practical internal stability of n d discrete systems this paper deals with the problem of asymptotic stability for n d discrete systems in the practical sense that the system input and output signals are unbounded in at most one dimension a definition of practical internal stability is introduced and necessary and sufficient conditions are derived the obtained results show that practical internal stability is less restrictive and more relevant for practical applications than the conventional two dimensional 2 d internal stability

technical communique iterative learning control of linear discrete time multivariable systems the authors present an iterative learning control rule for linear discrete time multivariable systems the control rule assures a zero output error for the whole desired trajectory after only one learning iteration the robustness of the control rule is studied and two numerical examples are used to validate the new control rule

size distribution of clusters in irreversible kinetic aggregation the size distribution of clusters is derived analytically at a given time in a process of irreversible kinetic aggregation of diffusive clusters which starts from a collection of a great number of individual particles a general reduced shape is given in the scaling regime where gelation does not occur the time evolution is derived by use of smoluchowski s equation then a scaling reasoning allows one to find the exponents in real cases and a quantitative comparison is made with numerical simulations

optimal control of 2 d systems two dimensional 2 d optimal control theory that parallels one dimensional 1 d optimal control is developed a generalized performance measure suited to 2 d systems is introduced the canonical equations associated with this performance measure and a general nonlinear model are obtained the 2 d linear quadratic regulator problem is formulated and its canonical equations are derived for the roesser model an earlier result by t kaczorek and j klamka 1986 for the solution of the minimum energy problem with fixed final local state is rederived using this approach a new problem minimum energy with fixed final pass local states is formulated and solved and a numerical example is given

iterative learning control synthesis based on 2 d system theory an algorithm is presented for iterative learning of the control input for a linear discrete time multivariable system necessary and sufficient conditions are stated for convergence of the proposed algorithm the algorithm synthesis and analysis are based on two dimensional 2 d system theory a numerical example is given

on profinite groups with engel like conditions abstract let g be a profinite group in which for every element x g there exists a natural number q q x such that x q is engel we show that g is locally virtually nilpotent further let p be a prime and g a finitely generated profinite group in which for every k value x g there exists a natural p power q q x such that x q is engel we show that k g is locally virtually nilpotent

on the orders of primitive groups with restricted nonabelian composition factors we prove that given c0 there exists c such that the following holds if g is a primitive permutation group of degree n no composition factor of which is an alternating group of degree greater than c0 or a classical group of dimension greater than c0 then g nc in particular if the nonabelian composition factors of g have bounded order then g is polynomially bounded

commutators and commutator subgroups in profinite groups abstract let g be a profinite group we prove that the commutator subgroup g is finite by procyclic if and only if the set of all commutators of g is contained in a union of countably many procyclic subgroups

very strong approximation for certain algebraic varieties let f be a global field in this work we show that the brauer manin condition on adelic points for subvarieties of a torus t f over f cuts out exactly the rational points if either f is a function field or if f mathbb q and t f is split as an application we prove a conjecture of harari voloch over global function fields which states roughly speaking that on any rational hyperbolic curve the local integral points with the brauer manin condition are the global integral points finally we prove for tori over number fields a theorem of stoll on adelic points of zero dimensional subvarieties in abelian varieties

bases for primitive permutation groups and a conjecture of babai abstract a base of a permutation group g is a sequence b of points from the permutation domain such that only the identity of g fixes b pointwise we show that primitive permutation groups with no alternating composition factors of degree greater than d and no classical composition factors of rank greater than d have a base of size bounded above by a function of d this confirms a conjecture of babai

edge property from end points for scattering hurwitz polynomials the counterparts of recent results on growth conditions and convex directions for univariate polynomials and quasipolynomials are obtained for multivariate polynomials with respect to the scattering hurwitz property the technical devices employed to develop the machineries leading to the main result are the bijective relationship between multivariate scattering hurwitz polynomials and multivariate reactance functions and a certain representation of such functions

data driven control for relative degree systems via iterative learning iterative learning control ilc is a kind of effective data driven method that is developed based on online and or offline input output data the main purpose of this paper is to supply a unified 2 d analysis approach for both continuous time and discrete time ilc systems with relative degree it is shown that the 2 d roesser system framework can be established for general ilc systems regardless of relative degree under which convergence conditions can be provided to guarantee both asymptotic stability and monotonic convergence of the ilc processes in particular conditions for the monotonic convergence of ilc can be given in terms of linear matrix inequalities and formulas for the updating law can be derived simultaneously simulation results are presented to illustrate the effectiveness of ilc determined through the 2 d design approach in dealing with the higher order relative degree problem of ilc systems as well as the robustness of such ilc against uncertainties

word maps word maps with constants and representation varieties of one relator groups abstract we consider word maps and word maps with constants on a simple algebraic group g we present results on the images of such maps in particular we prove a theorem on the dominance of general word maps with constants which can be viewed as an analogue of a well known theorem of borel on the dominance of genuine word maps besides we establish a relationship between the existence of unipotents in the image of the map induced by w f m and the structure of the representation variety r w g of the group w f m w

pro finite p adic lie algebras abstract let p be a prime number a finite nilpotent lie ring of characteristic a power of p is called finite p a pro p lie ring is an inverse limit of finite p lie rings pro p lie rings play a role in lie theory similar to that played by pro p groups in group theory every pro p lie ring admits the structure of a lie algebra over the p adic integers furthermore every p adic lie algebra of finite rank as a p adic module has an open pro p subalgebra we make a detailed study of pro p lie rings in terms of various properties including their topology prufer rank subring growth and p adic module structure in particular we prove the equivalence of the following conditions for a finitely generated pro p lie ring l l has finite prufer rank l is isomorphic to a closed subring of gl v for some p adic module v of finite rank and for sufficiently large n the lie f p subalgebra w n e 12 t e 22 gl 2 f p t t n is not an open section of l by reducing to the pro p lie ring case we also prove that all engelian pro finite lie rings are locally nilpotent this is a lie theoretic analogue of zelmanov s theorem which states that every periodic pro p group is locally finite

position space renormalisation group for aggregation fractals a simple position space renormalisation group scheme is used to find the fractal dimension of sutherland type aggregation clusters grown in low dimensional space clusters which move along paths of fractal dimension dw 0 1 2 are considered and a systematic variation in cluster fractal dimension matching that of recent computer simulation is found

decentralized iterative learning control for a class of large scale interconnected dynamical systems the problem of decentralized iterative learning control for a class of large scale interconnected dynamical systems is considered in this paper it is assumed that the considered large scale dynamical systems are linear time varying and the interconnections between each subsystem are unknown for such a class of uncertain large scale interconnected dynamical systems a method is presented whereby a class of decentralized local iterative learning control schemes is constructed it is also shown that under some given conditions the constructed decentralized local iterative learning controllers can guarantee the asymptotic convergence of the local output error between the given desired local output and the actual local output of each subsystem through the iterative learning process finally as a numerical example the system coupled by two inverted pendulums is given to illustrate the application of the proposed decentralized iterative learning control schemes

a monte carlo renormalisation group for aggregation processes a general monte carlo renormalisation group transformation for the static as well as the kinetic properties of aggregates is formulated and applied to different growth models particle aggregation dla cluster aggregation cici and invasion percolation ip the leading critical exponent is determined numerically for dla the first correction to scaling exponent has been estimated as well

fuchsian groups finite simple groups and representation varieties let be a fuchsian group of genus at least 2 at least 3 if is non oriented we study the spaces of homomorphisms from to finite simple groups g and derive a number of applications concerning random generation and representation varieties precise asymptotic estimates for hom g are given implying in particular that as the rank of g tends to infinity this is of the form g 1 o 1 where is the measure of we then prove that a randomly chosen homomorphism from to g is surjective with probability tending to 1 as g combining our results with lang weil estimates from algebraic geometry we obtain the dimensions of the representation varieties text hom gamma bar g where bar g is gl n k or a simple algebraic group over k an algebraically closed field of arbitrary characteristic a key ingredient of our approach is character theory involving the study of the zeta function g s 1 s where the sum is over all irreducible complex characters of g

two dimensional model and algorithm analysis for a class of iterative learning control systems the iterative learning control system ilcs has attracted considerable research interest in recent years however the theoretical results on some fundamental properties of the ilcs have not yet been established one of the common drawbacks shared by current approaches is lack of a suitable mathematical model which can clearly describe both the dynamics of the control system itself and the behaviour of the learning process as well in this paper a class of iterative learning control systems is analysed from the point of view of two dimensional 2 d system theory the 2 d model for a class of ilcs is established in the form of roessor s model based on which a general type of learning controller is proposed analysis of the 2 d error equation shows that the 2 d asymptotic stability of the 2 d model guarantees the learning convergence of an ilcs design criteria for a learning controller is also suggested the learning gain matrices are obtained from a recursive 2 d identifier using the input and output

on self similarity of p adic analytic pro p groups of small dimension abstract given a torsion free p adic analytic pro p group g with dim g p we show that the self similar actions of g on regular rooted trees can be studied through the virtual endomorphisms of the associated z p lie lattice we explicitly classify 3 dimensional unsolvable z p lie lattices for p odd and study their virtual endomorphisms together with lazard s correspondence this allows us to classify 3 dimensional unsolvable torsion free p adic analytic pro p groups for p 5 and to determine which of them admit a faithful self similar action on a p ary tree in particular we show that no open subgroup of s l 1 1 p admits such an action on the other hand we prove that all the open subgroups of s l 2 z p admit faithful self similar actions on regular rooted trees

comments on a robust mimo terminal sliding mode control scheme for rigid robotic manipulators the authors state that there are two errors in the above paper zhihong et al ibid vol 39 no 12 p 2464 9 1994 the purpose of this note is to give correct versions of remark 2 1 and remark 3 3

on a discrete time stochastic learning control algorithm the learning gain for a selected learning algorithm is derived based on minimizing the trace of the input error covariance matrix for linear time varying systems it is shown that if the product of the input output coupling matrices is a full column rank then the input error covariance matrix converges uniformly to zero in the presence of uncorrelated random disturbances however the state error covariance matrix converges uniformly to zero in presence of measurement noise moreover it is shown that if a certain condition is met then the knowledge of the state coupling matrix is not needed to apply the proposed stochastic algorithm the proposed algorithm is shown to suppress a class of nonlinear and repetitive state disturbance the application of this algorithm to a class of nonlinear systems is also considered a numerical example is included to illustrate the performance of the algorithm

on the structure of just infinite profinite groups abstract a profinite group g is just infinite if every closed normal subgroup of g is of finite index we prove that an infinite profinite group is just infinite if and only if for every open subgroup h of g there are only finitely many open normal subgroups of g not contained in h this extends a result recently established by barnea gavioli jaikin zapirain monti and scoppola 2009 in 1 who proved the same characterisation in the case of pro p groups we also use this result to establish a number of features of the general structure of profinite groups with regard to the just infinite property

on verbal subgroups of adelic groups abstract let g be a closed subgroup of p gl n z p if g is finitely generated as a profinite group and has no infinite virtually abelian quotients then every verbal subgroup in g is closed

abstract commensurators of profinite groups in this paper we initiate a systematic study of the abstract com mensurators of profinite groups the abstract commensurator of a profinite group g is a group comm g which depends only on the commensurability class of g we study various properties of comm g in particular we find two natural ways to turn it into a topological group we also use comm g to study topological groups which contain g as an open subgroup all such groups are totally disconnected and locally compact for instance we con struct a topologically simple group which contains the pro 2 completion of the grigorchuk group as an open subgroup on the other hand we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups several celebrated rigidity theorems like pink s analogue of mostow s strong rigidity theorem for simple algebraic groups defined over local fields and the neukirch uchida theorem can be re formulated as structure theorems for the commensurators of certain profinite groups

asymptotic tracking for 2 d and delay differential systems abstract a method for asymptotic tracking in 2 d systems is presented and is applied to discrete 2 d systems and delay differential systems the method consists of solving two linear equations in 2 d polynomials solvability conditions are derived and the questions of stability causality and properness are discussed

repetitive control system a new type servo system for periodic exogenous signals a control scheme called repetitive control is proposed in which the controlled variables follow periodic reference commands a high accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period l into the closed loop system sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small gain theorem and the stability theorem for time lag systems synthesis algorithms are presented by both the state space approach and the factorization approach in the former approach the technique of the kalman filter and perfect regulation is utilized while coprime factorization over the matrix ring of proper stable rational functions and the solution of the hankel norm approximation are used in the latter one

the design of practically stable n d feedback systems this paper deals with by using the matrix fractional description mfd approach the problem of feedback practical stabilization of nd multidimensional discrete systems whose input and output signals are unbounded in at most one dimension a constructive algorithm is first presented for solving the bezout equation over the ring of practically stable rational functions then a necessary and sufficient condition for an nd system to be practically stabilizable is derived and the parametrization of all nd practically stabilizing compensators is given these results make it clear that the nd practical stabilization problem can be essentially solved by using 1d approaches

diameters of finite simple groups sharp bounds and applications let g be a finite simple group and let s be a normal subset of g we determine the diameter of the cayley graph r g s associated with g and s up to a multiplicative constant many applications follow for example we deduce that there is a constant c such that every element of g is a product of c involutions and we generalize this to elements of arbitrary order we also show that for any word w w xl xd there is a constant c c w such that for any simple group g on which w does not vanish every element of g is a product of c values of w from this we deduce that every verbal subgroup of a semisimple profinite group is closed other applications concern covering numbers expanders and random walks on finite simple groups

bounded normal generation and invariant automatic continuity abstract we study the question of how quickly products of a fixed conjugacy class in the projective unitary group of a ii1 factor von neumann algebra cover the entire group our result is that the number of factors that are needed is essentially as small as permitted by the 1 norm in analogy to results of liebeck and shalev for non abelian finite simple groups as an application of the techniques we prove that every homomorphism from the projective unitary group of a finite factor to a polish sin group is continuous a result which is even new for pu n moreover we show that the projective unitary group of a ii1 factor carries a unique polish group topology

word maps conjugacy classes and a noncommutative waring type theorem let w w x xd 1 be a nontrivial group word we show that if g is a sufficiently large finite simple group then every element g e g can be expressed as a product of three values of w in g this improves many known results for powers commutators as well as a theorem on general words obtained in 19 the proof relies on probabilistic ideas algebraic geometry and character theory our methods which apply the zeta function g s x eirrg xo s giye rise to various additional results of independent interest including applications to conjectures of ore and thompson

on bounded conciseness of words in residually finite groups abstract we conjecture that any group word that is concise in the class of residually finite groups is in fact boundedly concise we show that this is true for the word w q where w is a multilinear commutator word and q is a prime power

stability of nonlinear parameter varying digital 2 d systems a general state space model for nonlinear parameter varying digital two dimensional 2 d systems is proposed sufficient conditions for system stability are given presented theorem can be considered as a generalization of the lyapunov stability theorem for one dimensional nonlinear systems given results can be useful in stability analysis for systems described by 2 d models

robust discrete time iterative learning control for nonlinear systems with varying initial state shifts this note is concerned with the robust discrete time iterative learning control ilc design for nonlinear systems with varying initial state shifts a two gain ilc law is considered using a 2d analysis approach sufficient conditions are derived to guarantee both convergence of the learning process for fixed initial condition and boundedness of the tracking error for variable initial condition it is shown that the error data with anticipation in time can well handle the varying initial state shifts in discrete time ilc

bounds on the number and sizes of conjugacy classes in finite chevalley groups with applications to derangements we present explicit upper bounds for the number and size of conjugacy classes in finite chevalley groups and their variations these results have been used by many authors to study zeta functions associated to representations of finite simple groups random walks on chevalley groups the final solution to the ore conjecture about commutators in finite simple groups and other similar problems in this paper we solve a strong version of the boston shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups the remaining cases are settled in two other papers of the authors and applications are given in a third

analysis of iterative learning control for a class of nonlinear discrete time systems abstract an output feedback iterative learning control scheme is proposed for a class of nonlinear discrete time systems in the presence of direct transmission from system inputs to outputs the convergence condition of the iterative learning control scheme is analyzed the effectiveness of suggested discrete time iterative learning control algorithm is demonstrated with a single link robotic manipulator as the illustrative example 1997 elsevier science ltd

closed subgroups of profinite groups theorem 1 asserts that emph in a finitely generated prosoluble group every subgroup of finite index is open this generalises an old result of serre about pro p groups it follows by a standard argument from theorem 2 emph in a d generator finite soluble group every element of the derived group is equal to a product of 72d 2 46d commutators this result about finite soluble groups is proved by induction on the order of the group and is elementary though rather intricate the essence of the proof lies in reducing the problem to one about the number of solutions of quadratic equations over a finite field corollaries include the following emph let gamma be a finitely generated prosoluble group then each term of the lower central series of gamma and each power subgroup gamma n is closed 1991 mathematics subject classification 20e18 20d10

a discrete time learning control algorithm for a class of linear time invariant systems a discretized version of the d type learning control algorithm is presented for a mimo linear discrete time system a necessary and sufficient condition for uniform convergence of the proposed learning algorithm is presented then we prove that the same condition is sufficient for the global robustness of the proposed learning algorithm to state disturbances measurement noise at the output and reinitialization error are present at each iteration a numerical example is given to illustrate the results

transparency effects in cluster cluster aggregation with linear trajectories two hierarchical models of cluster cluster aggregation whose aggregates follow random linear trajectories with model i or without model ii random impact parameter are investigated numerically the difference of the fractal dimension d of the clusters as a function of space dimension d is attributed to transparency effects which are less important in model ii a simple argument is given which suggests that in contrast with model i model ii should not have an upper critical dimension

growth in finite simple groups of lie type we prove that if l is a finite simple group of lie type and a a set of generators of l then either a grows i e vert a 3 vert vert a vert 1 varepsilon where varepsilon depends only on the lie rank of l or a 3 l this implies that for simple groups of lie type of bounded rank a well known conjecture of babai holds i e the diameter of any cayley graph is polylogarithmic we also obtain new families of expanders a generalization of our proof yields the following let a be a finite subset of sl n mathbb f mathbb f an arbitrary field satisfying big vert a 3 big vert le mathcal k vert a vert then a can be covered by mathcal k m i e polynomially many cosets of a virtually soluble subgroup of sl n mathbb f which is normalized by a where m depends on n see more at http www ams org journals jams 0000 000 00 s0894 0347 2014 00821 3 home html sthash lp65mzge dpuf

on bounded conciseness of engel like words in residually finite groups abstract given a group word w and a group g the verbal subgroup w g is the one generated by all w values in g the word w is said to be boundedly concise if for each positive integer m there exists a number depending only on m and w bounding the order of w g whenever the set of w values in a group g has size at most m in the present article we show that various generalizations of the engel word are boundedly concise in residually finite groups

design of practically stable n dimensional feedback systems a state space approach this paper investigates by using the state space approach some basic properties and state feedback compensation problem of n dimensional discrete systems in the practical sense that the input and output signals of the systems are unbounded in at most one dimension notions of practical controllability practical observability practical stabilizability by state feedback and practical detectability are introduced and corresponding necessary and sufficient conditions are derived moreover a connection between the state space representation and doubly coprime matrix fractional description on the ring of practically stable rational functions is shown the obtained results clarify that all the n dimensional problems considered in the practical sense are in fact equivalent to the corresponding problems of n one dimensional systems hence can be essentially solved by using one dimensional methods

congruence topologies on the mapping class group abstract let s be the pure mapping class group of a connected orientable surface s of negative euler characteristic for c a class of finite groups let 1 s c be the pro c completion of the fundamental group of s the c congruence completion s c of s is the profinite completion induced by the embedding s out 1 s c in this paper we begin a systematic study of such completions for different c we show that the combinatorial structure of the profinite groups s c closely resemble that of s a fundamental question is how c congruence completions compare with pro c completions even though in general e g for c the class of finite solvable groups s c is not even virtually a pro c group we show that for z 2 c g s 2 and s open there is a natural epimorphism from the c congruence completion s 2 c of the abelian level of order 2 to its pro c completion s 2 c in particular this is an isomorphism for the class of finite groups and for the class of 2 groups moreover in these two cases the result also holds for a closed surface

on finitely generated profinite groups ii products in quasisimple groups we prove two results 1 there is an absolute constant d such that for any finite quasisimple group s given 2d arbitrary automorphisms of s every element of s is equal to a product of d twisted commutators defined by the given automorphisms 2 given a natural number q there exist c c q and m m q such that if s is a finite quasisimple group with s z s c j j 1 m are any automorphisms of s and qj j 1 m are any divisors of q then there exist inner automorphisms j of s such that s m s j j q j these results which rely on the classification of finite simple groups are needed to complete the proofs of the main theorems of part i

hierarchical model for irreversible kinetic cluster formation a model is proposed to describe the growth of clusters by a mechanism of irreversible clustering of clusters the fractal exponent is extracted by means of numerical simulations on small systems both directly and in the form of a renormalisation group analysis the results are in excellent agreement with previous monte carlo simulations the numerical precision is better due to the simpler hierarchical formulation of the model in the present form the model looks like the diffusion limited aggregation model which for the sake of comparison is treated by the same renormalisation group method

generators and commutators in finite groups abstract quotients of compact groups the first part of the paper establishes results about products of commutators in a d generator finite group g for example if h g g1 gr then every element of the subgroup h g is a product of f r factors of the form h 1 g 1 h 1 prime g 1 1 ldots lbrack h r g r h r prime g r 1 with h 1 h 1 prime ldots allowbreak h r h r prime in h under certain conditions on h a similar conclusion holds with the significantly weaker hypothesis that g h g1 gr where f r is replaced by f1 d r the results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups and in more general compact groups results include the characterization of topologically finitely generated compact groups which have a countably infinite image and of those which have a virtually dense normal subgroup of infinite index as a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient

stochastic p type d type iterative learning control algorithms this paper presents stochastic algorithms that compute optimal and sub optimal learning gains for a p type iterative learning control algorithm ilc for a class of discrete time varying linear systems the optimal algorithm is based on minimizing the trace of the input error covariance matrix the state disturbance reinitialization errors and measurement errors are considered to be zero mean white processes it is shown that if the product of the input output coupling matrices c t 1 b t is full column rank then the input error covariance matrix converges to zero in presence of uncorrelated disturbances another sub optimal p type algorithm which does not require the knowledge of the state matrix is also presented it is shown that the convergence of the input error covariance matrices corresponding to the optimal and sub optimal p type and d type algorithms are equivalent and all converge to zero at a rate inversely proportional to the number of learning iterations a transient response pe

commutator maps measure preservation and t systems let g be a finite simple group we show that the commutator map g g g is almost equidistributed as g 1 this somewhat surprising result has many applications it shows that a for a subset x g we have 1 x g 2 x g o 1 namely is almost measure preserving from this we deduce that almost all elements g 2 g can be expressed as commutators g x y where x y generate g this enables us to solve some open problems regarding t systems and the product replacement algorithm pra graph we show that the number of t systems in g with two generators tends to infinity as g 1 this settles a conjecture of guralnick and pak a similar result follows for the number of connected components of the pra graph of g with two generators some of our results apply for more general finite groups and more general word maps our methods are based on representation theory combining classical character theory with recent results on character degrees and values in finite simple groups in particular the so called witten zeta function g s p 2irr g 1 s plays a key role in the proofs

word maps and waring type problems waring s problem asks whether every natural number is a sum of g k fcth powers where g is a suitable function this was solved affirmatively by hubert in 1909 optimizing g k has been a central problem in additive number theory ever since see na for more detail and background recently there has been considerable interest in group theoretic analogues of this phenomenon where the aim is to present group elements as short products of certain special elements these special elements can be powers or commutators or values of a general word w or elements of a special conjugacy class in the group the motivation for this is sometimes topological let g be a topologically finitely generated pro p group using expressions of elements of the commutator subgroup g1 as bounded products of commutators serre showed that g is closed and deduced that every finite index subgroup of g is open a recent deep result of nikolov and segal nsi ns2 shows that this holds for every finitely generated profinite group again the core of the proof is presenting group elements as short products of values of certain words

on finitely generated profinite groups i strong completeness and uniform bounds we prove that in every finitely generated profinite group every subgroup of finite index is open this implies that the topology on such groups is determined by the algebraic structure this is deduced from the main result about finite groups let w be a locally finite group word and d n then there exists f f w d such that in every d generator finite group g every element of the verbal subgroup w g is equal to a product of f w values an analogous theorem is proved for commutators this implies that in every finitely generated profinite group each term of the lower central series is closed the proofs rely on some properties of the finite simple groups to be established in part ii

on the lattice of normal subgroups in ultraproducts of compact simple groups we prove that the lattice of normal subgroups of ultraproducts of compact simple non abelian groups is distributive in the case of ultraproducts of finite simple groups or compact connected simple lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion 2010 mathematics subject classification 03c20 20d30 primary 20f69 20d06 22e46 secondary

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