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

biset transformations of tambara functors abstract if we are given an h g biset u for finite groups g and h then any mackey functor on g can be transformed by u into a mackey functor on h in this article we show that the biset transformation is also applicable to tambara functors when u is right free and in fact forms a functor between the category of tambara functors on g and h this biset transformation functor is compatible with some algebraic operations on tambara functors such as ideal quotients or fractions in the latter part we also construct the left adjoint of the biset transformation

the spectrum of the burnside tambara functor on a finite cyclic p group abstract for a finite group g a tambara functor on g is regarded as a g bivariant analog of a commutative ring in this analogy previously we have defined an ideal of a tambara functor in this article we calculate the prime spectrum of the burnside tambara functor when g is a finite cyclic p group for a prime integer p

tambarization of a mackey functor and its application to the witt burnside construction abstract for an arbitrary group g a semi mackey functor is a pair of covariant and contravariant functors from the category of g sets and is regarded as a g bivariant analog of a commutative semi group in this view a g bivariant analog of a semi ring should be a semi tambara functor a tambara functor is firstly defined by tambara which he called a tnr functor when g is finite as shown by brun a tambara functor plays a natural role in the witt burnside construction it will be a natural question if there exist sufficiently many examples of tambara functors compared to the wide range of mackey functors in the first part of this article we give a general construction of a tambara functor from any mackey functor on an arbitrary group g in fact we construct a functor from the category of semi mackey functors to the category of tambara functors this functor gives a left adjoint to the forgetful functor and can be regarded as a g bivariant analog of the monoid ring functor in the latter part when g is finite we investigate relations with other mackey functorial constructions crossed burnside ring elliott s ring of g strings jacobson s f burnside ring all these lead to the study of the witt burnside construction

diffusion in lorentz lattice gas cellular automata the honeycomb and quasi lattices compared with the square and triangular lattices we study numerically the nature of the diffusion process on a honeycomb and a quasi lattice where a point particle moving along the bonds of the lattice scatters from randomly placed scatterers on the lattice sites according to strictly deterministic rules for the honeycomb lattice fully occupied by fixed rotators two symmetric isolated critical points appear to be present with the same hyperscaling relation as for the square and the triangular lattices no such points appear to exist for the quasi lattice a comprehensive comparison is made with the behavior on the previously studied square and triangular lattices a great variety of diffusive behavior is found ranging from propagation superdiffusion normal quasi normal and anomalous to absence of diffusion the influence of the scattering rules as well as of the lattice structure on the diffusive behavior of a point particle moving on the all lattices studied so far is summarized

on the second solution to a critical growth robin problem abstract we investigate the existence of the second mountain pass solution to a robin problem where the equation is at critical growth and depends on a positive parameter more precisely we determine existence and nonexistence regions for this type of solutions depending both on and on the parameter in the boundary conditions

ideals of tambara functors abstract for a finite group g a tambara functor on g is regarded as a g bivariant analog of a commutative ring in this article we consider a g bivariant analog of the ideal theory for tambara functors

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

crossed burnside rings and bouc s construction of green functors abstract we use a formula for primitive idempotents of the crossed burnside ring given by f oda and t yoshida and a result from the theory of green functors obtained by s bouc to prove an induction theorem for the drinfel d double for a finite group over the complex field we obtain artin s induction theorem for group algebras as a corollary of the theorem

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

lattice lorentz gas in cellular automaton fluids where binary collision laws admit reflections the stosszahl ansatz and the boltzmann equation are no longer valid at low densities this is proven for a class of lorentz gas models defined on cubic lattices for which the diffusion coefficient is calculated from a self consistent ring kinetic equation or effective medium theory the calculated results are in excellent agreement with results from molecular dynamics simulations over the whole density interval

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

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

crossed burnside rings i the fundamental theorem abstract let g be a finite group and s a finite g monoid a crossed g set over s is a finite g set equipped with a g map into s called a weight function a crossed burnside ring x g s is the grothendieck ring of the category of crossed g sets with respect to disjoint unions and tensor products in this paper we prove the fundamental theorem of crossed burnside rings and an idempotent formula at characteristic 0

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

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

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

existence and nonexistence of least energy solutions of the neumann problem for a semilinear elliptic equation with critical sobolev exponent and a critical lower order perturbation let be a smooth bounded domain in rn with n 5 a 0 0 and 2 2nn 2 we show that the exponent q 2 n 1 n 2 plays a critical role regarding the existence of least energy or ground state solutions of the neumann problem u au u2 1 uq 1in u 0in u 0on namely we prove that when q 2 n 1 n 2 there exists an 0 0 such that the problem has a least energy solution if 0

novel phenomena in lorentz lattice gases abstract in lorentz lattice gases a point particle moves along the bonds of a lattice whose lattice sites are occupied by randomly placed scatterers which scatter the particle according to certain a priori given scattering rules we consider here mainly the simplest case of a square lattice fully or partly covered with two kinds of scatterers which scatter the particle to its right or its left respectively either according to probabilistic or strictly deterministic scattering rules although the global probability of the particle to be scattered to its right or left is equal for the two scattering rules the local probability is very different leading to a completely different diffusive behavior for the two cases while the probabilistic scattering rules lead to gaussian diffusion the deterministic rules lead either to anomalous or no diffusion depending on the concentrations of the scatterers on the lattice in the first case the mean square displacement is still proportional to the time so that a diffusion coefficient can be defined but the diffusion is non gaussian while in the second case the mean square displacement is bounded and all particles are trapped in a finite time in both cases this behavior is due to the existence of closed periodic orbits which in the first case can be of infinite extent extended closed orbits while in the second case they can only be of finite extent the lengths of the extended closed orbits show for a lattice fully occupied equally by right and left scatterers a scaling behavior identical to that of percolation clusters at the bond percolation threshold of a related bond percolation problem for special concentrations on a not fully occupied lattice the same scaling behavior is found leading to two critical lines but no known related percolation like problem exists in this case this critical behavior of the closed orbits in anomalous diffusion appears to obtain for all two dimensional lattices studied so far a number of extensions of the above results are indicated

the crossed burnside rings iii the dress construction for a tambara functor abstract we discuss the dress construction for a tambara functor associated to a finite commutative monoid with action of a finite group

partial tambara structure on the burnside biset functor induced from a derivator like system of adjoint triplets abstract in the previous article a mackey functor theoretic interpretation of biset functors we have constructed the 2 category s of finite sets with variable finite group actions in which bicoproducts and bipullbacks exist as shown in it biset functors can be regarded as a special class of mackey functors on s in this article we equip s with a system of adjoint triplets which satisfies properties analogous to a derivator this system encodes the six operations for finite groups as a corollary we show that the associated burnside rings satisfy analogous properties to a tambara functor in the context of biset functor theory

a kinetic theorist s look at lattice gas cellular automata abstract the diffusion behavior of a number of lorentz lattice gases is studied in its dependence on collision rules and its similarity to corresponding lorentz gases that are continuous in space but have the same discrete velocity space the difference in behavior resulting from probabilistic or deterministic collision rules is discussed

crossed burnside rings ii the dress construction of a green functor abstract we present some results about the dress construction of green functor associated to a g monoid in particular we see that the crossed burnside rings the representation rings of quantum double and hochschild cohomology rings are constructed by the dress construction of some green functors

induction formulae for mackey functors with applications to representations of the twisted quantum double of a finite group abstract in the theory of canonical induction formulae for mackey functors boltje 4 demonstrated that the plus constructions together with the mark morphism are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups in this paper we present a short exact sequence for the plus constructions derived from cauchy frobenius lemma and apply it to the proof of boltje s integrality result for canonical induction formulae the methods appearing in boltje s theory combined with the dress construction for mackey functors are applicable to induction theorems on representations of the twisted quantum double of a finite group as a sequel to such a research we describe canonical versions of two induction theorems whose origins are artin s induction theorem and brauer s induction theorem on c characters of a finite group

further study on the effect of boundary conditions abstract this paper concerns the least energy solutions to a semilinear elliptic equation the role of geometry of the boundary is studied when a parameter in the boundary condition tends to a critical index as well as to infinity

higher topological cyclic homology and the segal conjecture for tori abstract we investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory higher topological cyclic homology is constructed from the fixed points of a version of topological hochschild homology based on the n dimensional torus and we propose it as a computationally tractable cousin of n fold iterated algebraic k theory the fixed points of toral topological hochschild homology are related to one another by restriction and frobenius operators we introduce two additional families of operators on fixed points the verschiebung indexed on self isogenies of the n torus and the differentials indexed on n vectors we give a detailed analysis of the relations among the restriction frobenius verschiebung and differentials producing a higher analog of the structure hesselholt and madsen described for 1 dimensional topological cyclic homology we calculate two important pieces of higher topological cyclic homology namely topological restriction homology and topological frobenius homology for the sphere spectrum the latter computation allows us to establish the segal conjecture for the torus which is to say to completely compute the cohomotopy type of the classifying space of the torus

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

generalized burnside rings and group cohomology we define the cohomological burnside ring b n g m of a finite group g with coefficients in a zgmodule m as the grothendieck ring of the isomorphism classes of pairs x u where x is a g set and u is a cohomology class in a cohomology group h n x g m the cohomology groups h x g m are defined in such a way that h x g m i h hi m when x is the disjoint union of transitive g sets g hi i fa is an abelian group with trivial action then b 1 g a is the same as the monomial burnside ring over a and when m is taken as a g monoid then b 0 g m is equal to the crossed burnside ring b c g m we discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological burnside rings we also give an interpretation of b 2 g m in terms of twisted

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

recurrence properties of lorentz lattice gas cellular automata recurrence properties of a point particle moving on a regular lattice randomly occupied with scatterers are studied for strictly deterministic nondeterministic and purely random scattering rules

generalized witt vectors abstract the witt burnside ring as contrived by a dress and c siebeneicher adv in math 70 1988 87 132 is a generalisation of the ring of witt vectors in this paper we characterise this ring algorithmically generalising the carryless witt vectors described by n metropolis and g c rota adv in math 50 1983 95 125 this approach has the merit of simplifying dress main theorem which asserts the existence of this ring and a few properties that characterise it it also makes the definition of rota s carryless witt vectors more natural

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

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

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

constructing witt burnside rings abstract the witt burnside ring of a profinite group g over a commutative ring a generalizes both the burnside ring of virtual g sets and the rings of universal and p typical witt vectors over a the witt burnside ring of g over the monoid ring z m where m is a commutative monoid is proved isomorphic to the grothendieck ring of a category whose objects are almost finite g sets equipped with a map to m that is constant on g orbits in particular if a is a commutative ring and a denotes the set a as a monoid under multiplication then the witt burnside ring of g over z a is isomorphic to graham s ring of virtual g strings with coefficients in a this result forms the basis for a new construction of witt burnside rings and provides an important missing link between the constructions of dress and siebeneicher adv in math 70 1988 87 132 and graham adv in math 99 1993 248 263 with this approach the usual truncation frobenius verschiebung and teichmuller maps readily generalize to maps between witt burnside rings

probability of first return on cayley trees a lorentz lattice gas with a fraction of the scatterers being pure backscatterers the remaining being stochastic right and left rotators is considered the problem at hand is the evaluation of the probability x that the moving particle returns to its original site in the limit where the density of sites occupied by scatterers is small and the lattice becomes a cayley tree in the special case of deterministic collision rules gunn ortuno model on a cayley tree an exact cubic equation for the laplace transform of the distribution function of first return times is derived and its asymptotic form near the threshold value beyond which x 1 is obtained in the general case of stochastic models a mean field approximation is proposed for x and the mean return time the approximation reduces to the exact result in the case of the deterministic model as well as in the stochastic model without backscatterers comparison with monte carlo simulations shows a reasonable agreement in the dependence of x and on the relevance of the results to the development of approximate analytic expressions for the diffusion coefficient is discussed

diffusion and propagation in triangular lorentz lattice gas cellular automata the diffusion process of point particles moving on regular triangular and random lattices randomly occupied with stationary scatterers a lorentz lattice gas cellular automaton is studied for strictly deterministic scattering rules as a function of the concentration of the scatterers in addition to the normal and various kinds of retarded diffusion found before on the regular square lattice straight line propagation through the scatterers is observed

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

global structure of solutions for equations of brezis nirenberg type on the unit ball the brezis nirenberg equation and the scalar field equation on the three dimensional unit ball are studied under the robin condition we show the existence and uniqueness of radial solutions in a unified way in particular it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value

topological dynamics of flipping lorentz lattice gas models we study the topological dynamics of the flipping mirror model of ruijgrok and cohen with one or an infinite number of particles in particular we prove the topological transitivity and topological mixing up to a natural first integral for the one particle model

the representation ring of the quantum double of a finite group abstract we present some results about the representation ring of the quantum double of a finite group over fields of arbitrary characteristic we give a direct sum decomposition of this representation ring into ideals involving green rings of subgroups given characters of such green rings we construct characters of the representation ring of the quantum double and show that all characters arise in this fashion we prove that the characters thereby obtained from brauer characters separate modules of the quantum double up to composition factors

a generalization of the dress construction for a tambara functor and its relation to polynomial tambara functors abstract for a finite group g semi mackey functors and semi tambara functors are regarded as g bivariant analogs of semi groups and semi rings respectively in fact if g is trivial they agree with the ordinary semi groups and semi rings and many naive algebraic properties concerning rings and groups have been extended to these g bivariant analogous notions in this article we investigate a g bivariant analog of the semi group rings with coefficients just as a coefficient ring r and a monoid q yield the semi group ring r q our construction enables us to make a tambara functor t m out of a semi mackey functor m and a coefficient tambara functor t this construction is a composant of the tambarization and the dress construction as expected this construction is the one uniquely determined by the righteous adjoint property besides in analogy with the trivial group case if m is a mackey functor then t m is equipped with a natural hopf structure moreover as an application of the above construction we also obtain some g bivariant analogs of the polynomial rings

the hochschild cohomology ring of a group algebra there is a standard additive decomposition of the hochschild cohomology ring of the group algebra of a finite group g as the direct sum of the cohomology rings of the centralizers of representatives of the conjugacy classes of g a special case of our main result describes the cup product in terms of this decomposition as applications we determine presentations for the hochschild cohomology rings of 1 the mod 3 group algebra of the symmetric group s 3 2 the mod 2 group algebra of the alternating group a 4 and 3 the mod 2 group algebras of the dihedral 2 groups

the witt vectors for green functors abstract we define twisted hochschild homology for green functors this construction is the algebraic analogue of the relative topological hochschild homology thh c n and under flatness conditions it describes the e 2 term of the kunneth spectral sequence for relative thh applied to ordinary rings we obtain new algebraic invariants extending hesselholt s construction of the witt vectors of noncommutative rings we interpret our construction as providing witt vectors for green functors

on the fractions of semi mackey and tambara functors for a finite group g a semi mackey resp tambara functor is regarded as a g bivariant analog of a commutative monoid resp ring as such some naive algebraic constructions are generalized to this g bivariant setting in this article as a g bivariant analog of the fraction of a ring we consider fraction of a tambara and a semi mackey functor by a multiplicative semi mackey subfunctor

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

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