on language equations with invertible operations the paper studies language equations of the type x l r and l y r where l and r are given languages and is an invertible binary word language operation for most of the considered insertion and deletion operations the existence of both a solution and a singleton solution to these equations proves to be decidable for given regular l and r in case l is a context free language and r is a regular one the existence of a solution is generally undecidable the results can be extended to more complex linear equations systems of linear equations as well as for equations of higher degree

normal forms symmetry and linearization of dynamical systems we discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system or a parameter dependent family via the poincare normal form approach we discuss this formally at first and later pay attention to the convergence of the linearizing procedure we also discuss some generalizations of our main result

on convergent normal form transformations in presence of symmetries abstract we prove a convergence criterion for transformations to poincare dulac normal form that involves the centralizer of the given vector field

rapidly convergent iteration method for simultaneous normal forms of commuting maps we propose a new rapidly convergent iteration scheme under weak simultaneous arithmetic conditions for overdetermined systems related to simultaneous conjugation of commuting maps the first application is for commuting holomorphic maps in mathcal c n fixing the origin not necessarily diffeomorphisms if n geq 2 the second one is for orientation preserving gevrey circle mappings by means of nonlinear superposition estimates

testing dna code words properties of regular languages one aspect of dna computing is the possibility of using dna molecules for solving some complicated computational problems in this context the dna code word design problem assumes a fundamental role given a problem encoded in dna strands and biochemical processes the final computation is a concatenation of the input dna strands that must allow us to recover the solution of the given problem in terms of the input unique decipherability thus the initial set of dna strands must be a code in addition it should satisfy some restrictions called here dna properties in order to prevent them from interacting in undesirable ways so a new interest towards the design of efficient algorithms for testing whether a language x is a code has arisen from wet dna computing but as far as we know only when x is a finite set in this paper we provide an algorithm for testing whether an infinite but regular set of words is a code that avoids some dna properties among unwanted intermolecular and intramolecular hybridizations

two complementary operations inspired by the dna hairpin formation completion and reduction we consider two complementary operations hairpin completion introduced in d cheptea c martin vide v mitrana a new operation on words suggested by dna biochemistry hairpin completion in proc transgressive computing 2006 pp 216 228 with motivations coming from dna biochemistry and hairpin reduction as the inverse operation of the hairpin completion both operations are viewed here as formal operations on words and languages we settle the closure properties of the classes of regular and linear context free languages under hairpin completion in comparison with hairpin reduction while the class of linear context free languages is exactly the weak code image of the class of the hairpin completion of regular languages rather surprisingly the weak code image of the class of the hairpin completion of linear context free languages is a class of mildly context sensitive languages the closure properties with respect to the hairpin reduction of some time and space complexity classes are also studied we show that the factors found in the general cases are not necessary for regular and context free languages this part of the paper completes the results given in the earlier paper where a similar investigation was made for hairpin completion finally we briefly discuss the iterated variants of these operations

codes unambiguous automata and sofic systems we study the relationship between codes and unambiguous automata inside a sofic system we show that a recognizable set is a code in a sofic system if and only if a particular automaton associated to the set and the shift is unambiguous we discuss an example of a finite complete code in a sofic system in connection with the factorization conjecture

on the convergence of normalizing transformations in the presence of symmetries it is shown that under suitable conditions involving in particular the existence of analytic constants of motion the presence of lie point symmetries can ensure the convergence of the transformation of a vector field or dynamical system into normal form

normal forms and nonlinear symmetries we give some general theorems and extensions of previous results concerning the problem of transforming an algebra of vector fields into poincare normal form by means of a unifying algebraic language we show the possibility of obtaining either a parallel or joint normal form of the vector fields in a definite way which simplifies the construction of normal forms providing a precise restriction on their structure the application to finite dimensional dynamical systems and their lie point symmetries is also discussed

convergent normal forms of symmetric dynamical systems it is shown that the presence of lie point symmetries of non hamiltonian dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical system or vector field into poincare dulac normal form

coding properties of dna languages the computation language of a dna based system consists of all the words dna strands that can appear in any computation step of the system in this work we define properties of languages which ensure that the words of such languages will not form undesirable bonds when used in dna computations we give several characterizations of the desired properties and provide methods for obtaining languages with such properties the decidability of these properties is addressed as well as an application we consider splicing systems whose computation language is free of certain undesirable bonds and is generated by nearly optimal comma free codes

deletion along trajectories we describe a new way to model deletion operations on formal languages called deletion along trajectories we examine its closure properties which differ from those of shuffle on trajectories previously introduced by mateescu et al in particular we define classes of nonregular sets of trajectories such that the associated deletion operation preserves regularity our results give uniform proofs of closure properties of the regular languages for several deletion operations we also show that deletion along trajectories serves as an inverse to shuffle on trajectories this leads to results on the decidability of certain language equations including those of the form ltx r where l r are regular languages and x is unknown

orbital linearization around equilibria of vector fields in rn and lie symmetries abstract in this paper we study some relationships between analytic vector fields x in r n which are analytically orbitally linearizable in a neighborhood of an equilibrium point and the structure of the set n x of its locally analytic normalizers analytic infinitesimal generators of lie point orbitally symmetries in that neighborhood characterization and necessary and sufficient conditions for the orbital linearization of x via the existence of special elements in n x are given

integration of perturbed initial value problems through reduction theory we propose a method to integrate an initial value problem formed by the differential equations of a perturbed dynamical system plus the initial condition the approach consists of several steps first of all if the original system does not enjoy a continuous symmetry it is transformed up to a certain order of approximation m into an equivalent but equivariant one after truncation at order m second this new symmetry allows us to define a reduction map which produces in its whole phase space a splitting of the transformed equations specifically the latter system is decoupled into two subsystems also initial value problems one of them defined in the orbit space associated with the reduction and the other linear in its coordinates as a third step the reduced subsystems are analytically resolved or numerically integrated using standard methods finally the solution of the original system is recovered by inverting back the reduction map and thereafter the analytic transformations we apply our procedure to three examples

complexity and elementary abelian p groups one of the main differences between ordinary and modular representation theory of finite groups is the existence of nonprojective modules in the modular representation algebra however given a finitely generated module m over the group algebra kg where k is a field of characteristic p 0 there exists a uniquely determined up to kg isomorphism projective module p called the projective cover of m and an epimorphism f

transducer descriptions of dna code properties and undecidability of antimorphic problems this work concerns formal descriptions of dna code properties and related un decidability questions this line of research allows us to give a property as input to an algorithm in addition to any regular language which can then answer questions about the language and the property here we define dna code properties via transducers and show that this method is strictly more expressive than that of regular trajectories without sacrificing the efficiency of deciding the satisfaction question we also show that the maximality question can be undecidable our undecidability results hold not only for the fixed dna involution but also for any fixed antimorphic permutation moreover we also show the undecidability of the antimorphic version of the post correspondence problem for any fixed antimorphic permutation

on properties of bond free dna languages the input data for dna computing must be encoded into the form of single or double dna strands as complementary parts of single strands can bind together forming a double stranded dna sequence one has to impose restrictions on these sets of dna words languages to prevent them from interacting in undesirable ways we recall a list of known properties of dna languages which are free of certain types of undesirable bonds then we introduce a general framework in which we can characterize each of these properties by a solution of a uniform formal language inequation this characterization allows us among others to construct i a uniform algorithm deciding in polynomial time whether a given dna language possesses any of the studied properties and ii in many cases also an algorithm deciding whether a given dna language is maximal with respect to the desired property

aspects of shuffle and deletion on trajectories word and language operations on trajectories provide a general framework for the study of properties of sequential insertion and deletion operations a trajectory gives a syntactical constraint on the scattered insertion deletion of a word into from another one with an intuitive geometrical interpretation moreover deletion on trajectories is an inverse of the shuffle on trajectories these operations are a natural generalization of many binary word operations like catenation quotient insertion deletion shuffle etc besides they were shown to be useful e g in concurrent processes modelling and recently in biocomputing area we begin with the study of algebraic properties of the deletion on trajectories then we focus on three standard decision problems concerning linear language equations with one variable involving the above mentioned operations we generalize previous results and obtain a sequence of new ones particularly we characterize the class of binary word operations for which the validity of such a language equation is un decidable for regular and context free operands

a note on the cohomology of finite dimensional cocommutative hopf algebras abstract in the context of finite dimensional cocommutative hopf algebras we prove versions of various group cohomology results the quillen venkov theorem on detecting nilpotence in group cohomology chouinard s theorem on determining whether akg module is projective by restricting to elementary abelianp subgroups ofg and quillen s theorem which identifies the cohomology ofg modulo nilpotent elements this last result is only proved for graded connected hopf algebras

permutation complexes and modular representation theory hc g o iti has two basic functors a rg 2 di rh and a rh 2 rg given by restriction and induction which play an essential role in representation theory an important and elementary class of rg representations is permutation modules which are direct sums of modules indz r obtained by induction from the trivial rh module r for various hs g in another extreme one has rg modules which arise by induction from rh projective modules leading to the concept of relative projectivity and green s theory of vertices and sources cr gr the value of these subcategories of modules in representation theory and related areas is well known in a different direction influenced by algebraic geometry and topology one considers not only module categories but various categories of chain complexes of modules and their cohomologies this culminates in the more recent approaches to representation theory through the theory of derived categories see sc cps and their many references