the acyclicity of induction from some rank one modules for quantised borel schur subalgebras is deduced this is used to prove the exactness of the complexes recently constructed by boltje and maisch giving resolutions of the co specht modules for hecke algebras

on projective and injective polynomial modules let k denote a field for each n 0 we have a category pol n of finite dimensional polynomial modules and a category rat n of rational modules over k for r 0 we write pol n r for the full subcategory of pol n whose objects are the polynomial modules of degree r each v pol n has a unique decomposition v r 0 v r where v r is polynomial of degree r furthermore the category pol n r is naturally equivalent to the category of modules for the finite dimensional associative schur algebra s n r it follows that every polynomial module has a projective cover and an injective hull we here attempt to describe those polynomial modules which are both projective and injective we give a complete description in the cases n 2 3 and make a conjecture which if true gives a complete description in the general case a finite dimensional polynomial module which is injective and projective must be a tilting module and thanks to a natural duality on the category of polynomial modules our problem is equivalent to that of determining which

on bases of centres of iwahori hecke algebras of the symmetric group abstract in 1990 using norms the second author constructed a basis for the centre of the hecke algebra of the symmetric group s n over q trans amer math soc 317 1 1990 361 392 an integral minimal basis was later given by the first author in j algebra 221 1 1999 1 28 following m geck r rouquier centers and simple modules for iwahori hecke algebras in finite reductive groups luminy 1994 birkhauser boston ma 1997 pp 251 272 in principle one can then write elements of the norm basis as integral linear combinations of minimal basis elements in this paper we find an explicit non recursive expression for the coefficients appearing in these linear combinations these coefficients are expressed in terms of certain permutation characters of s n in the process of establishing this main theorem we prove the following items of independent interest a result on the projection of the norms onto parabolic subalgebras the existence of an inner product on the hecke algebra with some interesting properties and the existence of a partial ordering on the norms

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

on schur algebras and related algebras vi some remarks on rational and classical schur algebras abstract in a recent paper dipper and doty 4 introduced certain finite dimensional algebras associated with the natural module of the general linear group and its dual which they call rational schur algebras we give a proof via tilting modules that these algebras are in fact generalised schur algebras using the same technique we show that certain finite dimensional algebras associated with classical groups introduced by doty 20 are quasi hereditary algebras a generalised schur algebras may be viewed as a quotient of the algebra of distributions of a reductive group by a certain ideal we give generators for this ideal

the partition algebra and the kronecker coefficients we propose a new approach to study the kronecker coefficients by using the schur weyl duality between the symmetric group and the partition algebra we explain the limiting behaviour and associated bounds in the context of the partition algebra our analysis leads to a uniform description of the reduced kronecker coefficients when one of the indexing partitions is a hook or a two part partition

some remarks on gill s theorems on young modules abstract in a recent paper 10 gill using methods of the modular representation theory of finite groups describes some results on the tensor product of young modules for symmetric groups we here give an alternative approach using the polynomial representation theory of general linear groups and the schur functor the main result is a formula for the multiplicities of young modules in a tensor product in terms of the characters of the simple polynomial modules for general linear groups our approach is also valid for young modules for hecke algebras of type a at roots of unity and here the formula involves the characters of the simple polynomial modules for quantised general linear groups

representation type of schur algebras we give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type in combination with earlier work of some of the authors on semisimplicity and finiteness this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics

composition factors of tensor products of truncated symmetric powers let g be the general linear group of degree n over an algebraically closed field k of characteristic p 0 we study the m fold tensor product s e m of the truncated symmetric algebra s e of the symmetric algebra s e of the natural module e for g we are particularly interested in the set of partitions occurring as the highest weight of a composition factor of s e m we explain how the determination of these composition factors is related to the determination of the set of composition factors of the m fold tensor product s e m of the symmetric algebra we give a complete description of the composition factors of s e m in terms of distinguished partitions our main interest is in the classical case but since the quantised version is essentially no more difficult we express our results in the general context throughout

pade approximation for the exponential of a block triangular matrix abstract in this work we obtain improved error bounds for pade approximations to e a when a is block triangular as a result improved scaling strategies ensue which avoid some common overscaling difficulties

a poisson model for the coverage problem with a genomic application summary suppose a population has infinitely many individuals and is partitioned into unknown n disjoint classes the sample coverage of a random sample from the population is the total proportion of the classes observed in the sample this paper uses a nonparametric poisson mixture model to give new understanding and results for inference on the sample coverage the poisson mixture model provides a simplified framework for inferring any general abundance f coverage the sum of the proportions of those classes that contribute exactly k individuals in the sample for some k in with being a set of nonnegative integers a new moment based derivation of the well known turing estimators is presented as an application a gene categorisation problem in genomic research is addressed since turing s approach is a moment based method maximum likelihood estimation and minimum distance estimation are indicated as alternatives for the coverage problem finally it will be shown that any turing estimator is asymptotically fully efficient

tame representation type of infinitesimal q schur algebras abstract in this paper we give a classification of the infinitesimal q schur algebras which have tame representation type

polynomially and infinitesimally injective modules abstract the injective polynomial modules for a general linear group g of degree n are labelled by the partitions with at most n parts working over an algebraically closed field of characteristic p we consider the question of which partitions correspond to polynomially injective modules that are also injective as modules for the restricted enveloping algebra of the lie algebra of g the question is related to the index of divisibility of a polynomial module in general and an explicit answer is given for n 2

q tensor space and q weyl modules we obtain the irreducible representations of the q schur algebra motivated by the fact that these representations give all the irreducible representations of gln q in the nondescribing characteristic the irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction the theory of polynomial representations of general linear groups is equivalent to the representation theory of schur algebras see green s book 5 and the bibliography therein in 4 we defined q analogues of schur algebras when q 1 these are the usual schur algebras and when q is a prime power representations of q schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case including all irreducible representations of these groups and important information about decomposition numbers it is natural to ask what features of the classical schur algebras have qanalogues in this paper we define q analogues of tensor space of weyl modules and of weight spaces thereby generalizing the main reslllts which appear in green s book 5 for example we classify the irreducible modules for qschur algebras we determine bases for q weyl modules compatible with weight spaces and we give results on composition multiplicities of the irreducible modules in q weyl modules the proofs are largely self contained so by specializing q to 1 we recover the corresponding results in 5 this paper therefore is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 the q schur algebra let r be a natural number let r be an integral domain and let q be a unit in r we denote the symmetric group on r letters by er the hecke algebra z is the r free r algebra with basis twlw e er where the multiplication is determined by the following rule if a i i 1 is a basic transposition in received by the editors august 15 1989 1980 mathematics subject classification 198 5 revision primary 1 6a64 1 6a6 5 secondary 20c30 this research was supported in part by nsf grant no dms 8802290 the authors gratefully acknowledge support received from nato grant no 0222 87 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

good l filtrations for q gl 3 k abstract let k be an algebraically closed field of characteristic p possibly zero and g q gl 3 k the quantum group of three by three matrices as defined by dipper and donkin we may also take g to be gl 3 k we first determine the extensions between simple g modules for both g and g 1 the first frobenius kernel of g we then determine the submodule structure of certain induced modules z for the infinitesimal group g 1 b we induce this structure to g to obtain a good l filtration of certain induced modules for g we also determine the homomorphisms between these induced modules for g

nonparametric estimation of the probability of discovering a new species abstract a random sample is taken from a population consisting of an unknown number of distinct species a quantity of interest is the probability of discovering a new species when an additional draw from the population is made an estimator of this quantity was introduced by starr 1979 we prove a conjecture of starr s that the estimator is uniformly minimum variance unbiased and give various asymptotic properties of the estimator a nonparametric maximum likelihood estimator that has similar asymptotic properties is introduced a monte carlo study that suggests guidelines for choosing an estimator under various circumstances is given to amplify suppose that if we take a sample of size 1 from a population then the probability of drawing a representative of the ith species is p 1 if n draws are made with replacement the unconditional probability that species i will be observed for the first time on the n 1st draw is given simply as the term p i q n i q i 1 p i from a geometric distributio

estimating the number of classes via sample coverage abstract assume that a random sample is drawn from a population with unknown number of classes and possibly unequal class probabilities a nonparametric estimation technique is proposed to estimate the number of classes using the idea of sample coverage which is defined as the sum of the cell probabilities of the observed classes since expected sample coverage can be well estimated we were motivated to find its role in the estimation of the number of classes this work generalizes the result of esty to a nonparametric approach and extends darroch and ratcliff to incorporate the heterogeneity of the class probabilities the coefficient of variation of the class sizes is shown to play an important role in the recommended estimation procedures the performance of the proposed estimators is investigated by means of monte carlo simulations

predicting the conditional probability of discovering a new class consider a population comprising disjoint classes an important problem arising from various fields is prediction of the random conditional probability of discovering a new class the asymptotic normality of the discovery probability is established in a poisson model where the number of individuals from each class is a poisson process with a class specific rate a new derivation is presented for the well known good toulmin predictor as a moment based estimator for the asymptotic limit of the discovery probability the good toulmin predictor is also shown to be a nonparametric empirical bayes estimator for the expectation of the discovery probability given the rates of the poisson processes and an approximation to an unbiased estimator only for the identifiable part of the expectation of the discovery probability in a multinomial model the properties of the moment based estimator are investigated so that confidence and prediction intervals can be constructed the good toulmin predictor and the discovery

a normal limit law for a nonparametric estimator of the coverage of a random sample the coverage of a multinomial random sample is the sum of the probabilities of the observed classes a normal limit law is rigorously proved for good s 1953 coverage estimator the result is valid under very general conditions and all terms except the coverage itself are observable nevertheless the implied confidence intervals are not much wider than those developed under restrictive assumptions such as in the classical occupancy problem the asymptotic variance is somewhat unexpected the proof utilizes a method of holst 1979

on an algebraic approach to higher dimensional statistical mechanics we study representations of temperley lieb algebras associated with the transfer matrix formulation of statistical mechanics on arbitrary lattices we first discuss a new hyperfinite algebra the diagram algebra d underset raise0 3em hbox smash scriptscriptstyle n q which is a quotient of the temperley lieb algebra appropriate for potts models in the mean field case and in which the algebras appropriate for all transverse lattice shapesg appear as subalgebras we give the complete structure of this subalgebra in the case hat a n potts model on a cylinder the study of the full temperley lieb algebra of graphg reveals a vast number of infinite sets of inequivalent irreducible representations characterized by one or more complex parameters associated to topological effects such as links we give a complete classification in the hat a n case where the only such effects are loops and twists

on the good filtration dimension of weyl modules for a linear algebraic group let g be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible in this paper we calculate the weyl filtration di mension of the induced g modules and the simple g modules l fora regular weight we use this to calculate some ext groups of the form ext extl l and ext where are regular and is the weyl module of highest weight we then deduce the projective dimensions and injective dimensions for l and for a regular weight in associated generalised schur algebras we also deduce the global dimension of the schur algebras for gln s n r when p n and for s mp p with m an integer

on the cohomology of young modules for the symmetric group abstract the main result of this paper is an application of the topology of the space q x to obtain results for the cohomology of the symmetric group on d letters d with twisted coefficients in various choices of young modules and to show that these computations reduce to certain natural questions in representation theory the authors extend classical methods for analyzing the homology of certain spaces q x with mod p coefficients to describe the homology h d v d as a module for the general linear group gl v over an algebraically closed field k of characteristic p as a direct application these results provide a method of reducing the computation of ext d y y where y y are young modules to a representation theoretic problem involving the determination of tensor products and decomposition numbers in particular in characteristic two for many d a complete determination of h d y can be found this is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given for arbitrary d the authors determine h i d y for i 0 1 2 an interesting phenomenon is uncovered namely a stability result reminiscent of generic cohomology for algebraic groups for each i the cohomology h i p a d y p a stabilizes as a increases the methods in this paper are also powerful enough to determine for any p and precisely when h d y 0 such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus

decompositions of some specht modules i abstract we give a decomposition as a direct sum of indecomposable modules of several types of specht modules in characteristic 2 these include the specht modules labelled by hooks whose decomposability was considered by murphy 15 since the main arguments are essentially no more difficult for hecke algebras at parameter q 1 we proceed in this generality

representations of q schur superalgebras in positive characteristics abstract in 9 we obtained a classification of irreducible representations for a q schur superalgebra s k m n r at a root of unity in a field k of characteristic 0 under the assumption m n r in this paper we will generalise this result to fields of positive characteristics p 2

the quasi partition algebra abstract we introduce the quasi partition algebra q p k n as a centralizer algebra of the symmetric group this algebra is a subalgebra of the partition algebra and inherits many similar combinatorial properties we construct a basis for q p k n give a formula for its dimension in terms of the bell numbers and describe a set of generators for q p k n as a complex algebra in addition we give the dimensions and indexing set of its irreducible representations we also provide the bratteli diagram for the tower of quasi partition algebras constructed by letting k range over the positive integers

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

centralizers of iwahori hecke algebras to date integral bases for the centre of the iwahori hecke algebra of a finite coxeter group have relied on character theoretical results and the isomorphism between the iwahori hecke algebra when semisimple and the group algebra of the finite coxeter group in this paper we generalize the minimal basis approach of f1 to provide a way of describing and calculating elements of the minimal basis for the centre of an iwahori hecke algebra which is entirely combinatorial in nature and independent of both the above mentioned theories this opens the door to further generalization of the minimal basis approach to other cases in particular we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the coxeter group we show here that these properties hold for groups of type a and b giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of iwahori hecke algebra

rank varieties and projectivity for a class of local algebras we consider algebras open image in new window where k is an algebraically closed fields to any finite dimensional module for this algebra we associate a rank variety when char k 2 we recover carlson s rank variety the main result states that a module is projective if and only if its rank variety vanishes this has applications to other algebras including tensor products of certain brauer tree algebras and certain parabolic hecke algebras in addition the result has implications for the graph structure of the stable auslander reiten quiver mathematics subject classification 2000 16g10 18g05 20c20

modular representations of the supergroup q n i the representation theory of the algebraic supergroup q n has been studied quite intensively over the complex numbers in recent years especially by penkov and serganova 18 19 20 culminating in their solution 21 22 of the problem of computing the characters of all irreducible finite dimensional representations ofq n the characters of one important family of irreducible representations the so called polynomial representations had been determined earlier by sergeev 24 exploiting an analogue of schur weyl duality connecting polynomial representations of q n to the representation theory of the double covers n of the symmetric groups in 2 we used sergeev s ideas to classify for the first time the irreducible representations of n over fields of positive characteristic p 2 in the present article and its sequel we begin a systematic study of the representation theory of q n in positive characteristic motivated by its close relationship to n let us briefly summarize the main facts proved in this article by purely algebraic techniques let g q n defined over an algebraically closed field k of characteristic p 6 2 see 2 3 for the precise definition in 4 we construct the superalgebra dist g of distributions on g by reduction modulo p from a kostant z form for the enveloping superalgebra of the lie superalgebra q n c this provides one of the main tools in the remainder of the paper there is an explicit equivalence between the category of representations of g and the category of integrable dist g supermodules see corollary 5 7 in 6 we classify the irreducible representations of g by highest weight theory they turn out to be parametrized by the set x p n 1 n z 1 n with i i 1 only if p i for x p n the corresponding irreducible representation is denoted l and is constructed naturally as the simple socle of an induced representation h indbu where b is a borel subgroup of g and u is a certain irreducible representation of b of dimension a power of 2 the main difficulty here is to show that h0 6 0 for x p n which we prove by exploiting the main result of 2 classifying the irreducible polynomial representations of g so ultimately the proof that h0 6 0 depends on a counting argument involving p regular conjugacy classes in n

on the kronecker product of sn characters abstract let be irreducible s n characters and assume appears in the kronecker product with maximal first part 1 then 1 min i i a similar result holds for the maximal first column we also give a recursive formula for as an application we show that if n 1 1 1 then s n 2 3 2 3 2 3 s n 1 where denotes the outer tensor product these results are applied to study the character where runs through the partitions with no more then k parts this character is closely related to the polynomial identities of the algebra of k k matrices

invariants of specht modules abstract in 14 hemmer conjectures that the module of fixed points for the symmetric group m of a specht module for n with n m over a field of positive characteristic p has a specht series when viewed as a n m module we provide a counterexample for each prime p the examples have the same form for p 5 and we treat the cases p 3 and p 2 separately

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

the efficiency of good s nonparametric coverage estimator the asymptotic efficiency of good s nonparametric coverage estimator is obtained relative to the best estimator derived under the assumption that all classes are equally likely even when that assumption is true good s estimator is quite efficient with an asymptotic relative efficiency of greater than 85 in all cases and greater than 95 if the expected coverage is less than one half

stopping rules and estimation for recapture debugging with unequal failure rates summary a theoretical optimal stopping rule based on the minimization of the testing cost and expected penalty due to the unremoved bugs is derived without the equal failure rate assumption under nayak s 1988 recapture debugging procedure two adaptive stopping rules suitable for practical use are constructed and tested by simulation the estimation of the number of undetected bugs is also considered via remaining failure rate estimation

poincare series for tensor invariants and the mckay correspondence abstract for a finite group g and a finite dimensional g module v we prove a general result on the poincare series for the g invariants in the tensor algebra t v k 0 v k we apply this result to the finite subgroups g of the 2 2 special unitary matrices and their natural module v of 2 1 column vectors because these subgroups are in one to one correspondence with the simply laced affine dynkin diagrams by the mckay correspondence the poincare series obtained are the generating functions for the number of walks on the simply laced affine dynkin diagrams

on young modules of general linear groups abstract we study l permutation modules of finite general linear groups gl n q acting on partial flags in the natural module where the coefficient field of the modules has characteristic l for l q we call the indecomposable summands of these permutation modules linear young modules we determine their vertices and green correspondents by methods relying only on the representation theory of gl n q furthermore we show that when the multiplicative order of q modulo l is strictly greater than 1 the specht modules for gl n q in characteristic l form a stratifying system this implies in particular that for gl n q modules with specht filtration the filtration multiplicities are independent of the filtration this is an analogue of a recent theorem by hemmer and nakano

estimating population size for capture recapture data when capture probabilities vary by time and individual animal there have been no estimators of population size associated with the capture recapture model when the capture probabilities vary by time and individual animal this work proposes a nonparametric estimation technique that is appropriate for such a model using the idea of sample coverage which is defined as the proportion of the total individual capture probabilities of the captured animals a simulation study was carried out to show the performance of the proposed estimation procedure numerical results indicate that it generally works satisfactorily when the coefficient of variation of the individual capture probabilities is relatively large an example is also given for illustration