classification of the ring of witt vectors and the necklace ring associated with the formal group law x y qxy abstract in this paper we classify the ring of witt vectors and the necklace ring associated with the formal group law x y q x y up to strict isomorphism as q varies over the set of integers we also introduce a q deformation of the grothendieck ring of formal power series with constant term 1 over rings in which q 1 is a unit various bijections related to witt vectors necklaces and formal power series are given as an application we provide a q analogue of the celebrated cyclotomic identity and its bijective proof

formal group theoretic generalizations of the necklace algebra including aq deformation abstract n metropolis and g c rota adv math 50 1983 95 125 studied thenecklace polynomials and were lead to define thenecklace algebraas a combinatorial model for the classical ring ofwitt vectors which corresponds to the multiplicative formal group lawx y xy in this paper we define and study a generalized necklace algebra which is associated with an arbitrary formal group lawfover a torsion free ringa the map from the ring of witt vectors associated withfto the necklace algebra is constructed in terms of certain generalizations of the necklace polynomials we present a combinatorial interpretation for these polynomials in terms of words on a given alphabet the actions of theverschiebungandfrobeniusoperators as well as of thep typification idempotentare described and interpreted combinatorially aq analogue and other generalizations of thecyclotomic identityare also presented in general the necklace algebra can only be defined over the rationalizationa q nevertheless we show that for the family of formal group laws over the integersfq x y x y qxy q z we can define the corresponding necklace algebras over z we classify these algebras and define isomorphic ring structures on the groups of witt vectors and the groups of curves associated with the formal group lawsfq theq necklace polynomials which turn out to benumerical polynomialsin two variables can be interpreted combinatorially in terms of so calledq words and they satisfy an identity generalizing a classical one

a sub graph isomorphism algorithm for matching large graphs we present an algorithm for graph isomorphism and subgraph isomorphism suited for dealing with large graphs a first version of the algorithm has been presented in a previous paper where we examined its performance for the isomorphism of small and medium size graphs the algorithm is improved here to reduce its spatial complexity and to achieve a better performance on large graphs its features are analyzed in detail with special reference to time and memory requirements the results of a testing performed on a publicly available database of synthetically generated graphs and on graphs relative to a real application dealing with technical drawings are presented confirming the effectiveness of the approach especially when working with large graphs

on point based temporal disjointness abstract we address the problems of determining consistency and of finding a solution for sets of three point relations expressing exclusion of a point from an interval and for sets of four point relations expressing interval disjointness availability of these relations is an important requirement for dealing with the sorts of temporal constraints encountered in many ai applications such as plan reasoning we prove that consistency testing is np complete and finding a solution is np hard

a compression type mountain pass theorem in conical shells in this paper we present a compression type version of the mountain pass lemma in a conical shell with respect to two norms an application to second order ordinary differential equations is included

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

double linking theorem and multiple solutions abstract in this paper we establish a new double linking theorem if two sets a and b link each other then under suitable conditions we get two bounded ps sequences which produce two distinct critical points the critical points either have different critical values or belong to linking sets a and b separately the abstract result will be used to solve semilinear eigenvalue problems and problems with jumping nonlinearities which may or may not resonant with respect to fucik spectrum

congruence relations characterizing the representation ring of the symmetric group abstract using the theory of burnside rings various canonical families of congruence relations are exhibited which hold for characters of the symmetric group and characterize the ring generated by them as a subring of the ring of integer valued class functions as a corollary the irreducible characters are characterized as integer valued class functions n z which satisfy those congruence relations and the standard conditions 1 0 and n 2 n

critical point theorems in cones and multiple positive solutions of elliptic problems abstract we obtain critical point variants of the compression fixed point theorem in cones of krasnoselskii critical points are localized in a set defined by means of two norms in applications to semilinear elliptic boundary value problems this makes possible the use of local moser harnack inequalities for the estimations from below multiple solutions are found for problems with oscillating nonlinearity

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

witt vectors and the algebra of necklaces where u is the classical mobius function m a n is called a necklace polynomial we shall be concerned with some remarkable identities satisfied by the polynomials m a n which apparently have not been previously noticed specifically we prove that m a n can be expressed as a quadratic polynomial in m a i and mj that m a n can be expressed linearly and integrally in terms of m a i the proofs given below of these identities are entirely combinatorial theorems l 6

a mechanical solution of schubert s steamroller by many sorted resolution abstract we demonstrate the advantage of using a many sorted resolution calculus by a mechanical solution of a challenge problem this problem known as schubert s steamroller had been unsolved by automated theorem provers before our solution clearly demonstrates the power of a many sorted resolution calculus the proposed method is applicable to all resolution based inference systems

a bounded mountain pass lemma without the ps condition and applications we present a version of the mountain pass lemma which does not require the ps condition we apply this version to problems where the ps condition is not satisfied

the leray schauder boundary condition in critical point theory abstract we present a version of schechter s bounded mountain pass theorem in a hilbert space not identified to its dual an application to positive solutions of elliptic boundary value problems is included

a generalized cyclotomic identity abstract in this paper we prove a generalization of the cyclotomic identity for each group g satisfying suitable finiteness conditions we prove this generalized cyclotomic identity by counting coloured g set structures the cyclotomic identity corresponds to taking g the free abelian group of rank one for the free abelian group of rank two for example we get an analogous identity involving the generating function of the partition function interalia we show that studying g set structures gives a unified and conceptually simple approach to other classical combinatorial identities not to the author s knowledge previously linked in the literature

r analogue of the burnside ring of profinite groups and free lie algebras abstract let r be a finite dimensional torsion free special ring in this paper we generalize the results in dress and siebeneicher adv in math 70 1988 89 78 1989 1 by constructing r analogue r g of the burnside ring of profinite groups g in particular we remark that the grothendieck lie module denominator identity of free lie algebras in oh necklace rings and logarithmic functions preprint kias 2003 is closely related to the canonical isomorphism between r g and grothendieck s ring of formal power series with coefficients in r and constant term 1

the burnside ring of the infinite cyclic group and its relations to the necklace algebra λ rings and the universal ring of witt vectors abstract it is shown that well known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the burnside ring of the infinite cyclic group on the one hand and grothendieck s ring of formal power series with constant term 1 as well as the universal ring of witt vectors on the other hand

categorical aspects of generating functions i exponential formulas and krull schmidt categories abstract in this paper we study formal power series with exponents in a category for example the generating function of a category e with finite hom sets is defined by e t tx aut x where the summation is taken over all isomorphism classes of objects of e we can use such power series to enumerate the number of e structures along a faithful functor theorem 4 6 our theory is closely related to the theory of species joyal 1981 a species can be identified with a faithful functor from a groupoid to the category of finite sets theorem 3 6 we use mainly the concept of faithful functors with finite fibers instead of species so that we can separate the roles categories and functors play for example the exponential formula e t exp con e t means the unique coproduct decomposition property theorem 5 8 in the final section we give some applications of our theory to rather classical enumerations

fixed point indices of iterated maps on determine toutes les relations universelles qui existent entre les indices de point fixe i f n n 1 2 des iterees f n d une application f v y ou y est un enr vcy est un sous ensemble ouvert

q analog of the mobius function and the cyclotomic identity associated to a profinite group abstract let g be a profinite group and q an indeterminate in this paper we introduce and study a q analog of the mobius function and the cyclotomic identity arising from the lattice of open subgroups of g when q is any integer we show that they have close connections with the functors w g q nr g q and nr g q introduced in y t oh q deformation of witt burnside rings math z 257 2007 151 191 in particular we interpret the multiplicative property of the inverse of the table of marks and the mobius function of g as a composition property of certain functors classification of w g q nr g q and nr g q up to strict natural isomorphism as q varies over the set of integers and its application will be dealt with too

q deformation of witt burnside rings in this paper we construct a q deformation of the witt burnside ring of a profinite group over a commutative ring where q ranges over the set of integers when q 1 it coincides with the witt burnside ring introduced by dress and siebeneicher adv math 70 87 132 1988 to achieve our goal we first show that there exists a q deformation of the necklace ring of a profinite group over a commutative ring as in the classical case i e the case q 1 q deformed witt burnside rings and necklace rings always come equipped with inductions and restrictions we also study their properties as a byproduct we prove a conjecture due to lenart j algebra 199 703 732 1998 finally we classify mathbb w g q up to strict natural isomorphism in case where g is an abelian profinite group

the computational complexity of propositional strips planning abstract i present several computational complexity results for propositional strips planning i e strips planning restricted to ground formulas different planning problems can be defined by restricting the type of formulas placing limits on the number of pre and postconditions by restricting negation in pre and postconditions and by requiring optimal plans for these types of restrictions i show when planning is tractable polynomial and intractable np hard in general it is pspace complete to determine if a given planning instance has any solutions extremely severe restrictions on both the operators and the formulas are required to guarantee polynomial time or even np completeness for example when only ground literals are permitted determining plan existence is pspace complete even if operators are limited to two preconditions and two postconditions when definite horn ground formulas are permitted determining plan existence is pspace complete even if operators are limited to zero preconditions and one postcondition one of the interesting tractable problems is if each operator is restricted to positive preconditions and one postcondition only ground literals the blocks world problem slightly modified is a subproblem of this restricted planning problem these results in combination with previous analyses are not encouraging for domain independent planning

nested witt vectors and their q deformation abstract let n n be a truncation set we study the ring of n nested witt vectors and its q deformation given two arbitrary integers q and r we provide a necessary and sufficient condition of a so that w n q a and w n r a should be strictly isomorphic to each other also an isomorphism of functors w n q w m q w m n q will be established for coprime truncation sets m and n as a byproduct we deal with interesting connections between nested witt vectors and other areas such as generalized mobius functions and numerical polynomials i e polynomials which take integral values at integer arguments

decomposition of the witt burnside ring and burnside ring of an abelian profinite group abstract let g h be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers the aim of this paper is to establish an isomorphism of functors w g q w h q w g h q where w g q denotes the q deformed witt burnside ring functor of g introduced in y t oh q deformation of witt burnside rings math z 207 1 2007 151 191 to do this we first establish an isomorphism of functors b g q b h q b g h q where b g q denotes the q deformed burnside ring functor of g which was also introduced in y t oh q deformation of witt burnside rings math z 207 1 2007 151 191 as an application we derive a pseudo multiplicative property of the q mobius function associated to the lattice of open subgroups of the direct sum of g and h

planning for conjunctive goals abstract the problem of achieving conjunctive goals has been central to domain independent planning research the nonlinear constraint posting approach has been most successful previous planners of this type have been complicated heuristic and ill defined i have combined and distilled the state of the art into a simple precise implemented algorithm tweak which i have proved correct and complete i analyze previous work on domain independent conjunctive planning in retrospect it becomes clear that all conjunctive planners linear and nonlinear work the same way the efficiency and correctness of these planners depends on the traditional add delete list representation for actions which drastically limits their usefulness i present theorems that suggest that efficient general purpose planning with more expressive action representations is impossible and suggest ways to avoid this problem

witt vectors and tambara functors abstract we give a combinatorial description of the ring of g witt vectors on a polynomial algebra over the integers for every finite group g using this description we show that the functor which takes a commutative ring with trivial action of g to its ring of witt vectors coincides with the left adjoint of the algebraic functor from the category of g tambara functors to the category of commutative rings with an action of g

q deformed necklace rings and q mobius function abstract we introduce a q deformation of the classical mobius function and investigate its properties in connection with q deformed truncated necklace rings also we study the strictly natural isomorphism of q deformed necklace rings

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

on an elliptic boundary value problem at double resonance abstract let be a bounded domain in r n n 2 with smooth boundary we discuss the existence and multiplicity of solutions of the boundary value problem u p x u in u 0 on roughly speaking under the assumption that l lim t p x t t l 1 for two distinct consecutive eigenvalues l l 1 of the first boundary value problem of

decomposable functors and the exponential principle abstract we develop a combinatorial theory for the operation of forming the logarithm of generating functions associated with covariant functors on the category of finite sets and bijective maps this leads to a rather general and flexible exponential principle for labeled combinatorial structures we also introduce the concept of a topos like category which turns out to be a convenient framework for the discussion of a wide range of applications a few of which are studied in detail

kernel partial least squares regression in reproducing kernel hilbert space a family of regularized least squares regression models in a reproducing kernel hilbert space is extended by the kernel partial least squares pls regression model similar to principal components regression pcr pls is a method based on the projection of input explanatory variables to the latent variables components however in contrast to pcr pls creates the components by modeling the relationship between input and output variables while maintaining most of the information in the input variables pls is useful in situations where the number of explanatory variables exceeds the number of observations and or a high level of multicollinearity among those variables is assumed motivated by this fact we will provide a kernel pls algorithm for construction of nonlinear regression models in possibly high dimensional feature spaces we give the theoretical description of the kernel pls algorithm and we experimentally compare the algorithm with the existing kernel pcr and kernel ridge regression techniques we will demonstrate that on the data sets employed kernel pls achieves the same results as kernel pcr but uses significantly fewer qualitatively different components

necklace rings and logarithmic functions in this paper we develop the theory of the necklace ring and the logarithmic function regarding the necklace ring we introduce the necklace ring functor nr from the category of special rings into the category of special rings and then study the associated adams operators as far as the logarithmic function is concerned we generalize the results in bryant s paper free lie algebras and formal power series j algebra 253 1 2002 167 188 to the case of graded lie super algebras with a group action by applying the euler poincare principle

multiplicity results for a semilinear boundary value problem with the nonlinearity crossing higher eigenvalues on etudie le probleme aux valeurs limites au g u h x u 0 ou a est un operateur differentiel lineaire auto adjoint uniformement elliptique d ordre 2

a probabilistic analysis of prepositional strips planning abstract i present a probabilistic analysis of prepositional strips planning the analysis considers two assumptions one is that each possible precondition likewise postcondition of an operator is selected independently of other pre and postconditions the other is that each operator has a fixed number of preconditions likewise postconditions under both assumptions i derive bounds for when it is highly likely that a planning instance can be efficiently solved either by finding a plan or proving that no plan exists roughly if planning instances under either assumption have n propositions ground atoms and g goals and the number of operators is less than an o n in g bound then a simple efficient algorithm can prove that no plan exists for most instances if the number of operators is greater than an n in g bound then a simple efficient algorithm can find a plan for most instances the two bounds differ by a factor that is exponential in the number of pre and postconditions a similar result holds for plan modification i e solving a planning instance that is close to another planning instance with a known plan thus it appears that prepositional strips planning a pspace complete problem exhibits a easy hard easy pattern as the number of available operators increases with a narrow range of hard problems an empirical study demonstrates this pattern for particular parameter values because prepositional strips planning is pspace complete this extends previous phase transition analyses which have focused on np complete problems also the analysis shows that surprisingly simple algorithms can solve a large subset of the planning problem

an application of burnside rings in elementary finite group theory abstract a canonical map from the burnside ring c of a finite cyclic group c into the burnside ring g of any finite group g of the same order is exhibited and it is shown that many results from elementary finite group theory in particular those claiming certain congruence relations are simple consequences of the existence of this map in addition it is shown that this map defines an isomorphism from c onto the subring 0 g of g consisting of those virtual g sets x which have the same number of invariants for every two subgroups u v of g of the same order if and only if g is nilpotent finally a rather natural extension to profinite groups is indicated

kernel functions for case based planning case based planning can take advantage of former problem solving experiences by storing in a plan library previously generated plans that can be reused to solve similar planning problems in the future although comparative worst case complexity analyses of plan generation and reuse techniques reveal that it is not possible to achieve provable efficiency gain of reuse over generation we show that the case based planning approach can be an effective alternative to plan generation when similar reuse candidates can be chosen in this paper we describe an innovative case based planning system called oakplan which can efficiently retrieve planning cases from plan libraries containing more than ten thousand cases choose heuristically a suitable candidate and adapt it to provide a good quality solution plan which is similar to the one retrieved from the case library given a planning problem we encode it as a compact graph structure that we call planning encoding graph which gives us a detailed description of the topology of the planning problem by using this graph representation we examine an approximate retrieval procedure based on kernel functions that effectively match planning instances achieving extremely good performance in standard benchmark domains the experimental results point out the effect of the case base size and the importance of accurate matching functions for global system performance overall we show that oakplan is competitive with state of the art plan generation systems in terms of number of problems solved cpu time plan difference values and plan quality when cases similar to the current planning problem are available in the plan library

the burnside ring of profinite groups and the witt vector construction abstract the classical construction of witt vectors can be understood as a special case of a construction which can be defined relative to any given profinite group g in terms of its completed burnside ring lde g it specializes to the classical situation in case g is infinite cyclic

axiomatic theory of burnside rings i abstract in this paper we study abstract burnside rings of essentially finite categories under unique epi mono factorization property and the existence of coequalizers for some kind we prove the existence of a fundamental exact sequence for abr furthermore an abr can be embedded into a direct product of rational character rings

diophantine equations hilbert series and undecidable spaces summary in 1979 jan erik roos published a fascinating paper 22 making explicit connections between the poincare betti series of loop spaces or local rings and the hilbert series of finitely presented graded hopf algebras this paper develops a way to model diophantine equations within the category of graded hopf algebras combining this with roos work we obtain loop spaces and local rings whose series reflect the solution sets of arbitrary diophantine equations by matiyasevic s negative solution to hilbert s tenth problem 15 16 there is no algorithm for deciding in general whether or not a given diophantine equation has solutions for us one consequence is that no algorithm exists to decide whether a given finitely presented graded algebra is generic see 3 as to rings there is no finite procedure for evaluating whether an arbitrary ring s poincare series equals a given series even though by 10 the sequence of coefficients is recursive likewise given two finite simply connected cw complexes there is no guaranteed method to tell in general whether their loop spaces have the same rational homotopy type at the same time matiyasevic showed that there exist undecidable diophantine equations i e polynomial equations for which neither the existence nor the non existence of integer solutions may be proved in a given axiomatization of arithmetic we obtain undecidable local rings undecidable spaces and undecidable topological maps these concepts will be made precise in section 4 a few other consequences of the link between diophantine equations and hilbert series may be listed first a graded algebra resp loop space or local ring exists whose hilbert resp poincare series radius of convergence may be proved to be a transcendental number also the hilbert series can represent a transcendental function which solves no algebraic differential equation lastly we prove a stability theorem for diophantine solution sets of bounded complexity

witt burnside ring and burnside ring over a special λ ring abstract given an arbitrary group g we construct a covariant functor f g from the category of special rings to that of commutative rings with unity when g is a strongly complete profinite group it will be used to reconstruct the functor n r g which was first introduced in y t oh r analogue of the burnside ring of profinite groups and free lie algebras adv math 190 2005 1 46 to investigate the structure of witt burnside rings over special rings more precisely given a special ring r we show that n r g r is functorially isomorphic to f g r j g r for some ideal j g r of f g r the connection to the functor f g due to j j graham generalized witt vectors adv math 99 1993 248 263 will also be demonstrated

generalized burnside grothendieck ring functor and aperiodic ring functor associated with profinite groups abstract for every profinite group g we construct two covariant functors g and ap g which are equivalent to the functor w g introduced in a dress c siebeneicher the burnside ring of profinite groups and the witt vectors construction adv math 70 1988 87 132 we call g the generalized burnside grothendieck ring functor and ap g the aperiodic ring functor associated with g in case g is abelian we also construct another functor ap g from the category of commutative rings with identity to itself as a generalization of the functor ap introduced in k varadarajan k wehrhahn aperiodic rings necklace rings and witt vectors adv math 81 1990 1 29 finally it is shown that there exist q analogues of these functors i e w g g ap g and ap g in case g is the profinite completion of the multiplicative infinite cyclic group c