torsion in the cohomology of congruence subgroups of sl 4 z and galois representations abstract we report on the computation of torsion in certain homology theories of congruence subgroups of sl 4 z among these are the usual group cohomology the tate farrell cohomology and the homology of the sharbly complex all of these theories yield hecke modules we conjecture that the hecke eigenclasses in these theories have attached galois representations the interpretation of our computations at the torsion primes 2 3 5 is explained we provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels 31

existence of quasiperiodic solutions and littlewood s boundedness problem of super linear impact oscillators abstract so far most application of kolmogorov arnold moser kam theory has been restricted to smooth dynamical systems in this paper it is shown by a series of transformations that how kam theory can be used to analyze the dynamical behavior of duffing type equations with impact the analysis is carried out for the example 0 1 x x 2 n 1 p t for x t 0 x t 0 x t 0 x t 0 if x t 0 0 with p c 5 being periodic we prove that all solutions are bounded and that there are infinitely many periodic and quasiperiodic solutions in this case

on littlewood s boundedness problem for relativistic oscillators with anharmonic potentials abstract in this paper we are concerned with the littlewood s boundedness problem and the existence of quasi periodic solutions for the nonlinear differential equation of the motions of relativistic oscillators with anharmonic potentials d d t x 1 x 2 x 1 x p t where the 1 periodic function p t is a smooth function and is a positive constant

aubry mather sets for finite twist maps of a cylinder and semilinear duffing equations in this paper based on a generalized version of the aubry mather theorem about twist maps of an infinite cylinder without the usual infinite twist assumption we establish the existence of certain quasiperiodic solutions and unlinked periodic solutions in some semilinear duffing equations x g x p t where p t is a continuous periodic function of period 1 and the semilinearity means that 0 g x x k

boundedness of solutions for a class of reversible systems in this paper we are concerned with the boundedness of all solutions for the following differential systemx email protected p y e t y email protected q x f x f p y f x e t where f r s s r 2s r 1 f x is odd e t c 4 r z is even max 1 p 2 p 1

boundedness in asymmetric oscillations abstract in this paper we are concerned with the boundedness of all the solutions of the equation x ax bx f t wheref t is a smooth 2 periodic function aandbare positive constants a b

on littlewood s boundedness problem for relativistic oscillators with singular potentials abstract in this paper we study the following nonlinear differential equations of motions of relativistic oscillators with singular potentials d d t x 1 x 2 v x p t where v is a singular potential and p is a 1 periodic function we will prove the boundedness of all solutions and the existence of infinitely many quasi periodic solutions via moser s twist theorem

a comparison of sampford s sampling procedure versus unequal probability sampling with replacement summary the classical result that simple random sampling without replacement is better than simple random sampling with replacement holds in unequal probability sampling for the sampling procedure of sampford

reciprocity laws and galois representations recent breakthroughs suppose f x is a monic irreducible polynomial with integer coefficients if p is a prime number then reducing the coefficients of f x modulo p gives a new polynomial fp x which may be reducible we say that f x is split modulo p if fp x is the product of distinct linear factors example 1 1 1 the polynomial f x x2 1 is split modulo 5 since f5 x x 2 x 3 mod 5 but it is not split modulo 7 since f7 x is irreducible nor is it split modulo 2 since f2 x x 1 2 mod 2 has a repeated factor the first few p for which x2 1 is split modulo p are 5 13 17 29 37 41 53

cohomology with twisted one dimensional coefficients for congruence subgroups of sl4 z and galois representations abstract we extend the computations in 2 3 4 to find the cohomology in degree five of a congruence subgroup of sl 4 z with coefficients in a field twisted by a nebentype character along with the action of the hecke algebra on the cohomology this is the top cuspidal degree for each hecke eigenclass we find we produce the unique galois representation that appears to be attached to it the computations require serious modifications to our previous algorithms nontrivial coefficients add a layer of complication to our data structures new possibilities must be taken into account in the galois finder the code that finds the galois representations we have improved the galois finder to report when the attached galois representation is uniquely determined by our data

galois representations and torsion in the coherent cohomology of hilbert modular varieties let f be a totally real number eld and let p be a prime unramied in f we prove the existence of galois pseudo representations attached to modp m hecke eigenclasses of paritious weight occurring in the coherent cohomology of hilbert modular varieties forf of level prime to p

boundedness in nonlinear oscillations at resonance abstract in this paper we are concerned with the boundedness of all the solutions for the semilinear duffing equation x n 2 x x p t with the lazer leach condition where p t is a smooth 2 periodic function and the perturbation x is bounded

higher operations in string topology of classifying spaces examples of non trivial higher string topology operations have been regrettably rare in the literature in this paper working in the context of string topology of classifying spaces we provide explicit calculations of a wealth of non trivial higher string topology operations associated to a number of different lie groups as an application of these calculations we obtain an abundance of interesting homology classes in the twisted homology groups of automorphism groups of free groups the ordinary homology groups of holomorphs of free groups and the ordinary homology groups of affine groups over the integers and the field of two elements

long time stability estimates for the non periodic littlewood boundedness problem we consider the nonlinear oscillator equation for 3 and non periodic forcing p for any solution x which is unbounded in the phase plane it is shown that for e 0 small enough there is a solution x e which is bounded in the phase plane and such that the actions of x e and x remain close on a time interval of length e 2 further precise information is available on the location of the zeros of x e in addition it is possible to take the bounded solution x e from a fixed countable family of such solutions

on torsion in the cohomology of locally symmetric varieties the main result of this paper is the existence of galois representations associated with the mod p or mod p m cohomology of the locally symmetric spaces for gln over a totally real or cm eld proving conjectures of ash and others following an old suggestion of clozel recently realized by harris lan taylor thorne for characteristic 0 cohomology classes one realizes the cohomology of the locally symmetric spaces for gln as a boundary contribution of the cohomology of symplectic or unitary shimura varieties so that the key problem is to understand torsion in the cohomology of shimura varieties thus we prove new results on the p adic geometry of shimura varieties of hodge type namely the shimura varieties become perfectoid when passing to the inverse limit over all levels at p and a new period map towards the ag variety exists on them called the hodge tate period map it is roughly analogous to the embedding of the hermitian symmetric domain which is roughly the inverse limit over all levels of the complex points of the shimura variety into its compact dual the hodge tate period map has several favorable properties the most important being that it commutes with the hecke operators away from p for the trivial action of these hecke operators on the ag variety and that automorphic vector bundles come via pullback from the ag variety

boundedness of solutions for duffing s equations with semilinear potentials abstract in this paper we will prove the boundedness of all the solutions and the existence of quasiperiodic solutions for the second order scalar differential equation x arctan x ee t where e is a small parameter and the 1 periodic function e t is a smooth function

boundedness of solutions for reversible systems in this paper we study the boundedness of all the solutions for the following second order nonlinear differential equation p x f x p x g x e t ddtwhere p u u p 2u p 1 f g c r 0 and odd e c r z is odd g is subquasilinear sgn x g x g x p x 0 as x

invariant tori for semilinear reversible systems abstract in this paper we consider the problem of the boundedness of all solutions for the oscillator x f x x 2 x g x p t where 0 and n g x is bounded and p t p t 2 since the system in general is non hamiltonian we have to introduce reversibility assumptions to apply the twist theorem for reversible maps under some suitable conditions we then obtain the existence of invariant tori and thus the boundedness of all solutions

invariant tori of duffing type equations abstract in this paper we prove the existence of invariant tori and thus the boundedness of all solutions and the existence of quasiperiodic solutions for a class of duffing type equations d 2 x dt 2 x 2n 1 j 0 2n p j t x j 0 n 1 wherepj t j 0 1 n are 1 periodicc 1 functions and wherepj t j n 1 2n are 1 periodicc 2 functions

the construction of pi ps sampling designs through a method of emptying boxes we present a simple but universal technique for the construction of iips sampling designs a tool that is used in the construction consists of playing a game in which objects are removed from n boxes n at a time and at most one from each box at a time it is shown that every iips design can be derived from such a game sampling designs with additional properties are obtained through additional restrictions on emptying the boxes the emphasis is on controlling sample selection probabilities and inequalities for the first and second order inclusion probabilities

spatially balanced sampling through the pivotal method a simple method to select a spatially balanced sample using equal or unequal inclusion probabilities is presented for populations with spatial trends in the variables of interest the estimation c

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

efficient balanced sampling the cube method a balanced sampling design is defined by the property that the horvitz thompson estimators of the population totals of a set of auxiliary variables equal the known totals of these variables therefore the variances of estimators of totals of all the variables of interest are reduced depending on the correlations of these variables with the controlled variables in this paper we develop a general method called the cube method for selecting approximately balanced samples with equal or unequal inclusion probabilities and any number of auxiliary variables

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

a variational method for the existence of bounded solutions of a sublinear forced oscillator we prove that for every bounded and measurable forcing p t the differential equation ddot u u 1 3 p t has bounded solutions with arbitrarily large amplitude in general it is not possible to say that all solutions are bounded as shown by an example due to littlewood the proof is based on a variational method which can be seen as a dual version of nehari s method for boundary value problems on compact intervals

modular representations of gl 3 fp symmetric squares and mod p cohomology of gl 3 z abstract we prove certain cases of nonabelian reciprocity between mod p galois representations and hecke eigenclasses in the mod p cohomology of gl 3 z using the symmetric square liftings from gl 2 and congruences on hecke eigenvalues we need to develop some modular representation theory for gl 3 z p in some cases we also need to invoke certain computer calculations to finish the proof

boundedness of solutions of nonlinear p laplacian in this paper we prove the boundedness of all the solutions for the p laplacian f p x g x e t where f p u u p 2u p 2 g satisfies some subquasilinear conditions and e t c 6 r z

boundedness problem for solutions of semilinear asymmetric equations in this paper we consider the boundedness problem for the solutions of the following semilinear differential equation f p x p 1 email protected p x email protected f p x g x e t where f p u u p 2u p 2 a b are positive constants satisfying a 1 p email protected 1 p email protected 1 g x c r is semilinear and e t c r z

biometrika centenary sample surveys summary despite karl pearson s stated aim that the evolutionist has to become in the widest sense of the words a registrar general for all forms of life the papers published in biometrika on sample surveys have been mainly theoretical and do not start to appear until the 1940s there has been little work on sampling in the life sciences with the exception of capture recapture methods the major areas have been in variable probability designs the foundations of inference for finite populations analytic surveys and the role of selection mechanisms in inference in recent years the work on survey analysis has been the outcome of an interaction between sample selection mechanisms and mainstream theory with interesting consequences for both areas

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

an analogue of serre s conjecture for galois representations and hecke eigenclasses in the mod p cohomology of gl n ℤ

optimal controlled sampling designs optimal controlled sampling designs that minimize the probability of selecting a nonpreferred sample are obtained these plans satisfy either of the following two requirements i match specified joint inclusion probabilities ij associated with an uncontrolled sampling plan having certain desirable properties ii match specified first order inclusion probabilities and provide stable nonnegative unbiased variance estimators limitations of other approaches using methods of experimental designs are demonstrated

galois representations with conjectural connections to arithmetic cohomology in this paper we extend a conjecture of a ash and w sinnott relating niveau 1 galois representations to the mod p cohomology of congruence subgroups ofsln z to include galois representations of higher niveau we then present computational evidence for our conjecture in the case n 3 in the form of three dimensional galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture our examples include galois representations with nontrivial weight and level as well as irreducible three dimensional representations that are in no obvious way related to lower dimensional representations in addition we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture

existence of quasi periodic solutions and littlewood s boundedness problem of sub linear impact oscillators abstract in this paper it is shown by a series of transformations that how moser s invariant curve theorem can be used to analyze the dynamical behavior of sub linear duffing type equations with impact we prove that all solutions are bounded and that there are infinitely many periodic and quasi periodic solutions in this impact case

on chao s unequal probability sampling plan summary we give necessary and sufficient conditions for the second order inclusion probabilities to be strictly positive under chao s 1982 sampling plan for some order of arrangement of the population units we also prove that the variance of the horvitz thompson estimator of the population total based on this sampling plan is smaller than the variance of the hansen hurwitz estimator based on a probability proportional to size with replacement sample involving the same number of draws

applying the horvitz thompson criterion in complex designs a computer intensive perspective for estimating inclusion probabilities a modification of the horvitz thompson estimator is proposed for complex sampling designs the inclusion probabilities are estimated by means of independent replications of the sampling scheme the properties of the resulting estimator are derived guidelines for choosing the appropriate number of replications are given and some applications are considered

perfect forms k theory and the cohomology of modular groups abstract for n 5 6 and 7 using the classification of perfect quadratic forms we compute the homology of the voronoi cell complexes attached to the modular groups sl n z and g l n z from this we deduce the rational cohomology of those groups and some information about k m z when m 5 6 and 7

a general purpose unequal probability sampling plan summary we present a general purpose unequal probability without replacement sampling plan with fixed sample size in contrast to existing such plans our scheme keeps the sample size fixed and lets the population units enter the sample one at a time through a carefully designed random mechanism consequently all high order inclusion probabilities can be easily computed

existence of quasiperiodic solutions and littlewood s boundedness problem of duffing equations with subquadratic potentials where p t is periodic has been extensively investigated due to its relevance in applications much work has been carried out concerning the existence of periodic solutions see 3 8 also for further references in this paper we pay attention to more complicated solutions and the lagrangian stability problem proposed by littlewood 5 as remarked in 5 it has been conjectured that all solutions of eq 1 1 are bounded if either

strata hasse invariants hecke algebras and galois representations we construct group theoretical generalizations of the hasse invariant on strata closures of the stacks mathop g hbox mathtt zip nolimits mu restricting to zip data of hodge type we obtain a group theoretical hasse invariant on every ekedahl oort stratum closure of a general hodge type shimura variety a key tool is the construction of a stack of zip flags mathop g hbox mathtt zipflag nolimits mu fibered in flag varieties over mathop g hbox mathtt zip nolimits mu it provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by griffiths schmid and the flag space for siegel varieties studied by ekedahl van der geer four applications are obtained 1 pseudo representations are attached to the coherent cohomology of hodge type shimura varieties modulo a prime power 2 galois representations are associated to many automorphic representations with non degenerate limit of discrete series archimedean component 3 it is shown that all ekedahl oort strata in the minimal compactification of a hodge type shimura variety are affine thereby proving a conjecture of oort 4 part of serre s letter to tate on mod p modular forms is generalized to general hodge type shimura varieties

cohomology of congruence subgroups of sl4 z iii in two previous papers we computed cohomology groups h 5 0 n c for a range of levels n where 0 n is the congruence subgroup of sl 4 consisting of all matrices with bottom row congruent to 0 0 0 mod n in this note we update this earlier work by carrying it out for prime levels up to n 211 this requires new methods in sparse matrix reduction which are the main focus of the paper our computations involve matrices with up to 20 million nonzero entries we also make two conjectures concerning the contributions to h 5 0 n for n prime coming from eisenstein series and siegel modular forms

symmetric invariants and cohomology of groups some time ago nakaoka in determined the additive structure of h z f2 around the same time the ring structure of h 274 f2 was calculated and it was assumed that the same type of simple multiplicative relations would appear for n 4 since then most work has concentrated on the embedding 2 o using the polynomial ring structure of h xo f2 in particular most of the recent work in hi was explicitly or implicitly described in m m more than a decade ago and provides no new information on the multiplicative structure of the mod 2 cohomology of finite symmetric groups in this paper we use invariant theory to detect unexpected multiplicative relations in h z f2 the complicated nature of the numerical evidence which we present explains the absence of complete calculations for n 16 the rings of invariants which we study are at the core of any computation of h 2 f2 they build up sucessively yielding relations rich in symmetry but of a highly convoluted type the following problem lies at the heart of all this let p be a polynomial ring k over f2 on n variables and let p k p then ek acts by permuting blocks of generators 1

second order inclusion probabilities in sequential sampling without replacement with unequal probabilities chao 1982 presents a sequential procedure for sampling without replacement with unequal probabilities his sampling plan combines the simplicity of sequential processing with the possibility of computation of second order inclusion probabilities the second order inclusion probabilities produced by this method depend on the order in which the elements are processed and are not always strictly positive this paper presents necessary and sufficient conditions for the order of the elements in the population giving positive second order probabilities thus enabling variance estimation

modular characteristic classes for representations over finite fields abstract the cohomology of the degree n general linear group over a finite field of characteristic p with coefficients also in characteristic p remains poorly understood for example the lowest degree previously known to contain nontrivial elements is exponential in n in this paper we introduce a new system of characteristic classes for representations over finite fields and use it to construct a wealth of explicit nontrivial elements in these cohomology groups in particular we obtain nontrivial elements in degrees linear in n we also construct nontrivial elements in the mod p homology and cohomology of the automorphism groups of free groups and the general linear groups over the integers these elements reside in the unstable range where the homology and cohomology remain mysterious

unequal probability sampling without replacement through a splitting method summary a very general class of sampling methods without replacement and with unequal probabilities is proposed it consists of splitting the inclusion probability vector into several new inclusion probability vectors one of these vectors is chosen randomly thus the initial problem is reduced to another sampling problem with unequal probabilities this splitting is then repeated on these new vectors of inclusion probabilities at each step the sampling problem is reduced to a simpler problem the simplicity of this technique allows one to generate easily new sampling procedures with unequal probabilities the splitting method also generalises well known methods such as the midzuno method the elimination procedure and the chao procedure next a sufficient condition is given in order that a splitting method satisfies the sen yates grundy condition finally it is shown that the elimination procedure satisfies the gabler sufficient condition

modularity lifting beyond the taylor wiles method we prove new modularity lifting theorems for p adic galois representations in situations where the methods of wiles and taylor wiles do not apply previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology in practice this imposes several restrictions one must be in a shimura variety setting and the automorphic forms must be of regular weight at infinity in this paper we essentially show how to remove these restrictions our most general result is a modularity lifting theorem which on the automorphic side applies to automorphic forms on the group mathrm gl n over a general number field it is contingent on a conjecture which in particular predicts the existence of galois representations associated to torsion classes in the cohomology of the associated locally symmetric space we show that if this conjecture holds then our main theorem implies the following if e is an elliptic curve over an arbitrary number field then e is potentially automorphic and satisfies the sato tate conjecture in addition we also prove some unconditional results for example in the setting of mathrm gl 2 over mathbf q we identify certain minimal global deformation rings with the hecke algebras acting on spaces of p adic katz modular forms of weight 1 such algebras may well contain p torsion moreover we also completely solve the problem for p odd of determining the multiplicity of an irreducible modular representation overline rho in the jacobian j 1 n where n is the minimal level such that overline rho arises in weight two

boundedness for sublinear reversible systems with a nonlinear damping and periodic forcing term abstract in this paper we are concerned with the sublinear reversible systems with a nonlinear damping and periodic forcing term x f x g x x 1 x p t where f x p t are odd functions p t is smooth 1 periodic function and 0 is a constant a sufficient and necessary condition for the boundedness of all solutions of the above equation is established moreover we show the existence of aubry mather sets as well

an elimination procedure for unequal probability sampling without replacement summary an exact without replacement sampling procedure with fixed sample size and unequal inclusion probabilities is presented the proposed method eliminates a unit of the population at each of the n n steps of the algorithm in order finally to retain n units a simple expression is given for the joint inclusion probabilities

on the range of non vanishing p torsion cohomology for gln fp abstract the range of non vanishing of h gl n f p f p is not know in general in this paper we construct a cohomology class in h gl n f p f p of very low degree namely 2 p 2 and we prove that it is nonzero if p n

smith theory and hecke operators abstract of the connections between the cohomology of arithmetic groups and representations of galois groups some are known and more are conjectured this paper proves one of these conjectures for certain monomial galois representations let p be a prime we show that the representation of the absolute galois group of q induced from an f p valued ray class character of the cyclotomic field q p is attached to a hecke eigenclass in the mod p cohomology of a torsion free congruence subgroup m of gl p 1 z the method involves constructing an action of the hecke algebra on the fundamental exact sequence of smith theory arising from the action by conjugation of an element of order p in gl p 1 z on m

invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term abstract in this paper the boundedness of all solutions of the oscillator x f x x 2 x x p t is discussed where is a positive constant f x y x and p t are smooth functions and p t p t 2

boundedness for solutions of nonlinear hill s equations with periodic forcing terms via moser s twist theorem we consider the nonlinear hill s equation with periodic forcing term x x2n 1 a1 ea t x p t n 1 where a t and p t are continuous and 1 periodic a1 and are positive constants and e is a small parameter the purpose of this paper is to prove that all solutions of the above mentioned equation are bounded for t r and that there are infinitely many quasi periodic solutions and an infinity of periodic solutions of minimal period m for each positive integer m

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

on a conjecture of ash abstract in this paper we prove a particular case of a conjecture of ash that states the existence of galois representations associated to hecke eigenclasses in cohomology

homological stability for completed homology we prove that the completed homology groups of mathrm gl n mathbf z in fixed degree stabilize as n goes to infinity we also prove that the action of hecke operators on stable cohomology is trivial in a precisely defined sense

on unequal probability sampling sufficient conditions for the superiority of sampling without replacement summary a challenging theoretical problem in unequal probability sampling is the question which design sampling with or without replacement should be implemented in this context we compare the horvitz thompson estimator with the hansen hurwitz estimator and derive conditions under which the variance of the horvitz thompson estimator is smaller than the variance of the hansen hurwitz estimator examples are given showing the validity of these conditions

boundedness of solutions for semilinear duffing equations in this paper we are concerned with the boundedness of all the solutions and the existence of quasi periodic solutions for some semilinear duffing equationx g x e t wheree t is a smooth 2 periodic function and0 g x x k