complexity and multiple complexes in this chapter we introduce the notion of the complexity of a module and explore some related ideas complexity was first defined by jon alperin in the late 1970 s and it helped to motivate much of the development of the homological properties of modules in fact the complexity of a finitely generated kg module m is a rather crude invariant in that it only measures the polynomial rate of growth of a minimal projective resolution of the module the original theorem of alperin and evens which extended quillen s dimension theorem to modules was proved only in terms of complexity later it was extended to support varieties as well the theorem computes the complexity of a kg module m as the maximum of the complexities of the restrictions of m to the elementary abelian subgroups of the group g

order restricted score parameters in association models for contingency tables abstract the row effects and column effects models for two way contingency tables have parameters for the row and column categories pertaining to the association between the variables for classifications having ordered categories it is often reasonable to assume that the association parameters have a corresponding ordering this article proposes order restricted estimates of the association parameters in these models the maximum likelihood solution can be determined by the solution of a simple isotonic regression of some of the model sufficient statistics the models are primarily log linear in form and can be expressed in terms of odds ratios for 2 2 subtables consisting of adjacent rows and adjacent columns for the order restricted solution these local log odds ratios have uniform sign goodness of fit statistics for this solution are related to corresponding statistics for collapsed tables and to statistics for testing equality of sets of the parameters the row effects model discussed in this art

integrable systems and reductions of the self dual yang mills equations many integrable equations are known to be reductions of the self dual yang mills equations this article discusses some of the well known reductions including the standard soliton equations the classical painleve equations and integrable generalizations of the darboux halphen system and chazy equations the chazy equation first derived in 1909 is shown to correspond to the equations studied independently by ramanujan in 1916

the order restricted rc model for ordered contingency tables estimation and testing for fit the rc model has been proposed as a model for ordered contingency tables it involves parametric scores that are assigned to the rows and columns of the table so that these scores reflect the ordinality of the row and column categories efficient estimation of these parameters subject to order constraints remained an open problem mainly due to severe difficulties in computing these estimates and difficulties in deriving an appropriate asymptotic goodness of fit test a nonstandard yet very simple algorithm is derived which produces the desired order restricted maximum likelihood estimates with probability converging to 1 testing the order restricted rc model for fit is also discussed

fusion systems and constructing free actions on products of spheres we show that every rank two p group acts freely and smoothly on a product of two spheres this follows from a more general construction given a smooth action of a finite group g on a manifold m we construct a smooth free action on m times mathbb s n 1 times dots times mathbb s n k when the set of isotropy subgroups of the g action on m can be associated to a fusion system satisfying certain properties another consequence of this construction is that if g is an almost extra special p group of rank r then it acts freely and smoothly on a product of r spheres

analysis of contingency tables by correspondence models subject to order constraints abstract inferential correspondence analysis which has gained much attention in recent years is applied here to contingency tables with ordered categories to reflect such order the parameters of the underlying correspondence models are constrained to follow the order induced by the categories of the analyzed table a reparameterization of the correspondence model in terms of a latent variable model is presented this allows a simple and straightforward use of the em algorithm to obtain efficient order restricted estimates a goodness of fit test is also discussed and an example is analyzed a small monte carlo example is presented

lax representation and quadratic first integrals for a family of non autonomous second order differential equations abstract we consider a family of non autonomous second order differential equations which generalizes the lienard equation we explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic with the respect to the first derivative first integrals we show that these conditions are equivalent to the conditions for equations in the family under consideration to possess lax representations this provides a connection between the existence of a quadratic first integral and a lax representation for the studied dissipative differential equations which may be considered as an analogue to the theorem that connects lax integrability and arnold liouville integrability of hamiltonian systems we illustrate our results by several examples of dissipative equations including generalizations of the van der pol and duffing equations each of which have both a quadratic first integral and a lax representation

lectures on classical integrability we review some essential aspects of classically integrable systems the detailed outline of the sections consists of 1 introduction and motivation with historical remarks 2 liouville theorem and action angle variables with examples harmonic oscillator kepler problem 3 algebraic tools lax pairs monodromy and transfer matrices classical r matrices and exchange relations non ultralocal poisson brackets with examples non linear schrodinger model principal chiral field 4 features of classical r matrices belavin drinfeld theorems analyticity properties and lift of the classical structures to quantum groups 5 classical inverse scattering method to solve integrable differential equations soliton solutions spectral properties and the gel fand levitan marchenko equation with examples kdv equation sine gordon model prepared for the durham young researchers integrability school organised by the gatis network this is part of a collection of lecture notes

modular solutions to equations of generalized halphen type solutions to a class of differential systems that generalize the halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group these functions uniformize riemann surfaces of genus zero and have q series with integral coefficients rational maps between these functions are derived implying subgroup relations between their automorphism groups as well as symmetrization maps between the associated differential systems

group actions on spheres with rank one isotropy let g be a rank two finite group and let h denote the family of all rank one p subgroups of g for which rankp g 2 we show that a rank two finite group g which satisfies certain explicit group theoretic conditions admits a finite g cw complex x s n with isotropy in h whose fixed sets are homotopy spheres our construction provides an infinite family of new non linear g cw complex examples

subgroup conditions for groups acting freely on products of spheres let d and d be integers such that either d 2 and h 2 d 1 or d 4 and h 5 suppose that the group g contains an elementary abelian 2 subgroup e d of rank d with an element of order h in its normalizer we show that if g admits a free and f 2 cohomologically trivial action on s n d then some nontrivial power of a centralizes e 4 the cohomology ring h e d f 2 f 2 y 1 y d is a module over the steenrod algebra a 2 let f 2 y 1 y d and let c d 2 be an integer

some reductions of the self dual yang mills equations to integrable systems in 2 1 dimensions a reduction of the self dual yang mills sdym equations is studied by imposing two space time symmetries and by requiring that the connection one form belongs to a lie algebra of formal matrix valued differential operators in an auxiliary variable in this article the scalar case and the canonical cases for 2 2 matrices are examined in the scalar case it is shown that the field equations can be reduced to the forced burgers equation in the matrix case several well known 2 1 integrable equations are obtained also examined are certain transformation properties between the solutions of some of these 2 1 equations

periodic complexes and group actions in this paper we show that the cohomology of a connected cw complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional as applications we characterize those discrete groups that act freely and properly on in x sm we construct nonstandard free actions of rank two simple groups on finite complexes y sn x sm and we prove that a finite p group p acts freely on such a complex if and only if it does not contain a subgroup isomorphic to z p 3

free actions of finite groups on s n x s n let p be an odd prime we construct a non abelian extension of s by z p z p and prove that any finite subgroup of acts freely and smoothly on s2p 1 s2p 1 in particular for each odd prime p we obtain free smooth actions of infinitely many non metacyclic rank two p groups on s2p 1 s2p 1 these results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres

estimation and inference for the causal effect of receiving treatment on a multinomial outcome this article considers the analysis of two arm randomized trials with noncompliance which have a multinomial outcome we first define the causal effect in these trials as some function of outcome distributions of compliers with and without treatment e g the complier average causal effect the measure of stochastic superiority of treatment over control for compliers then estimate the causal effect with the likelihood method next based on the likelihood ratio lr statistic we test those functions of or the equality of the outcome distributions of compliers with and without treatment although the corresponding lr statistic follows a chi squared chi 2 distribution asymptotically when the true values of parameters are in the interior of the parameter space under the null its asymptotic distribution is not chi 2 when the true values of parameters are on the boundary of the parameter space under the null therefore we propose a bootstrap double bootstrap version of a lr test for the causal effect in these trials the methods are illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices

on some similarities between canonical correlation models and latent class models for two way contingency tables summary relationships between parameters of canonical correlations models and latent class models for two way contingency tables are explored it is found that under certain conditions canonical models and latent models are equivalent as a result a new interpretation of canonical variables in contingency tables is proposed let p be an i x j probability matrix describing the joint distribution of two categorical random variables x and y having i and j categories respectively in canonical analysis e g fisher 1940 williams 1952 and lancaster 1958 p is decomposed into a sum of r 2 which is discussed in 3

all p local finite groups of rank two for odd prime p in this paper we give a classification of the rank two p local finite groups for odd p this study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible f radical subgroups also for each case in the classification either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p local finite group getting some new examples of these for p 3

first integrals of a generalized darboux halphen system a third order system of nonlinear ordinary differential equations depending on three arbitrary parameters is analyzed the system arises in the study of su 2 invariant hypercomplex manifolds and is a dimensional reduction of the self dual yang mills equation the general solution first integrals and the nambu poisson structure of the system are explicitly derived it is shown that the first integrals are multi valued on the phase space even though the general solution of the system is single valued for special choices of parameters

cohomology theories based on flats abstract let a be an associative ring with identity k flat a the homotopy category of flat modules and k p flat a the full subcategory of pure complexes the quotient category k flat a k p flat a called here the pure derived category of flats was introduced by neeman in this category flat resolutions are unique up to homotopy and so can be used to compute cohomology we develop theories of tate and complete cohomology in the pure derived category of flats these theories extend naturally to sheaves over semi separated noetherian schemes where there are not always enough projectives but we do have enough flats as applications we characterize rings with finite sfli and schemes which are locally gorenstein

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

canonical structures on anti self dual four manifolds and the diffeomorphism group an unusual and attractive system is studied that arises from the anti self dual asd yang mills equations with maximal translational symmetry and with gauge group the volume preserving diffeomorphisms of an auxiliary four manifold m the resulting equations lead to a system consisting of a volume form together with four independent vector fields on m satisfying three simple lie bracket relations this structure is shown to give rise to a two sphere s worth of closed simple two forms which in turn lead to the standard hyper kahler structure of an anti self dual metric on m the system determines not only the asd metric but also a frame that is proportional to an orthonormal frame it is shown that the freedom in the choice of frame is related to a pair of solutions of a modified laplacian and can always be chosen so that the proportionality factor is unity the plebanski first and second heavenly forms for general asd metrics are written out in terms of these structures the first being the standard description in terms of a complex structure and kahler scalar one of the scalars is interpreted as the generating function for the diffeomorphisms symplectomorphism in line with the origin of the system as the asd yang mills equations with the volume preserving diffeomorphisms as gauge group

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

the order restricted association model two estimation algorithms and issues in testing this paper presents a row column rc association model in which the estimated row and column scores are forced to be in agreement with an a priori specified ordering two efficient algorithms for finding the order restricted maximum likelihood ml estimates are proposed and their reliability under different degrees of association is investigated by a simulation study we propose testing order restricted rc models using a parametric bootstrap procedure which turns out to yield reliablep values except for situations in which the association between the two variables is very weak the use of order restricted rc models is illustrated by means of an empirical example

bayesian analysis of stochastically ordered distributions of categorical variables abstract this article considers a finite set of discrete distributions all having the same finite support the problem of interest is to assess the strength of evidence produced by sampled data for a hypothesis of a specified stochastic ordering among the underlying distributions and to estimate these distributions subject to the ordering we present a bayesian approach that is an alternative to using the posterior probability of the hypothesis and the bayes factor in favor of the hypothesis we develop computational methods for the implementation of bayesian analyses we analyze examples to illustrate inferential and computational developments the methodology used for testing a hypothesis is seen to apply to a wide class of problems in bayesian inference and has some distinct advantages

an algebraic scheme associated with the noncommutative kp hierarchy and some of its extensions a well known ansatz trace method for soliton solutions turns the equations of the non commutative kp hierarchy and those of certain extensions into families of algebraic sum identities we develop an algebraic formalism in particular involving a mixable shuffle product to explore their structure more precisely we show that the equations of the non commutative kp hierarchy and its extension xnckp in the case of a moyal deformed product as derived in previous work correspond to identities in this algebra furthermore the moyal product is replaced by a more general associative product this leads to a new even more general extension of the non commutative kp hierarchy relations with rota baxter algebras are established

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

on the relation between k flatness and k projectivity abstract krause 19 has proved that the homotopy category k r pureproj of pure projective modules over an associative ring r is compactly generated and equivalent to the verdier quotient k r t r where t r is the triangulated subcategory of the homotopy category k r consisting of the pure acyclic complexes we provide an alternative proof of this result when restricted to the homotopy category of projective modules this equivalence reduces to that obtained by neeman 26 as an application of this description of k r pureproj we show that the k flat complexes defined by spaltenstein 29 are up to homotopy equivalence the filtered colimits of k projective complexes of pure projective modules

integrable systems and modular forms of level 2 a set of nonlinear differential equations associated with the eisenstein series of the congruent subgroup 0 2 of the modular group is constructed these nonlinear equations are analogues of the well known ramanujan equations as well as the chazy and darboux halphen equations associated with the modular group the general solutions of these equations can be realized in terms of the schwarz triangle function s 0 0 1 2 z

an integral homological characterization of finite groups abstract we show that a group g is finite if and only if every injective z g module has projective length one

a connection between the einstein and yang mills equations it is our purpose here to show an unusual relationship between the einstein equations and the yang mills equations we give a correspondence between solutions of the self dual einstein vacuum equations and the self dual yang mills equations with a special choice of gauge group the extension of the argument to the full yang mills equations yields einstein s unifield equations we try to incorporate the full einstein vacuum equations but the approach is incomplete we first consider yang mills theory for an arbitrary lie algebra with the condition that the connection 1 form and curvature are constant on minkowski space this leads to a set of algebraic equations on the connection components we then specialize the lie algebra to be the infinite dimensional lie algebra of a group of diffeomorphisms of some manifold the algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act in the self dual case if we choose the connection components from the lie algebra of the volume preserving 4 dimensional diffeomorphism group the resulting equations are the same as those obtained by ashtekar jacobsen and smolin in their remarkable simplification of the self dual einstein vacuum equations an alternative derivation of the same equations begins with the self dual yang mills connection now depending only on the time then choosing the lie algebra as that of the volume preserving 3 dimensional diffeomorphisms when the reduced full yang mills equations are used in the same context we get einstein s equations for his unified theory based on absolute parallelism to incorporate the full einsteinvacuum equations we use as the lie group the semi direct product of the diffeomorphism group of a 4 dimensional manifold with the group of frame rotations of anso 1 3 bundle over the 4 manifold this last approach however yields equations more general than the vacuum equations

reduced rank models for contingency tables summary reduced rank models for the analysis of two way contingency tables are introduced two classes of reduced rank models are discerned with well known exponents canonical analysis and latent class analysis the relation between these two classes is discussed results on the subject mentioned earlier in the literature are shown to be either redundant or inaccurate

a fast improvement to the em algorithm on its own terms the em algorithm is a numerical technique for the evaluation of maximum likelihood estimates for parameters describing incomplete data models it is easy to apply in many problems and is stable but slow the algorithm fails to provide a consistent estimator of the standard errors of the maximum likelihood estimates unless the additional analysis required by the louis method is performed newton type or other gradient methods are faster and provide error estimates but tend to be unstable and require the analytical evaluation of likelihoods to derive expressions for the score function and at least approximations to the fisher information matrix the purpose of this paper is to expand on a result by fisher that permits a unification of em methodology and newton methods the evaluation of the individual observation by observation score functions of the incomplete data is a byproduct of the application of the e step of the em algorithm once these become available the fisher information matrix may be consistently estimated and the m step may be replaced by a fast newton type step

bayesian model comparison for the order restricted rc association model association models constitute an attractive alternative to the usual log linear models for modeling the dependence between classification variables they impose special structure on the underlying association by assigning scores on the levels of each classification variable which can be fixed or parametric under the general row column rc association model both row and column scores are unknown parameters without any restriction concerning their ordinality however when the classification variables are ordinal order restrictions on the scores arise naturally under such restrictions we adopt an alternative parameterization and draw inferences about the equality of adjacent scores using the bayesian approach to achieve that we have constructed a reversible jump markov chain monte carlo algorithm for moving across models of different dimension and estimate accurately the posterior model probabilities which can be used either for model comparison or for model averaging the proposed methodology is evaluated through a simulation study and illustrated using actual datasets

a self dual yang mills hierarchy and its reductions to integrable systems in 1 1 and 2 1 dimensions the self dual yang mills equations play a central role in the study of integrable systems in this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self dual yang mills flow as the first member we show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1 1 and 2 1 dimensions and can be considered as a universal integrable hierarchy prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators symmetries and conservation laws are discussed

on certain cohomological invariants of groups abstract let r be a left and right 0 noetherian ring we show that if all projective left and all projective right r modules have finite injective dimension then all injective left and all injective right r modules have finite projective dimension using this result we prove that the invariants silp z g and spli z g which were introduced by gedrich and gruenberg 1987 15 are equal for any group g as an application of the latter equality we show that a group g is finite if and only if cd g 0 where cd is the generalized cohomological dimension of groups introduced by ikenaga 1984 21

a characterization of cohomological dimension for a big class of groups abstract it was shown by cornick and kropholler 1998 in 5 that if a group g is in h f then the finitistic dimension of z g findim z g is equal to the supremum of the projective dimensions of the z g modules m such that pd z h m for all finite subgroups h of g here we show that a locally h f group g is of type if and only if findim z g a group g is of type if for any z g module m pd z g m if and only if pd z h m for every finite subgroup h of g type was proposed in talelli 2007 18 as an algebraic characterization for those groups g that admit a finite dimensional model for e g we also deduce a number of corollaries for locally h f groups

the analysis of multivariate contingency tables by restricted canonical and restricted association models abstract restricted canonical models and restricted association models are proposed and applied to multiway contingency tables these models have been previously applied to two way contingency tables however multivariate generalization has been impeded in the past since canonical and association models both depend on singular value decompositions that apply only to two way arrays in this article this restriction to two way arrays is overcome by division of the cross classified variables into explanatory and response variables the explanatory variables are treated as a single polytomous variable and the response variables are treated as a second single polytomous variable in this fashion the multiway table is reduced to a two way array to which traditional canonical and association models may be applied use of linear restrictions on parameters in canonical and association models is especially important in multiway tables if useful models are to be constructed the class of models considered in th

seiberg witten equations in r4 lie symmetries particular solutions integrability it is shown that the 11 parameter automorphism group of lie algebra of four dimensional euclidean group is a maximal lie symmetries group of the seiberg witten equations in r4 particular explicit solutions which are invariant under so 3 subgroups of the maximal lie symmetries group are constructed it is established that seiberg witten equations do not possess the painleve property nevertheless so 3 invariant solutions obtained are turned to admit a characteristic singularity structure

canonical analysis of contingency tables by maximum likelihood abstract canonical analysis has often been employed instead of log linear models to analyze the relationship of two polytomous random variables however until the last few years analysis has been informal in this article models are examined that place nontrivial restrictions on the values of the canonical parameters so that a parsimonious description of association is obtained maximum likelihood is used to obtain parameter estimates for these restricted models approximate confidence intervals are derived for parameters and chi squared tests are used to check adequacy of models the resulting models may be used to determine the appropriateness of latent class analysis or to determine whether a set of canonical scores has specified patterns results are illustrated through analysis of two tables previously analyzed in the statistical literature comparisons are made with alternate methods of analysis based on a log linear parameterization of cell probabilities it is shown that canonical analysis wh