test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

on binary equality sets and a solution to the test set conjecture in the binary case abstract k culik ii and j karhumaki rairo inform theor 14 1980 349 369 conjectured that the equality set of two injective morphisms over a binary alphabet is of the form u v for some possibly empty words u and v here we show that such an equality set is always either of the above form or of the form uw v for some words u w and v as an application we give a simple proof for the test set conjecture in the binary case cf k culik ii and a salomaa j comput system sci 20 1980 379 396 in fact we show that a test set can always be chosen to contain no more than three words

the ehrenfeucht conjecture a algebra framework for its proof abstract we describe an algebraic framework including in particular the affine monoid of a given ring within this framework it is a simple matter to deduce ehrenfeucht s conjecture from hilbert s basis theorem

multiple factorizations of words and defect effect we prove that if x is a finite prefix set and w is a non periodic bi infinite word possessing 3 disjoint x factorizations then the combinatorial rank of x is at most card x gamma 2 this is one of the rare cases when a cumulative defect effect is known to hold finally connections to the critical factorization theorem are discussed

a note on decidability questions on presentations of word semigroups we apply automata theoretic tools and some recently established compactness properties in the study of f semigroups that is subsemigroups of free semigroups with each f semigroup we associate an f presentation which turns out to be finite for all finitely generated f semigroups connections between f presentations and ordinary presentations of semigroups are also pointed out it is also shown that it is undecidable whether two finitely generated f semigroups satisfy a common relation in their f presentation

a compactness property of the k abelian monoids abstract the k abelian equivalence of words counting the numbers of occurrences of factors of length at most k has been analyzed in recent years from several different directions we continue this analysis the k abelian equivalence classes are known to constitute a monoid hence equations over these monoids are well defined we show that these monoids satisfy a compactness property each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems we give two proofs for this compactness result one is based the fact that the monoid can be embedded into the multiplicative monoid of matrices and the other directly on linear algebra the former method allows the application of hilbert s basis theorem the latter one in turn allows to conclude an upper bound for the size of the finite subsystem

fractal boundaries of basin of attraction of newton raphson method in helicopter trim a key problem in helicopter aeroelastic analysis is the enormous computational time required for a numerical solution of the nonlinear system of algebraic equations required for trim particularly when free wake models are used trim requires calculation of the main rotor and tail rotor controls and the vehicle attitude which leads to the six steady forces and moments about the helicopter center of gravity to be zero an appropriate initial estimate of the trim state is needed for successful helicopter trim this study aims to determine the control inputs that can have considerable effect on the convergence of trim solution in the aeroelastic analysis of helicopter rotors by investigating the basin of attraction of the nonlinear equations set of initial guess points from which the nonlinear equations converge it is illustrated that the three main rotor pitch controls of collective pitch longitudinal cyclic pitch and lateral cyclic pitch have a significant contribution to the convergence of the trim solution trajectories of the newton iterates are shown and some ideas for accelerating the convergence of a trim solution in the aeroelastic analysis of helicopters are proposed it is found that the basins of attraction can have fractal boundaries

algebraic properties of word equations abstract the question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words recently aleksi saarela has introduced a new approach to the problem that is based on linear algebraic properties of polynomials encoding the equations and their solutions in this paper we develop further this approach and take into account other algebraic properties of polynomials namely their factorization this in particular allows to improve the bound for the number of independent equations with maximal rank from quadratic to linear

unique factorization results for semigroups of l functions in this paper we prove the unique factorization of certain subsemigroups of the extended selberg class of l functions these semigroups are generated by the functions of degree 0 and 1 we also consider extensions of such results and applications to classical l functions

a remark on the uniqueness of the dirichlet series with a riemann type function equation abstract we show that if for a nonzero complex number c the inverse images l 1 1 c and l 2 1 c of two functions in the extended selberg class are the same then l 1 s and l 2 s must be identical

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

many aspects of defect theorems we give a survey and a unified presentation of the defect theorem its generalizations and recent aspects of interest in its basic form the defect theorem states that if a set of n words satisfies a nontrivial relation then these words can be expressed simultaneously as products of at most n 1 words in other words dependency of words causes a defect effect there does not exist just one defect theorem but several ones depending on the restrictions that are put to the n 1 words the defect theorem is closely related to equations of words and in this way to the compactness theorem for systems of word equations

analysis of velocity flux least squares principles for the navier stokes equations part ii this paper continues the development of the least squares methodology for the solution of the incompressible navier stokes equations started in part i here we again use a velocity flux first order navier stokes system but our focus now is on a practical algorithm based on a discrete negative norm

equations over finite sets of words and equivalence problems in automata theory abstract several observations concerning equations over monoids of finite sets of words are made monoids considered are the monoid of finite prefix codes the monoid of all finite sets and the monoid of all finite multisets as well as some others as an application some known decidability results in automata theory are extended finally several open questions are raised

newton s algorithm for magnetohydrodynamic equations with the initial guess from stokes like problem abstract the magnetohydrodynamic equations are second order nonlinear partial differential equations which are coupled by fluid velocity and magnetic fields and we consider to apply the newton s algorithm to solve them it is well known that the choice of a proper initial guess is critical to assure the convergence of newton s iterations in solving nonlinear partial differential equations in this paper we provide a good initial guess for newton s algorithm when it is applied for solving magnetohydrodynamic equations

on common zeros of l functions we will prove that two l functions l1 l2 satisfying the same functional equation must be identically equal if sufficiently many zeros of l1 are also zeros of l2

on the structure of the selberg class i 0 d 1 the selberg class s is a rather general class of dirichlet series with functional equation and euler product and can be regarded as an axiomatic model for the global l functions arising from number theory and automorphic representations one of the main problems of the selberg class theory is to classify the elements of s such a classication is based on a real valued invariant d called degree and the degree conjecture asserts that d 2 n for every l function in s the degree conjecture has been proved for d 5 3 and in this paper we extend its validity to d 2 the proof requires several new ingredients in particular a rather precise description of the properties of certain nonlinear twists associated with the l functions in s

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

test sets for finite substitutions abstract ehrenfeucht s conjecture that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s has recently been verified in this paper we extend this result from endomorphisms to certain types of finite substitutions by applying konig s lemma and embeddings of the monoid into a free algebra

a uniqueness theorem for functions in the extended selberg class we study the uniqueness of functions in the extended selberg class it was shown in ki adv math 231 2484 2490 2012 that if for a nonzero complex number c the inverse images l 1 1 c and l 2 1 c of two functions satisfying the same functional equation in the extended selberg class are the same then l 1 s and l 2 s are identical here we prove that this holds even without the assumption that they satisfy the same functional equation

the expressibility of languages and relations by word equations classically several properties and relations of words such as being a power of the same word can be expressed by using word equations this paper is devoted to a general study of the expressive power of word equations as main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations in particular the primitiveness and the equal length are such properties as well as being any word over a proper subalphabet

value distribution for the derivatives of the logarithm of l functions from the selberg class in the half plane of absolute convergence abstract in the present paper we show that for every 0 the function log l s m where m n 0 and l s n 1 a n n s is an element of the selberg class s takes any value infinitely often in the strip 1 re s 1 provided p x a p 2 x for some 0 in particular l s takes any non zero value infinitely often in the strip 1 re s 1 and the first derivative of l s has infinitely many zeros in the half plane re s 1

on systems of word equations with simple loop sets consider the infinite system s of word equations x 0u 1 ix 1u 2 ix 2 u m ix m y 0v 1 iy 1v 2 iy 2 v n iy n email protected n for each email protected n let t k be the subsystem of s given by email protected k k 1 k 2 we prove two properties of the above system 1 let k 1 if f is a solution of t k such that primitive roots of f u 1 f u 2 f u m are of equal length as well as primitive roots of f v 1 f v 2 f v n then f is a solution of the whole s 2 if n 1 then for any k 2 a solution f of t k is also a solution of s

l functions as distributions we define an axiomatic class of l functions extending the selberg class we show in particular that one can recast the traditional conditions of an euler product analytic continuation and functional equation in terms of distributional identities akin to weil s explicit formula the generality of our approach enables some new applications for instance we show that the l function of any cuspidal automorphic representation of mathrm gl 3 mathbb a mathbb q has infinitely many zeros of odd order

multiplicities a deterministic view of nondeterminism abstract we discuss four important problems in automata theory each of which has three natural variants a deterministic one a nondeterministic one and as an in between case a nondeterministic one with multiplicities our goal is to support a general view that the third variant is much closer to the first one than to the second one

local and global cyclicity in free semigroups it is shown that the system of equations x1r xnr s x1s xns r r s n is equivalent to its two element subset x1a xna b x1b xnb a x1a xna c x1c xnc a whenever a b c are integers such that 1 1

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

efficient constructions of test sets for regular and context free languages we present a simple construction of linear size test sets for regular languages and of single exponential test sets for context free languages in the case of regular sets the size of our test set is exactly the number of transitions of the automaton this improves the best known upper bounds exponential for regular and doubly exponential for context free languages we give also an o n log n time algorithm for the morphism equivalence and an o n3log n time algorithm to test the gsm equivalence on a regular language an o n2log n time algorithm is given to test the equivalence of two deterministic gsm s as well as that of two deterministic finite transducers

on complex zeros off the critical line for non monomial polynomial of zeta functions in this paper we show that any polynomial of zeta or l functions with some conditions has infinitely many complex zeros off the critical line this general result has abundant applications by using the main result we prove that the zeta functions associated to symmetric matrices treated by ibukiyama and saito certain spectral zeta functions and the euler zagier multiple zeta functions have infinitely many complex zeros off the critical line moreover we show that the lindelof hypothesis for the riemann zeta function is equivalent to the lindelof hypothesis for zeta functions mentioned above despite of the existence of the zeros off the critical line next we prove that the barnes multiple zeta functions associated to rational or transcendental parameters have infinitely many zeros off the critical line by using this fact we show that the shintani multiple zeta functions have infinitely many complex zeros under some conditions as corollaries we show that the mordell multiple zeta functions the euler zagier hurwitz type of multiple zeta functions and the witten multiple zeta functions have infinitely many complex zeros off the critical line

on the system of word equations x o u i i x 1 u i 2 x 2 u i 3 x 3 y 0 v i 1 y 1 v i 2 y 2 v i 3 y 3 i 0 1 2 l in a free monoid abstract we prove that if x 0 u 1 i x 1 u 2 i x 2 y 0 v 1 i y 1 v 2 i y 2 holds for all i 0 1 2 3 then it is true for each i 0 1 2 and that if x 0 u 1 i x 1 u 2 i x 2 u 3 i x 3 y 0 v 1 i y 1 v 2 i y 2 v 3 i y 3 holds for all i 0 1 2 3 4 5 then it is true for each i 0 1 2

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

a uniqueness theorem for dirichlet series satisfying a riemann type functional equation abstract we will prove a uniqueness theorem for l functions in terms of the pre images of two values in the complex plane

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

some remarks on the convergence of the dirichlet series of l functions and related questions first we show that the abscissae of uniform and absolute convergence of dirichlet series coincide in the case of l functions from the selberg class mathcal s we also study the latter abscissa inside the extended selberg class indicating a different behavior in the two classes next we address two questions about majorants of functions in mathcal s showing links with the distribution of the zeros and with independence results

newton s method for the navier stokes equations with finite element initial guess of stokes equations abstract it is shown that finite element solutions of stokes equations may be chosen as the initial guess for the quadratic convergence of newton s algorithm applied to navier stokes equations provided there are sufficiently small mesh size h and the moderate reynold s number we provide also a mixed convergence analysis in terms of iterations and finite error estimates of the initial guess with a regularity estimate and error analysis for each newton s step

detecting squarefree numbers we present an algorithm based on the explicit formula for l functions and conditional on grh for proving that a given integer is squarefree with little or no knowledge of its factorization we analyze the algorithm both theoretically and practically and use it to prove that several rsa challenge numbers are not squarefull f a non zero element of k x then it is well known that f is squarefree if and only if gcd f f 0 1 and the latter condition may be checked quickly in deterministic polynomial time by the euclidean algorithm it is a long standing question in algorithmic number theory whether there is a correspondingly simple procedure to test if a given integer is squarefree in particular can one determine whether n 2 z is squarefree more rapidly than by factoring it in this paper we describe an algorithm conditional on the generalized riemann hy pothesis grh for proving an integer squarefree with little or no knowledge of its factor ization and analyze the complexity of the algorithm both theoretically and practically in particular we present some heuristic evidence based on random matrix theory and other probabilistic calculations that our algorithm runs in deterministic subexponential time o exp logn 2 3 o 1 although this is poorer than the performance expected of the cur rent best known factoring algorithms our method is able to give partial results that one does not obtain from a failed attempt at factoring in particular we show the following see

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach