reduction of a nilpotent fuzzy matrix abstract a problem of reducing fuzzy matrices is considered and some results are obtained with regard to nilpotent matrices first reduction of irreflexive and transitive matrices is examined and then the results are applied to nilpotent matrices nilpotent matrices are matrices which represent acyclic graphs and the graphs are used to represent consistent systems the results are useful for simplification of various systems with fuzzy transitivity

the undecidability of some equivalence problems concerning ngsm s and finite substitutions abstract it is shown to be undecidable whether an e free finite substitution and a two state simple e free ngsm are equivalent on b consequently the state minimization of such ngsm s and the nonterminal minimization of linear grammars having two nonterminals are undecidable applications to some other decision problems concerning e free finite substitutions and linear grammars are also given finally it is shown that the time equivalence problem is undecidable for propagating ol systems i e for iterated e free finite substitutions this solves an open problem presented in 4

congruences for the andrews spt function ramanujan type congruences for the andrews spt n partition function have been found for prime moduli 5 l 37 in the work of andrews andrews ge 2008 j reine angew math 624 133 142 and garvan garvan f 2010 int j number theory 6 1 29 we exhibit unexpectedly simple congruences for all l 5 confirming a conjecture of garvan we show that if l 5 is prime and l 1 then spt l2 ln 1 24 0 mod l this congruence gives l 1 2 arithmetic progressions modulo l 3 which support a mod l congruence this result follows from the surprising fact that the reduction of a certain mock theta function modulo l for every l 5 is an eigenform of the hecke operator t l 2

on nilpotency of matrices over antirings abstract an antiring is a semiring which is zerosumfree i e a b 0 implies a b 0 for any a b in this semiring in this paper we study the nilpotency of matrices over commutative antirings we first provide some properties and characterizations of the nilpotent matrices in terms of principal permanental minors main diagonals and permanental adjoint matrices when a family of matrices are simultaneously considered we establish some characterizations of the simultaneous nilpotence for a family of matrices

invertible incline matrices and cramer s rule over inclines inclines are the additively idempotent semirings in which the products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines in this paper the complete description of the invertible incline matrices is given some necessary and sufficient conditions for an incline matrix to be invertible are studied cramer s rule over inclines is presented and the group of invertible incline matrices is investigated the main results in the present paper generalize and develop the corresponding results in the literatures for the boolean matrices the fuzzy matrices and the lattice matrices

on a special class of primitive words when representing dna molecules as words it is necessary to take into account the fact that a word u encodes basically the same information as its watson crick complement q u where q denotes the watson crick complementarity function thus an expression which involves only a word u and its complement can be still considered as a repeating sequence in this context we define and investigate the properties of a special class of primitive words called pseudo primitive words relative to q or simply q primitive words which cannot be expressed as such repeating sequences for instance we prove the existence of a unique q primitive root of a given word and we give some constraints forcing two distinct words to share their q primitive root also we present an extension of the well known fine and wilf theorem for which we give an optimal bound

conjugacy of finite biprefix codes two languages x and y are called conjugates if they satisfy the conjugacy equation xz zy for some non empty language z we will compare solutions of this equation with those of the corresponding equation of words and study the case of finite biprefix codes x and y we show that the maximal z in this case is rational we will also characterize x and y in the case where they are both finite biprefix codes this yields the decidability of the conjugacy of two finite biprefix codes

on compositions of lattice matrices in this paper we give some properties of compositions of matrices whose elements belong to a distributive lattice we also construct an idempotent matrix from a given matrix and factor a matrix into a product of a square matrix and a rectangular matrix of the same dimension this square matrix has reflexivity and transitivity the main results obtained in this paper are generalizations of previous results on fuzzy matrices by ragab emam and hashimato

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

partitions durfee symbols and the atkin garvan moments of ranks atkin and garvan introduced the moments of ranks of partitions in their work connecting ranks and cranks here we consider a combinatorial interpretation of these moments this requires the introduction of a new representation for partitions the durfee symbol and subsequent refinements this in turn leads us to a variety of new congruences for our marked durfee symbols much in the spirit of dyson s original conjectures on the ranks of partitions

higher order spt functions abstract andrews spt function can be written as the difference between the second symmetrized crank and rank moment functions using the machinery of bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions this implies an inequality between crank and rank moments that was only known previously for sufficiently large n and fixed order this combinatorial interpretation is in terms of a weighted sum of partitions a number of congruences for higher order spt functions are derived

playing with conway s problem the centralizer of a language is the maximal language commuting with it the question raised by conway in j h conway regular algebra and finite machines chapman hall 1971 whether the centralizer of a rational language is always rational recently received a lot of attention in kunc m kunc the power of commuting with finite sets of words in proc of stacs 2005 in lncs vol 3404 springer 2005 pp 569 580 a strong negative answer to this problem was given by showing that even complete co recursively enumerable centralizers exist for finite languages using a combinatorial game approach we give here an incremental construction of rational languages embedding any recursive computation in their centralizers

on the andrews stanley refinement of ramanujan s partition congruence modulo 5 and generalizations in a recent study of sign balanced labelled posets stanley introduced a new integral partition statistic srank o o where o denotes the number of odd parts of the partition and is the conjugate of in a forthcoming paper andrews proved the following refinement of ramanujan s partition congruence mod 5 p 0 5n 4 p 2 5n 4 0 mod 5 p n p 0 n p 2 n where p i n i 0 2 denotes the number of partitions of n with srank i mod 4 and p n is the number of unrestricted partitions of n andrews asked for a partition statistic that would divide the partitions enumerated by p i 5n 4 i 0 2 into five equinumerous classes in this paper we discuss three such statistics the st crank the 2 quotient rank and the 5 core crank the first one while new is intimately related to the andrews garvan 1988 crank the second one is in terms of the 2 quotient of a partition the third one was introduced by garvan kim and stanton in 1990 we use it in our combinatorial proof of the andrews refinement remarkably the andrews result is a simple consequence of a stronger refinement of ramanujan s congruence mod 5 this more general refinement uses a new partition statistic which we term the bg rank we employ the bg rank to prove new partition congruences modulo 5 finally we discuss some new formulas for partitions that are 5 cores and discuss an intriguing relation between 3 cores and the andrews garvan crank

on simultaneously nilpotent fuzzy matrices abstract nilpotent fuzzy matrices play a crucial role in the study of fuzzy matrices in this paper we shall extend the nilpotence to the notion of simultaneous nilpotence for a finite set of fuzzy matrices the notion of simultaneous nilpotence relates to the infinite products of a finite number of fuzzy matrices which converge to the zero matrix properties of the simultaneous nilpotence will be established in the study of consecutive powers of a fuzzy matrix a controllable fuzzy matrix can be characterized by an associated nilpotent fuzzy matrix in this paper we propose the notion of simultaneously controllable fuzzy matrices which can be thought of as a generalization of the notion of controllable fuzzy matrices similar to the nilpotent characterization for a controllable fuzzy matrix the simultaneously controllable fuzzy matrices can be characterized by a finite set of associated simultaneously nilpotent fuzzy matrices

the transitive closure convergence of powers and adjoint of generalized fuzzy matrices generalized fuzzy matrices are considered as matrices over a special type of semiring which is called an incline and their transitive closure powers determinant and adjoint matrices are studied an expression for the transitive closure of a matrix a as a sum of its powers and some su2cient conditions for powers of a matrix to converge are given if the incline is commutative a su2cient condition for nilpotency of a matrix is obtained namely the determinants of the principal submatrices of the matrix are all equal to zero element in addition it is proved that a n 1 is equal to the adjoint matrix of a if the matrix a satis5es a in

twin roots of words and their properties in this paper we generalize the notion of an i symmetric word from an antimorphic involution to an arbitrary involution i as follows a nonempty word w is said to be i symmetric if email protected email protected email protected for some words a b we propose the notion of i twin roots x y of an i symmetric word w we prove the existence and uniqueness of the i twin roots of an i symmetric word and show that the left factor a and right factor b of any factorization of w as email protected email protected email protected can be expressed in terms of the i twin roots of w in addition we show that for any involution i the catenation of the i twin roots of w equals the primitive root of w we also provide several characterizations of the i twin rots of a word for i being a morphic or antimorphic involution

indices and periods of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines this paper studies the power sequence of incline matrices in detail a necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated it is proved that the period of n n incline matrix with index is a divisor of n and that the set of periods of n n incline matrices with indices is not bounded from above in the sense of a power of n for all n an equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established the stability of the orbits of an incline matrix is considered and a theorem in fuzzy sets and systems 81 1996 227 is pointed out being false the results in the present paper include some previous results in the literatures which were obtained for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

inclines and incline matrices a survey inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean and fuzzy algebra we outline results obtained in a book with cao cao kim roush incline algebra and applications john wiley sons new york 1984 on algebraic structure of inclines themselves matrices over inclines topological and convergence results and some applications this survey paper represents our talk at the 10th annual ilas conference at auburn university june 2002

partition congruences and the andrews garvan dyson crank in 1944 freeman dyson conjectured the existence of a crank function for partitions that would provide a combinatorial proof of ramanujan s congruence modulo 11 forty years later andrews and garvan successfully found such a function and proved the celebrated result that the crank simultaneously explains the three ramanujan congruences modulo 5 7 and 11 this note announces the proof of a conjecture of ono which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function

commutativity in free inverse monoids abstract this article characterizes the solutions of the commutativity equation xy yx in free inverse monoids the main result implies the following interesting property that is the natural generalization to free inverse monoids of the solutions of the same equation in free monoids let x and y be non idempotent elements of a free inverse monoid such that xy yx then there exist some elements and z such that x and y are conjugate by to some positive powers of z namely x zn and y zm with n m 1 we also show that the centralizer of a given non idempotent element is a rational non recognizable subset of the free inverse monoid

on nilpotency of generalized fuzzy matrices in this paper generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra some properties and characterizations for nilpotent generalized fuzzy matrices are established reduction of generalized fuzzy matrices is defined and some properties of reduction of nilpotent generalized fuzzy matrices are obtained partial results obtained in this paper generalize the corresponding ones on fuzzy matrices on lattice matrices and on incline matrices

an application of incline matrices in dynamic analysis of generalized fuzzy bidirectional associative memories this paper studies an application of the incline matrix theory in the dynamic analysis of incline valued fuzzy bidirectional associative memories l fbams for short it is proved that the strong convergence and the strong stability of an l fbam are equivalent to the existence of indices and the convergence in finite steps of the product matrices of connection incline matrices of the l fbam respectively it is shown that the convergence index the period of limit cycles the stable states and equilibria of a strongly convergent l fbam are expressed by the indices the periods and the standard eigenvectors of the product matrices of connection incline matrices of the l fbam respectively

on the rank and the crank modulo 4 and 8 in this paper we prove some identities conjectured by lewis for the rank and crank of partitions concerning the modulo 4 and 8 these identities are similar to dyson s identities for the rank modulo 5 and 7 which give a combinatorial interpretation to ramanujan s partition congruences for this we use multisection of series and some of the results that watson established for the third order mock theta functions

on simultaneously nilpotent antiring matrices antirings are an important type of semirings which generalize boolean algebra fuzzy algebra distributive lattice and incline in this paper we study the issue of nilpotent antiring matrices provide some properties and characterizations of the simultaneous nilpotence for a finite number of antiring matrices present some methods for calculating the simultaneously nilpotent index of a finite number of antiring matrices partial results obtained in this paper generalize and develop the corresponding ones on nilpotent antiring matrices and on simultaneously nilpotent fuzzy matrices lattice matrices

an extension of the hardy ramanujan circle method and applications to partitions without sequences we develop a generalized version of the hardy ramanujan circle method in order to derive asymptotic series expansions for the products of modular forms and mock theta functions classical asymptotic methods including the circle method do not work in this situation because such products are not modular and in fact the error integrals that occur in the transformations of the mock theta functions can and often do make a significant contribution to the asymptotic series the resulting series include principal part integrals of bessel functions whereby the main asymptotic term can also be identified to illustrate the application of our method we calculate the asymptotic series expansion for the number of partitions without sequences andrews showed that the generating function for such partitions is the product of the third order mock theta function chi and a modular infinite product series the resulting asymptotic expansion for this example is particularly interesting because the error integrals in the modular transformation of the mock theta function component appear in the exponential main term

on nilpotent matrices over antirings in this paper the nilpotent matrices over commutative antirings are characterized in terms of principal permanental minors main diagonals and permanental adjoint matrices and a necessary and sufficient condition for a nilpotent matrix over a commutative antiring which has a given nilpotent index is obtained also a method for calculating the nilpotent index of any nilpotent matrix over a commutative entire antiring is given

pushdown processes a pushdown game is a two player perfect information infinite game on a transition graph of a pushdown automaton a winning condition in such a game is defined in terms of states appearing infinitely often in the play it is shown that if there is a winning strategy in a pushdown game then there is a winning strategy realized by a pushdown automaton an exptime procedure for finding a winner in a pushdown game is presented the procedure is then used to solve the model checking problem for the pushdown processes and the propositional calculus the problem is shown to be dexptime complete

pseudopalindrome closure operators in free monoids we consider involutory antimorphisms of a free monoid a and their fixed points called palindromes or pseudopalindromes a palindrome reduces to a usual palindrome when is the reversal operator for any word w a the right resp left palindrome closure of w is the shortest palindrome having w as a prefix resp suffix we prove some results relating palindrome closure operators with periodicity and conjugacy and derive some interesting closure properties for the languages of finite sturmian and episturmian words in particular a finite word w is sturmian if and only if both its palindromic closures are so moreover in such a case both the palindromic closures of w share the same minimal period of w a new characterization of finite sturmian words follows in terms of periodicity and special factors of their palindromic closures some weaker results can be extended to the episturmian case by using the right palindrome closure we extend the construction of standard episturmian words via directive words in this way one obtains a family of infinite words called standard words which are morphic images of episturmian words as well as a wider family of infinite words including the thue morse word on two symbols

conway s problem for three word sets we prove two results on commutation of languages first we show that the maximal language commuting with a three element language i e its centralizer is rational thus giving an affirmative answer to a special case of a problem proposed by conway in 1971 second we characterize all languages commuting with a three element code the characterization is similar to the one proved by bergman for polynomials over noncommuting variables see trans am math soc 137 1969 327 and algebraic combinatorics on words cambridge university press cambridge 2000 a language commutes with a three element code x if and only if it is a union of powers of x

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

on nilpotent fuzzy matrices abstract in this paper we study the issue of nilpotent fuzzy matrices we first provide some properties of nilpotent fuzzy matrices in terms of eigenvalues when a finite number of fuzzy matrices are simultaneously considered we establish some characterizations of the simultaneous nilpotence for a finite number of fuzzy matrices on the other hand it is well known that the max min algebra is one kind of lattice we shall extend the results on simultaneous nilpotence to matrices on a bounded distributive lattice

standard eigenvectors of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to either factor thus inclines generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the standard eigenvectors of incline matrices i e matrices over inclines in detail as the main results for an incline matrix with index and for a lattice matrix the structures of all the standard eigenvectors are given and the greatest standard eigenvector is determined also it is proved that for an incline matrix with index the set of all the standard eigenvectors of it coincides with the set of all the standard eigenvectors of its transitive closure the power convergence of an incline matrix with index is characterized in terms of the sets of standard eigenvectors the results in the present paper include some previous results for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

on the explicit construction of higher deformations of partition statistics the modularity of the partition generating function has many important consequences for example asymptotics and congruences for p n in a series of papers the author and ono 12 13 connected the rank a partition statistic introduced by dyson to weak maass forms a new class of functions which are related to modular forms and which were rst considered in 11 here we do a further step towards understanding how weak maass forms arise from interesting partition statistics by placing certain 2 marked durfee symbols introduced by andrews 1 into the framework of weak maass forms to do this we construct a new class of functions which we call quasiweak maass forms because they have quasimodular forms as components as an application we prove two conjectures of andrews it seems that this new class of functions will play an important role in better understanding weak maass forms of higher weight themselves and also their derivatives as a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices

on transitivity of generalized fuzzy matrices abstract in this paper generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra some elementary properties and characterizations for transitive generalized fuzzy matrices are established and transitivity of powers of a generalized fuzzy matrix is discussed also the transitive closure of a generalized fuzzy matrix is considered and some properties of the transitive closures are obtained partial results obtained in this paper generalize the corresponding ones on nilpotent matrices and on matrices with periods

asymptotic formulas for spt crank of partitions abstract in this paper using a variant of the hardy ramanujan circle method originally due to e wright we prove asymptotic formulas for spt crank of ordinary partitions

on nilpotent incline matrices inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the nilpotent incline matrices in detail it is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector the nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors main diagonals nilpotent indices and adjoint matrices also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established the results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references

maximum order index of matrices over commutative inclines an answer to an open problem abstract this paper proves that the maximum order index of n n matrices over an arbitrary commutative incline equals n 1 2 1 this is an answer to an open problem compute the maximum order index of a member of m n l proposed by cao kim and roush in a monograph incline algebra and applications 1984 where m n l is the set of all n n matrices over an incline l

a simple undecidable problem the inclusion problem for finite substitutions on ab c as an evidence of the power of finite unary substitutions we show that the inclusion problem for finite substitutions on the language l ab c is undecidable i e it is undecidable whether for two finite substitutions and the relation w w holds for all w in l

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

inequalities for ranks of partitions and the first moment of ranks and cranks of partitions abstract we prove two monotonicity properties of n m n the number of partitions of n with rank m they are i for any nonnegative integers m and n n m n n m 2 n and ii for any nonnegative integers m and n such that n 12 n m 2 n m n n m n 1 g e andrews b kim and the first author introduced ospt n a function counting the difference between the first positive rank and crank moments they proved that ospt n 0 in another article k bringmann and k mahlburg gave an asymptotic estimate for ospt n the two monotonicity properties for n m n lead to stronger inequalities for ospt n that imply the asymptotic estimate

on the nilpotent matrices over d01 lattice abstract in this paper we study nilpotent matrices over d 01 lattice a distributive lattice l is said to be d 01 lattice if 0 1 l and 0 x 1 for all x l a necessary and sufficient condition for a d 01 lattice matrix a to be a nilpotent matrix will be given we also establish some important properties of nilpotent matrices

inverse monoids decidability and complexity of algebraic questions this paper investigates the word problem for inverse monoids generated by a set c subject to relations of the form e f where e and f are both idempotents in the free inverse monoid generated by c it is shown that for every fixed monoid of this form the word problem can be solved both in linear time on a ram as well as in deterministic logarithmic space which solves an open problem of margolis and meakin for the uniform word problem where the presentation is part of the input exptime completeness is shown for the cayley graphs of these monoids it is shown that the first order theory with regular path predicates is decidable regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language as a corollary the decidability of the generalized word problem is deduced

asymptotic inequalities for positive crank and rank moments andrews chan and kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments in this paper we prove the asymptotic behavior of these moments in all cases our main result states that the two families of moment functions are asymptotically equal but the crank moments are also asymptotically larger than the rank moments andrews chan and kim also gave a combinatorial description for the differences of the first crank and rank moments that they named the ospt function our main results therefore also give the asymptotic behavior of the ospt function and its analogs for higher moments and we further determine the behavior of the ospt function modulo 2 by relating its parity to andrews spt function

commutation with codes the centralizer of a set of words x is the largest set of words c x commuting with x xc x c x x it has been a long standing open question due to j h conway regular algebra and finite machines chapman hall london 1971 whether the centralizer of any rational set is rational while the answer turned out to be negative in general see m kunc proc of icalp 2004 lecture notes in computer science vol 3142 springer berlin 2004 pp 870 881 we prove here that the situation is different for codes the centralizer of any rational code is rational and if the code is finite then the centralizer is finitely generated this result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by b ratoandromanana rairo inform theor 23 4 1989 425 444 many of the techniques we use in this paper follow her ideas we also give in this paper an elementary proof for the prefix case

on the complexity of decidable cases of the commutation problem of languages we investigate the complexity of basic decidable cases of the commutation problem for languages testing the equality xy yx for two languages x and y we show that it varies from co nexptime complete through pspace complete and co np complete to deterministic polynomial time when y is an explicitly given finite language and x is given by a cf grammar generating a finite language a nondeterministic finite automaton or a regular expression an acyclic nondeterministic finite automaton or an explicitly given finite language respectively interestingly in most cases the complexity status does not change if instead of explicitly given finite y we consider general y of the same type as x for deterministic finite automata the problem remains open due to the asymmetry of the catenation

on the existence of prime decompositions we investigate factorizations of regular languages in terms of prime languages a language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps we give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition we show that there exist non regular unary languages that do not have prime decompositions we also consider infinite factorizations of unary languages

proof of the andrews dyson rhoades conjecture on the spt crank abstract the spt crank of a vector partition or an s partition was introduced by andrews garvan and liang let n s m n denote the net number of s partitions of n with spt crank m that is the number of s partitions 1 2 3 of n with spt crank m such that the length of 1 is odd minus the number of s partitions 1 2 3 of n with spt crank m such that the length of 1 is even andrews dyson and rhoades conjectured that n s m n m is unimodal for any n and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions they obtained an asymptotic formula for the difference between the rank and crank of ordinary partitions which implies n s m n n s m 1 n for sufficiently large n and fixed m in this paper we introduce a representation of an ordinary partition called the m durfee rectangle symbol which is a rectangular generalization of the durfee symbol introduced by andrews we give a proof of the conjecture of andrews dyson and rhoades we also show that this conjecture implies an inequality between the positive rank and crank moments obtained by andrews chan and kim

parity of the partition function abstract although much is known about the partition function little is known about its parity for the polynomials d x d x 2 1 24 where d 23 mod 24 we show that there are infinitely many m resp n for which p d m is even resp p d n is odd if there is at least one such m resp n we bound the first m and n if any in terms of the class number h d for prime d we show that there are indeed infinitely many even values to this end we construct new modular generating functions using generalized borcherds products and we employ galois representations and locally nilpotent hecke algebras

conjugacy relations of prefix codes it is shown that if x and y are prefix codes and z is a non empty language satisfying the condition x z z y then z is the union of a non empty family p n i i of pairwise disjoint prefix sets such that x p i p i y for all i i consequently the conjugacy relations of prefix codes are explored and under the restriction that both of x and y are prefix codes the solutions of the conjugacy equation x z z y for languages are determined also the decidability of the conjugacy problem for finite prefix codes is confirmed

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

the commutation of finite sets a challenging problem we prove that given a set x of two nonempty words a set y of nonempty words commutes with x if and only if either y is a union of powers of x or x y subseteq t for some primitive word t we also show that the same holds for certain special types of codes but does not hold in general for sets of cardinality at least four

the spt function of andrews recently andrews introduced the function s n spt n which counts the number of smallest parts among the integer partitions of n we show that its generating function satisfies an identity analogous to ramanujan s mock theta identities as a consequence we are able to completely determine the parity of s n using another type of identity one based on hecke operators we obtain a complete multiplicative theory for s n modulo 3 these congruences confirm unpublished conjectures of garvan and sellers our methods generalize to all integral moduli

on the equivalence problem for binary dol systems it is shown that to test whether two dol sequences in the binary case coincide it is enough to test whether four first words of these sequences are the same the result is optimal

powers of matrices over distributive lattices a review in this paper we review the results on powers of matrices over a distributive lattice we stress the significance of the graph theoretical approach and show how several previous results for matrices of special types can be obtained in a unified way

on generalized fuzzy matrices with periods in this paper generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra some properties and characterizations for generalized fuzzy matrices with periods are established and an expression for the transitive closure of a generalized fuzzy matrix with a period as a sum of its powers is shown reduction of generalized fuzzy matrices is defined and some properties for reduction of generalized fuzzy matrices with periods are obtained also the structure of all standard eigenvectors for a generalized fuzzy matrix with a period is determined

on dyson s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions in this paper we prove a longstanding conjecture by freeman dyson concerning the limiting shape of the crank generating function we fit this function in a more general family of inverse theta functions which play a key role in physics

on nilpotent matrices over distributive lattices abstract in this paper nilpotent matrices over distributive lattices are considered some properties and characterizations for nilpotent matrices are established and in particular a necessary and sufficient condition for an n n nilpotent matrix to have the nilpotent index n is given also some properties of the reduction of a nilpotent matrix are obtained partial results obtained in this paper are results on nilpotent fuzzy matrices by h hashimoto

on the powers of matrices over a distributive lattice abstract in this paper the index and the period for a lattice matrix are estimated some necessary and sufficent conditions for the convergence of the powers of a lattice matrix are obtained also some equivalent conditions for a nilpotent lattice matrix are discussed

the semigroup of incline hall matrices this paper investigates the semigroup of incline hall matrices an incline hall matrix means an incline matrix which is greater than or equal to an incline invertible matrix it is proved that any incline hall matrix has index some necessary and sufficient conditions for an incline hall matrix to be regular in the semigroup of incline hall matrices are given and green s relations on the such semigroup are described also the sandwich semigroup of incline hall matrices is considered the main results in the present paper generalize some previous results on boolean matrices fuzzy matrices and lattice matrices

on structure of domains with quadratic growth abstract let f be a finite field and let r be an affine f algebra which is a domain of gelfand kirillov dimension smaller than 3 let m n be natural numbers assume that x r is transcendental over f and y 1 y n r are such that i j m i j x i y k x j 0 for some i j f not all equal to 0 and each k n it is shown that either r satisfies a polynomial identity or else the subalgebra of r generated by y 1 y 2 y n and x has gelfand kirillov dimension 1 from this we deduce that a finitely generated domain over f with quadratic growth and with an infinite centre satisfies a polynomial identity is a pi domain moreover the centralizer of a non algebraic element in a finitely generated domain with quadratic growth over finite field is a pi domain

the spt crank for ordinary partitions the spt function spt n was introduced by andrews as the weighted counting of partitions of n with respect to the number of occurrences of the smallest part andrews garvan and liang defined the spt crank of an s partition which leads to combinatorial interpretations of the congruences of spt n mod 5 and 7 let ns m n denote the net number of s partitions of n with spt crank m andrews garvan and liang showed that ns m n is nonnegative for all integers m and positive integers n and they asked the question of finding a combinatorial interpretation of ns m n in this paper we introduce the structure of doubly marked partitions and define the spt crank of a doubly marked partition we show that ns m n can be interpreted as the number of doubly marked partitions of n with spt crank m moreover we establish a bijection between marked partitions of n and doubly marked partitions of n a marked partition is defined by andrews dyson and rhoades as a partition with exactly one of the smallest parts marked they consider it a challenge to find a definition of the spt crank of a marked partition so that the set of marked partitions of 5n 4 and 7n 5 can be divided into five and seven equinumerous classes the definition of spt crank for doubly marked partitions and the bijection between the marked partitions and doubly marked partitions leads to a solution to the problem of andrews dyson and rhoades

asymptotic inequalities for k ranks and their cumulation functions abstract asymptotic formulas for the positive moments of rank and crank of partitions were obtained by k bringmann and k mahlburg recently motivated by their works in this paper we prove asymptotic formulas for the k ranks and their cumulation functions asymptotic inequalities between these combinatorial objects are also discovered in particular we show that for fixed integer l and sufficiently large n m l n n l n and m l n n l n where m l n resp n l n denotes the number partitions of n with crank resp rank l

on effective construction of the greatest solution of language inequality xa bx in this paper we consider effective constructions of the greatest solution of the language inequality xa bx it has been proved by kunc in 2005 that the greatest solution of xa bx is regular provided that b is regular no matter what a is however this proof is based on kruskal s tree theorem and does not provide any effective way to construct the greatest solution we focus on this gap in this paper we give an effective construction of the greatest solution for the following two cases i a b are regular and there exists k 1 such that pref b a k b kpref b a where pref b is the set of prefixes of words in b ii a b are regular and b is a code with finite decoding delay our construction takes the point of view of games as shown by kunc in his regularity proof the construction of the greatest solution can be reduced to the construction of the winning region of a two player game our contribution is to show that the winning regions of the two player game for the two cases can be constructed effectively the main ingredient of the construction for the first case is a shrinking lemma for the words on which one of the players has a winning strategy while the construction for the second case is based on the observation that the two player game can be reduced to a two player reachability game played on the transition graph of a one counter machine