existence and uniqueness of periodic solutions for a p laplacian duffing equation with a deviating argument abstract by means of mawhin s continuation theorem we study a class of p laplacian duffing type differential equations of the form p x t c x t g t x t x t t e t some new results on the existence and uniqueness of periodic solutions for the above equation are obtained it is significant that the growth degree with respect to the variables u v imposed on g t u v is allowed to be greater than p 1 so our results generalize and improve on the corresponding results in related papers

graded modalities in strategy logic abstract strategy logic sl is a logical formalism for strategic reasoning in multi agent systems its main feature is that it has variables for strategies that are associated to specific agents using a binding operator in this paper we introduce graded strategy logic graded sl an extension of sl by graded quantifiers over tuples of strategy variables i e there exist at least g different tuples x 1 x n of strategies where g is a cardinal from the set n 0 1 2 0 we prove that the model checking problem of graded sl is decidable we then turn to the complexity of fragments of graded sl when the g s are restricted to finite cardinals written graded n sl the complexity of model checking is no harder than for sl i e it is non elementary in the quantifier block rank we illustrate our formalism by showing how to count the number of different strategy profiles that are nash equilibria ne by analysing the structure of the specific formulas involved we conclude that the important problem of checking for the existence of a unique ne can be solved in 2 exptime which is not harder than merely checking for the existence of such an equilibrium

existence and uniqueness of periodic solutions for a kind of lienard equation with two deviating arguments in this paper a kind of lienard equation with two deviating arguments of the formx t f x t x t g 1 t x email protected 1 t g 2 t x email protected 2 t p t is considered we derive some new sufficient conditions for checking uniqueness of t periodic solutions of this equation by using coincidence degree theory we also establish some new results on the existence and uniqueness of t periodic solutions for this equation which are new and complement previously known results

on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

alternating time temporal logics with linear past abstract we investigate the succinctness gap between two known equally expressive and different linear past extensions of standard atl we establish by formal non trivial arguments that the memoryful linear past extension the history leading to the current state is taken into account can be exponentially more succinct than the standard local linear past extension the history leading to the current state is forgotten as a second contribution we consider the atl like fragment denoted atl l p of the known memoryful linear past extension of atl we show that atl l p is strictly more expressive than atl and interestingly it can be exponentially more succinct than the more expressive logic atl moreover we prove that both satisfiability and model checking for the logic atl l p are exptime complete

on temporal logics with data variable quantifications although data values are available in almost every computer system reasoning about them is a challenging task due to the huge data size or even infinite data domains temporal logics are the well known specification formalisms for reactive and concurrent systems various extensions of temporal logics have been proposed to reason about data values mostly in the last decade among them one natural idea is to extend temporal logics with variable quantifications ranging over an infinite data domain grumberg kupferman and sheinvald initiated the research on this topic recently and obtained several interesting results however there is still a lack of systematic investigation on the theoretical aspects of the variable extensions of temporal logics our goal in this paper is to fill this gap around this goal we make the following choices 1 we consider the variable extensions of two widely used temporal logics linear temporal logic ltl and computation tree logic ctl and allow arbitrary nestings of variable quantifications with boolean and temporal operators the resulting logics are called respectively variable ltl in brief vltl and variable ctl in brief vctl 2 we investigate the decidability and complexity of both the satisfiability and model checking problems over both finite and infinite words trees in particular we obtain the following results existential variable quantifiers or one single universal quantifier in the beginning already entail the undecidability of the satisfiability problems of both vltl and vctl over both finite and infinite words trees if only existential path quantifiers are used in vctl then the satisfiability problem over finite trees is decidable no matter which variable quantifiers are available for vltl formulae with one single universal variable quantifier in the beginning if the occurrences of the non parameterized atomic propositions are guarded by the positive occurrences of the quantified variables then its satisfiability problem becomes decidable over both finite and infinite words based on these results of the satisfiability problem we deduce the un decidability results of the model checking problem

probabilistic temporal logics via the modal mu calculus this paper presents a mu calculus based modal logic for describing properties of reactive probabilistic labeled transition systems rpltss and develops a model checking algorithm for determining whether or not states in finite state rpltss satisfy formulas in the logic the logic is based on the distinction between probabilistic systems and nonprobabilistic observations using the modal mu calculus one may specify sets of observations and the semantics of our logic then enable statements to be made about the measures of such sets at various system states the logic may be used to encode a variety of probabilistic modal and temporal logics in addition the model checking problem for it may be reduced to the calculation of solutions to systems of non linear equations finally the logic induces an equivalence on rpltss that coincides with accepted notions of probabilistic bisimulation in the literature

imprimitive ninth degree number fields with small discriminants we present tables of ninth degree nonprimitive i e containing a cubic subfield number fields each table corresponds to one signature except for fields with signature 3 3 for which we give two different tables depending on the signature of the cubic subfield details related to the computation of the tables are given as well as information about the cpu time used the number of polynomials that we deal with etc for each field in the tables we give its discriminant the discriminant of its cubic subfields the relative polynomial generating the field over one of its cubic subfields the corresponding irreducible polynomial over q and the galois group of the galois closure fields having interesting properties are studied in more detail especially those associated with sextic number fields having a class group divisible by 3

results on the propositional μ calculus abstract in this paper we define and study a propositional calculus l which consists essentially of propositional modal logic with a least fixpoint operator l is syntactically simpler yet strictly more expressive than propositional dynamic logic pdl for a restricted version we give an exponential time decision procedure small model property and complete deductive system theory subsuming the corresponding results for pdl

shimura curves of genus at most two we enumerate all shimura curves x d 0 r of genus at most two there are exactly 858 such curves up to equivalence

decidability of model checking with the temporal logic ef abstract the branching time temporal logic ef is a simple but natural fragment of computation tree logic ctl and the modal calculus we study the decidability of the model checking problem for ef and infinite state systems we use process rewrite systems prs to describe infinite state systems and define a hierarchy of subclasses of prs that includes petri nets pushdown processes basic parallel processes bpp context free processes and pa processes then we establish the exact limits of the decidability of model checking with ef in this hierarchy model checking with ef is undecidable for petri nets and even for parallel pushdown automata the pushdown extension of basic parallel processes on the other hand model checking with ef is decidable for pad a process model that subsumes both pa processes and pushdown processes

simulation relations for alternating buchi automata quotienting by simulation equivalences is a well established technique for reducing the size of nondeterministic buchi automata we adapt this technique to alternating buchi automata to this end we suggest two new quotients namely minimax and semi elective quotients prove that they preserve the recognized languages and show that computing them is not more difficult than computing quotients for nondeterministic buchi automata our approach is game theoretic the proofs rely on a specifically tailored join operation for strategies in simulation games which is interesting in its own right we explain the merits of our quotienting procedures with respect to converting alternating buchi automata into nondeterministic ones

existence of periodic solutions for a lienard type p laplacian differential equation with a deviating argument by means of mawhin s continuation theorem a class of p laplacian type differential equation with a deviating argument of the form p x t f x t x t t g t x t t x e t is studied a new result related to t and the deviating argument t x is obtained it is significant that the growth degree with respect to the variable x in g t x is allowed to be greater than p 1 which could be achieved infrequently in previous papers

computing all power integral bases of cubic fields applying baker s effective method and the reduction procedure of baker and davenport we present several lists of solutions of index form equations in totally real and complex cubic algebraic number fields these solutions yield all power integral bases of these fields 1 introduction let k be a cubic algebraic number field and denote by zk the ring of integers of k a power integral basis of k is an integral basis of the form 1 2 with some a e zk if there exists such an a then we say that k is monogenic since zk z e obviously if a has this property then it holds also for a k with any k e z from a practical point of view it is important to know whether there exists a power integral basis of k and if so what are the numerical values of a if the galois group of k is cyclic then the discriminant of k is a full square the problem of monogeneity in cyclic cubic fields was considered by m n gras 12 13 archinard 1 and dummit and kisilevsky 4 m n gras and archinard gave necessary and sufficient conditions for monogeneity and tested for several numerical examples whether or not the field is monogenic moreover dummit and kisilevsky proved that there exist infinitely many cyclic cubic fields with power integral bases for arbitrary algebraic number fields l it was proved by gyory 14 that up to obvious translations by elements of z there are only finitely many a e zl with zl z and he gave effective but rather large bounds for the sizes of these a in this paper we present a method which allows us to determine all possible values of a up to translation with rational integers such that 1 2 is a power integral

improved model checking of hierarchical systems we present a unified game based approach for branching time model checking of hierarchical systems such systems are exponentially more succinct than standard state transition graphs as repeated sub systems are described only once early work on model checking of hierarchical systems shows that one can do better than a naive algorithm that flattens the system and removes the hierarchy given a hierarchical system s and a branching time specification j for it we reduce the model checking problem does s satisfy j to the problem of solving a hierarchical game obtained by taking the product of s with an alternating tree automaton a j for j our approach leads to clean uniform and improved model checking algorithms for a variety of branching time temporal logics in particular by improving the algorithm for solving hierarchical parity games we are able to solve the model checking problem for the m calculus in pspace and time complexity that is only polynomial in the depth of the hierarchy our approach also leads to an abstraction refinement paradigm for hierarchical systems the abstraction maintains the hierarchy and is obtained by merging both states and sub systems into abstract states

on real cyclic sextic fields a table of units and class numbers of real cyclic sextic fields with conductor 16 1 leopoldt s unit index and the signature rank of the unit group

constructing complete tables of quartic fields using kummer theory we explain how to construct tables of quartic fields of discriminant less than or equal to a given bound in an efficient manner using kummer theory instead of the traditional and much less efficient method using the geometry of numbers as an application we describe the computation of quartic fields of discriminant up to 107 the corresponding table being available by anonymous ftp

some new results on the existence of periodic solutions to a kind of rayleigh equation with a deviating argument abstract by employing the continuation theorem of coincidence degree theory we study a kind of rayleigh equation with a deviating argument as follows x t f x t g x t t p t some new results on the existence of periodic solutions are obtained

periodic solutions for a fourth order p laplacian differential equation with a deviating argument abstract by means of mawhin s continuation theorem a kind of fourth order p laplacian differential equation with a deviating argument as follows p u t f u t u t g t u t u t t e t is studied under various assumptions the existence of periodic solutions is obtained

calculation of the class numbers of imaginary cyclic quartic fields any imaginary cyclic quartic field can be expressed uniquely in the form k q ja d b d where a is squarefree odd and negative d b2 c2 is squarefree b 0 c 0 and a d 1 explicit formulae for the discriminant and conductor of k are given in terms of a b c d the calculation of tables of the class numbers h k of such fields k is described let q denote the field of rational numbers and let k be a cyclic extension of q of degree 4 the unique quadratic subfield of k is denoted by k the class number of k resp k is denoted by h k resp h k the conductor of k is denoted by k in the case of real cyclic quartic fields k gras 3 has given a table of the values of h k for all such fields with 4000 recently the authors have carried out the calculation of the class numbers of imaginary cyclic quartic fields 4 in this note we give a brief description of the computation of the tables given in 4 the following explicit representation of a cyclic quartic field is proved in 4

the verification and control of interacting similar discrete event systems this paper explore issues related to the control and verification of similar module systems in the discrete event systems framework similar module systems are distributed systems comprised of subsystem modules that exhibit isomorphic local behavior coordinated on global event occurrences when given a global model of these systems it is shown how to decompose the global model into the component subsystems in polynomial time it is also shown how to perform various verification tasks for these interacting systems while mitigating common state explosion difficulties by taking advantage of the special similar module system structure control properties of the similar module systems are also discussed it is assumed that the local modules are supervised by exactly one local controller and the controllers enforce the same local control policy necessary and sufficient conditions for achieving local and global control specifications in this setting are identified

on the real cubic fields in this paper the authors announce a table of the 4753 totally real nonconjugate nonabelian cubic fields with discriminant less than 100000 each field is given by its discriminant the coefficients of a generating polynomial and the index of this polynomial over the field a basis of the integers of the field is also given some differences with other tables are pointed out godwin and samet 3 have described the construction of a table of real cubic fields with discriminants d 20000 using similar methods angell 1 extends these results up to d 100000 using a different method the authors 5 have constructed a table of the 4753 real nonabelian cubic fields with discriminants d 100000 here and in the sequel triplets of conjugate fields are counted once only the method developed in 5 generalizes the one used in 4 and the table constructed there gives for each cubic field k its discriminant an irreducible polynomial f x which defines k the index of f x over k and a basis for the integers of k in this work we present some consequences derived from 5 in particular we have discovered that ten fields are missing from angell s table in table 1 we give each of these ten fields and its corresponding class numbers h the field k with discriminant d is generated by a root 0 of the polynomial f x x3 ax b s is the index of 0 and 1 0 a is a basis of the integers of k where 02 to t 2 a a

the regular viewpoint on pa processes pa is the process algebra allowing non determinism sequential and parallel compositions and recursion we suggest viewing pa processes astrees and usingtree automata techniques for verification problems on pa our main result is that the set of iterated predecessors of a regular set of pa processes is a regular tree language and similarly for iterated successors furthermore the corresponding tree automata can be built effectively in polynomial time this has many immediate applications to verification problems for pa processes among which a simple and general model checking algorithm

alternating finite automata on ω words abstract alternating finite automata on words are introduced as an extension of nondeterministic finite automata which process infinite sequences of symbols the classes of languages defined by alternating finite automata are investigated and characterized under four types of acceptance conditions it is shown that for one type of acceptance condition alternation increases the power in comparison with nondeterminism and for other three acceptance conditions nondeterministic finite automata on words have the same power as alternating ones

new results on the nonexistence of generalized bent functions several new results on the nonexistence of generalized bent functions are proved by using properties of the decomposition law of primes and the class group of imaginary abelian number fields

an automata theoretic approach to branching time model checking translating linear temporal logic formulas to automata has proven to be an effective approach for implementing linear time model checking and for obtaining many extensions and improvements to this verification method on the other hand for branching temporal logic automata theoretic techniques have long been thought to introduce an exponential penalty making them essentially useless for model checking recently bernholtz and grumberg 1993 have shown that this exponential penalty can be avoided though they did not match the linear complexity of non automata theoretic algorithms in this paper we show that alternating tree automata are the key to a comprehensive automata theoretic framework for branching temporal logics not only can they be used to obtain optimal decision procedures as was shown by muller et al but as we show here they also make it possible to derive optimal model checking algorithms moreover the simple combinatorial structure that emerges from the automata theoretic approach opens up new possibilities for the implementation of branching time model checking and has enabled us to derive improved space complexity bounds for this long standing problem

on the resolution of inhomogeneous norm form equations in two dominating variables applying baker s well known method and the reduction procedure described by baker and davenport we give a numerical algorithm for finding all solutions of inhomogeneous thue equations of type nk q x ay x 1 in the variables x y e z and a e zic with a max x y ll2 where k q a is a totally real cubic field

isomorphism of k contractible graphs a generalization of bounded valence and bounded genus a polynomial time isomorphism test for graphs called k contractible graphs for fixed k is included the class of k contractible graphs includes the graphs of bounded valence and the graphs of bounded genus the algorithm uses several new ideas including 1 it removes portions of the graph and replaces them with groups which are used to keep track of the symmetries of these portions 2 it maintains with each group a tower of equivalence relations which allow a decomposition of the group these towers are called towers of k actions

uniform strategies rational relations and jumping automata a general concept of uniform strategies has recently been proposed as a relevant notion in game theory for computer science which subsumes various notions from the literature it relies on properties involving sets of plays in two player turn based arenas equipped with arbitrary binary relations between plays these properties are expressed in a language based on ctl with a quantifier over related plays there are two semantics for our quantifier a strict one and a full one that we study separately regarding the strict semantics the existence of a uniform strategy is undecidable for rational binary relations but introducing jumping tree automata and restricting attention to recognizable relations allows us to establish a 2 exptime complete complexity and still capture a class of two player imperfect information games with epistemic temporal objectives regarding the full semantics relying on information set automata we establish that the existence of a uniform strategy is decidable for rational relations and we provide a nonelementary synthesis procedure we also exhibit an essentially optimal subclass of rational relations for which the problem becomes 2 exptime complete considering rich classes of relations makes the theory of uniform strategies powerful it directly entails various results in logics of knowledge and time some of them already known and others new

periodic solutions for n dimensional generalized lienard type p laplacian functional differential system abstract we consider the n dimensional generalized lienard system d d t p x t c x t d d t f x t g x t t e t driven by the scalar p laplacian c is an n n symmetric matrix of constants using the degree theory we establish some criteria to guarantee the existence of periodic solutions for the above system which generalize and improve on the corresponding results in the related literature

periodic solutions for a rayleigh type p laplacian equation with sign variable coefficient of nonlinear term as p laplacian equations have been widely applied in the field of fluid mechanics and nonlinear elastic mechanics it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p laplacian by using lu s continuation theorem which is an extension of manasevich mawhin we study the existence of periodic solutions for a rayleigh type p laplacian equation f p x t f x t g 1 x email protected 1 t x email protected t g 2 x email protected 2 t x e t it is significant that the growth degree with respect to the variable u in g 1 u is allowed to be greater than p 1 and the coefficient b t of g 2 x email protected 2 t x can change sign in this paper which could be achieved rarely in the previous literature

precise interprocedural dependence analysis of parallel programs it is known that interprocedural detection of copy constants and elimination of faint code in parallel programs are undecidable problems if base statements are assumed to execute atomically we show that these problems become decidable if this assumption is abandoned so the unrealistic idealization from program verification atomic execution of base statements introduced in order to simplify matters actually increases the difficulty of these problems from the point of view of program analysis amazingly these problems become more tractable if we adopt a less idealized more realistic model of execution we introduce an effective abstract domain of antichains of dependence traces that allows us to perform a precise interprocedural dependence analysis in non atomically executing parallel programs the main idea is to trace sequences of dependences exhibited successively by program executions we define operations on antichains of dependence traces and show that they precisely abstract the corresponding operations on sets of non atomic program executions using these operations we can analyze dependences by means of an abstract interpretation of constraint systems that characterize sets of program executions of interest the result of the dependence analysis can in turn be used to detect all copy constants and to eliminate faint code while the run time of the algorithms is exponential in the number of program variables it is polynomial in the program size hence they are polynomial time algorithms if the number of program variables is bounded in order to justify their overall exponential run time we show that both detection of copy constants and elimination of faint code are intractable np hard even when the atomic execution idealization is abandoned this holds already for parallel programs without loops or procedures

periodic solutions for higher order differential equations with deviating argument abstract by using the coincidence degree theory of mawhin we study the existence of periodic solutions for higher order differential equations with deviating argument x n t i 1 n 1 b i x i t f t x t x t 1 t x t m t p t some new results on the existence of periodic solutions of the equations are obtained meanwhile an example is given to illustrate our results

tenth degree number fields with quintic fields having one real place in this paper we enumerate all number fields of degree 10 of discriminant smaller than 10 11 in absolute value containing a quintic field having one real place for each one of the 21509 resp 18167 found fields of signature 0 5 resp 2 4 the field discriminant the quintic field discriminant a polynomial defining the relative quadratic extension the corresponding relative diseriminant the corresponding polynomial over q and the galois group of the galois closure are given in a supplementary section we give the first coincidence of discriminant of 19 resp 20 nonisomorphic fields of signature 0 5 resp 2 4

computing ray class groups conductors and discriminants we use the algorithmic computation of exact sequences of abelian groups to compute the complete structure of z k m for an ideal m of a number field k as well as ray class groups of number fields and conductors and discriminants of the corresponding abelian extensions as an application we give several number fields with discriminants less than previously known ones

complexity results on branching time pushdown model checking the model checking problem of pushdown systems pmc problem for short against standard branching temporal logics has been intensively studied in the literature in particular for the modal m calculus the most powerful branching temporal logic used for verification the problem is known to be exptime complete even for a fixed formula the problem remains exptime complete also for the logic ctl which corresponds to a fragment of the alternation free modal m calculus for the logic ctl the problem is known to be in 2exptime in this paper we show that the complexity of the pmc problem for ctl is in fact 2exptime complete moreover we give a new optimal algorithm to solve this problem based on automata theoretic techniques finally we prove that the program complexity of the pmc problem against ctl i e the complexity of the problem in terms of the size of the system is exptime complete

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem

an axiomatization of bisimulation quantifiers via the µ calculus this paper deals with bisimulation quantifiers logic bql that is the extension of propositional dynamic logic pdl with the so called bisimulation quantifiers this logic is expressively equivalent to the calculus an extension of modal logic with extremal fixpoints albeit its formulas are easier to understand in this work we provide a complete axiomatization of bql based on certain normal form results for the calculus obtained by janin and walukiewicz

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

periodic solutions for a kind of lienard equation with a deviating argument abstract by means of continuation theorem of coincidence degree theory we study a kind of lienard equation with a deviating argument as follows x t f x t x t g x t t x t e t some new results on the existence of periodic solutions are obtained

on the resolution of relative thue equations an efficient algorithm is given for the resolution of relative thue equations the essential improvement is the application of an appropriate version of wildanger s enumeration procedure based on the ellipsoid method of fincke and pohst recently relative thue equations have gained an important application e g in computing power integral bases in algebraic number fields the presented methods can surely be used to speed up those algorithms the method is illustrated by numerical examples

reachability is decidable for weakly extended process rewrite systems process rewrite systems prs are widely accepted as a formalism for the description of infinite state systems it is known that the reachability problem for prs is decidable the problem becomes undecidable when prs are extended with a finite state control unit in this paper we show that the problem remains decidable when prs are extended with a weak i e acyclic except for self loops finite state control unit we also present some applications of this decidability result

computing all power integral bases in orders of totally real cyclic sextic number fields an algorithm is given for determining all power integral bases in orders of totally real cyclic sextic number fields the orders considered are in most cases the maximal orders of the fields the corresponding index form equation is reduced to a relative thue equation of degree 3 over the quadratic subfield and to some inhomogeneous thue equations of degree 3 over the rationals at the end of the paper numerical examples are given

on invariant polynomials and their application in field theory certain polynomials invariant under a permutation group g so called g polynomials play an important role in several computational methods of galois theory since their practical value depends on the degree it is important to know g polynomials of smallest possible degree a reasonable technique to find such g polynomials is presented and for certain classes of groups an explicit description is obtained the list of g polynomials given by stauduhar in vol 27 of this journal is thereby enlarged and improved as an application of g polynomials three important resolvents of quintic and sextic algebraic equations are computed and a parametric family of sextic equations with given galois group is exhibited introduction let k be a field and x1 x indeterminates let sm be the group of all permutations of 1 m which acts on the polynomial ring k xl xmi by s o xi xs i s e sm i 1 m let g be a subgroup of sm several problems in field theory e g the determination of the galois group or the computation of a galois resolvent of a given algebraic equation can be solved explicitly if one can do the following see 7 p 276 16 11 2 3 and references in the last paper i find a polynomial p belonging to g also called a g polynomial i e a polynomial p in k x1 xm whose stabilizer s e sm i p p is equal to g ii let z be an additional indeterminate and a m am the elementary symmetric functions defined by z x1 zxm zm a1zm l am let r e k x xm z be the polynomial of z degree sm g whose zeros in k x1 xm are the elements of s o p i s e sm find the unique polynomial r e k x xm z such that r r a am z suppose that problems i and ii have been solved for g 5 sm if f zm a1zm l am e k z is a polynomial with coefficients in k and splitting field k f then r a1 am z is a certain resolvent of f i e its splitting field is a subfield of k f the prime factor decomposition of r a1 am z in k z for example gives insight into the relation between g and the galois group g f gal k f k of f by this method one can determine the group g f loc cit up to now actual solutions of problem ii exist only in very few cases if g is the alternating group am take r z2 d where d is the discriminant of zm xizm xm for subgroups g of sm m 4 see 10 p 72 ff if m 5 and received september 16 1982 1980 mathematics subject classification primary 12 04 12f10 20b99 1987 american mathematical society 0025 5718 87 1 00 25 per page

enumeration of quartic fields of small discriminant with the mixed type case now completed all algebraic number fields of degree 4 with absolute discriminant 106 have been enumerated methods from the totally real and totally complex cases were used without major mod ification isomorphism of fields was determined by a method similar to one of lenstra the t2 criterion of pohst was applied to reduce the number of redundant examples

model checking higher order programs we propose a novel verification method for higher order functional programs based on higher order model checking or more precisely model checking of higher order recursion schemes recursion schemes for short the most distinguishing feature of our verification method for higher order programs is that it is sound complete and automatic for the simply typed calculus with recursion and finite base types and for various program verification problems such as reachability flow analysis and resource usage verification we first show that a variety of program verification problems can be reduced to model checking problems for recursion schemes by transforming a program into a recursion scheme that generates a tree representing all the interesting possible event sequences of the program we then develop a new type based model checking algorithm for recursion schemes and implement a prototype recursion scheme model checker to our knowledge this is the first implementation of a recursion scheme model checker experiments show that our model checker is reasonably fast despite the worst case time complexity of recursion scheme model checking being hyperexponential in general altogether the results provide a new promising approach to verification of higher order functional programs

decidable first order transition logics for pa processes we show the decidability of model checking pa processes against several first order logics based upon the reachability predicate the main tool for this result is the recognizability by tree automata of the reachability relation the tree automata approach and the transition logics we use allow a smooth and general treatment of parameterized model checking for pa this approach is extended to handle a quite general notion of costs of pa steps in particular when costs are parikh images of traces we show decidability of a transition logic extended by some form of first order reasoning over costs

sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument abstract by means of mawhin s continuation theorem we study some second order differential equations with a deviating argument x t f t x t x t t x t e t some new results on the existence of periodic solutions are obtained the interest is that we allow the degree with respect to the variables x 0 x 1 x 2 of f t x 0 x 1 x 2 to be greater than 1 and also the result theorem 3 2 is related to the deviating argument t meanwhile we give an example to demonstrate our result

a targeted martinet search constructing number fields with prescribed ramification is an important problem in computational number theory in this paper we consider the problem of computing all imprimitive number fields of a given degree which are unramified outside of a given finite set of primes s by combining the techniques of targeted hunter searches with martinet s relative version of hunter s theorem we then carry out this algorithm to generate complete tables of imprimitive number fields for degrees 4 through 10 and certain sets s of small primes

a table of quintic number fields all algebraic number fields f of degree 5 and absolute discriminant less than 2 x 107 totally real fields respectively 5 x 106 other signatures are determined we describe the methods which we applied and list significant data

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

a fast algorithm to compute cubic fields we present a very fast algorithm to build up tables of cubic fields real cubic fields with discriminant up to 10 11 and complex cubic fields down to 10 11 have been computed

existence of periodic solutions to a p laplacian lienard differential equation with a deviating argument abstract by means of mawhin s continuation theorem a kind of p laplacian lienard differential equation with a deviating argument as follows p y t f y t y t h y t g y t t e t is studied a new result on the existence of periodic solutions is obtained the interest is that the relation between the existence of periodic solutions and the deviating argument t is investigated meanwhile the approaches used to estimate a priori bounds of periodic solutions are different from the corresponding ones in the known literature

a thue equation with quadratic integers as variables in an earlier paper we determined all the solutions in z of a cubic thue equation with coefficients in a quadratic number field it is now shown that the method used there can be used to solve the more general problem of determining all the solutions of the thue equation in a ring of quadratic integers

on totally real cubic fields with discriminant 10 the authors have constructed a table of the 592923 nonconjugate totally real cubic number fields of discriminant d 107 thereby extending the existing table of fields with d 5 x 105 constructed by ennola and turunen 4 each field is given by its discriminant and the coefficients of a generating polynomial the method used is an improved version of the method developed in 8 the article contains an exposition of the modified method statistics and examples the decomposition of the rational primes is studied and the relative frequency of each type of decomposition is compared with the corresponding density given by davenport and heilbronn 2

alternating time temporal logic temporal logic comes in two varieties linear time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system branching time temporal logic allows explicit existential and universal quantification over all paths we introduce a third more general variety of temporal logic alternating time temporal logic offers selective quantification over those paths that are possible outcomes of games such as the game in which the system and the environment alternate moves while linear time and branching time logics are natural specification languages for closed systems alternating time logics are natural specification languages for open systems for example by preceding the temporal operator eventually with a selective path quantifier we can specify that in the game between the system and the environment the system has a strategy to reach a certain state the problems of receptiveness realizability and controllability can be formulated as model checking problems for alternating time formulas depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators we obtain the two alternating time temporal logics atl and atl atl and atl are interpreted over concurrent game structures every state transition of a concurrent game structure results from a choice of moves one for each player the players represent individual components and the environment of an open system concurrent game structures can capture various forms of synchronous composition for open systems and if augmented with fairness constraints also asynchronous composition over structures without fairness constraints the model checking complexity of atl is linear in the size of the game structure and length of the formula and the symbolic model checking algorithm for ctl extends with few modifications to atl over structures with weak fairness constraints atl model checking requires the solution of 1 pair rabin games and can be done in polynomial time over structures with strong fairness constraints atl model checking requires the solution of games with boolean combinations of buchi conditions and can be done in pspace in the case of atl the model checking problem is closely related to the synthesis problem for linear time formulas and requires doubly exponential time

power integral bases in a parametric family of totally real cyclic quintics we consider the totally real cyclic quintic fields k n q v n generated by a root v n of the polynomial f n x x 5 n 2 x 4 2n 3 6n 2 10n 10 x 3 n 4 5n 3 11n 2 15n 5 x 2 n 3 4n 2 10n 10 x 1 assuming that m n 4 5n 3 15n 2 25n 25 is square free we compute explicitly an integral basis and a set of fundamental units of k n and prove that k n has a power integral basis only for n 1 2 for n 1 2 both values presenting the same field all generators of power integral bases are computed

on model checking for the m calculus and its fragments in this paper we consider the model checking problem for the calculus and show that it is succinctly equivalent to the non emptiness problem of finite state automata on infinite binary trees with the parity acceptance condition we also present efficient model checking algorithms for two rich subclasses of the calculus formulas and relate their expressive power to well known extensions of branching time temporal logics

the computation of sextic fields with a cubic subfield and no quadratic subfield we describe six tables of sixth degree fields k containing a cubic subfield k and no quadratic subfield one for totally real sextic fields one for sextic fields with four real places two for sextic fields with two real places and two for totally imaginary sextic fields depending on whether the cubic subfield is totally real or not the tables provide for each possible discriminant dg of k a quadratic polynomial which defines k k the discriminant of the cubic subfield and the galois group of a galois closure at q of k q

new results on the existence of periodic solutions to a p laplacian differential equation with a deviating argument by means of mawhin s continuation theorem a kind of p laplacian differential equation with a deviating argument as follows p x t f t x t x t t x t e t is studied some new results on the existence of periodic solutions are obtained the main results theorems 3 2 and 3 3 are all related to the deviating argument t meanwhile the degrees with respect to the variables x0 x1 of f t x0 x1 x2 are allowed to be grater than p 1 which is different from the corresponding conditions of known literature

quadratic extensions of totally real quintic fields in this work we establish lists for each signature of tenth degree number fields containing a totally real quintic subfield and of discriminant less than 10 13 in absolute value for each field in the list we give its discriminant the discriminant of its subfield a relative polynomial generating the field over one of its subfields the corresponding polynomial voer q and the galois group of its galois closure we have examined the existence of several non isomorphic fields with the same discriminants and also the existence of unramified extensions and cyclic extensions

generalising automaticity to modal properties of finite structures we introduce a complexity measure of modal properties of finite structures which generalises the automaticity of languages it is based on graph automata like devices called labelling systems we define a measure of the size of a structure that we call rank and show that any modal property of structures can be approximated up to any fixed rank n by a labelling system the function that takes n to the size of the smallest labelling system doing this is called the labelling index of the property we demonstrate that this is a useful and fine grained measure of complexity and show that it is especially well suited to characterise the expressive power of modal fixed point logics from this we derive several separation results of modal and non modal fixed point logics some of which are already known whereas others are new

on the computation of totally real quartic fields of small discriminant all totally real quartic fields of discriminant less than 106 are computed the method used to generate the fields is derived from delone and faddeev with corrections and improvements a new method for deciding field i8omorphism is used to eliminate redundant examples integral bases and galois groups are given for each field 1 introduction in 3 pp 184 200 delone and faddeev give a number geometric method for computing generating equations for all totally real quartic fields of discriminant not exceeding a fixed upper bound l e r 0 this method is well suited for implementation on a computer unfortunately the original paper contains many mistakes and the proofs are sometimes hard to understand in the present paper we give a corrected version of the delone faddeev algorithm we describe its implementation and we give a table of all totally real quartic fields of discriminant less than 1 000 000 including their integral bases and the galois groups of their normal closures a different method for computing totally real quartic fields was given by god win 6 and for general algorithms for computing fields of small discriminants see martinet 8 and pohst 9 10

on the expressivity and complexity of quantitative branching time temporal logics we investigate extensions of ctl allowing to express quantitative requirements about an abstract notion of time in a simple discrete time framework and study the expressive power of several relevant logics when only subscripted modalities are used polynomial time model checking is possible even for the largest logic we consider while the introduction of freeze quantifiers leads to a complexity blow up

periodic solutions for a kind of lienard equation using inequality techniques and coincidence degree theory new results are provided concerning the existence and uniqueness of t periodic solutions for a lienard equations with delay an illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results and are more effective

a table of totally real quintic number fields we give a table of the 1077 totally real number fields of degree five having a discriminant less than 2 000 000 there are two nonisomorphic fields of discriminant 1 810 969 and two nonisomorphic fields of discriminant 1 891 377 all the other number fields in the table are characterized by their discriminant among these fields three are cyclic and four have a galois closure whose galois group is the dihedral group d5 the galois closure for all the other fields in the table has a galois group isomorphic to s5

tables of octic fields with a quartic subfield we describe the computation of extended tables of degree 8 fields with a quartic subfield using class field theory in particular we find the minimum discriminants for all signatures and for all the possible galois groups we also discuss some phenomena and statistics discovered while making the tables such as the occurrence of 11 non isomorphic number fields having the same discriminant or several pairs of non isomorphic number fields having the same dedekind zeta function