algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

presymplectic geometry and fermat s principle for anisotropic media the tools of presymplectic geometry are used to study light ray trajectories in anisotropic media

wavelet thresholding via a bayesian approach we discuss a bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression a prior distribution is imposed on the wavelet coefficients of the unknown response function designed to capture the sparseness of wavelet expansion that is common to most applications for the prior specified the posterior median yields a thresholding procedure our prior model for the underlying function can be adjusted to give functions falling in any specific besov space we establish a relationship between the hyperparameters of the prior model and the parameters of those besov spaces within which realizations from the prior will fall such a relationship gives insight into the meaning of the besov space parameters moreover the relationship established makes it possible in principle to incorporate prior knowledge about the function s regularity properties into the prior model for its wavelet coefficients however prior knowledge about a function s regularity properties might be difficult to elicit with this in mind we propose a standard choice of prior hyperparameters that works well in our examples several simulated examples are used to illustrate our method and comparisons are made with other thresholding methods we also present an application to a data set that was collected in an anaesthesiological study

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

computations of complex equivariant bordism rings we give explicit computations of the coefficients of homotopical complex equivariant cobordism theory mu g when g is abelian we present a set of generators which is complete for any abelian group we present a set of relations which is complete when g is cyclic and which we conjecture to be complete in general we proceed by first computing the localization of mu g obtained by inverting euler classes of representations we then define a family of operations which essentially divide by euler classes and use these operations to define our generating sets we give geometric applications of these computations to the study of equivariant genera circle actions on four manifolds and cobordism relations between lens spaces

fer s factorization as a symplectic integrator in this paper we analyze the main features of the so called fer expansion as a new method for integrating the ordinary differential equations derived from explicitly time dependent hamiltonian dynamical systems this method is based on a factorization of the evolution operator as an infinite product of exponentials of lie operators and thus exactly preserves the poincare integral invariants third and fourth order expansions are considered and numerical results are presented for a quadratic hamiltonian with various time dependent frequencies comparison is done with other numerical integration schemes

convergence of the exponential lie series improved results on convergence of the magnus expansion are reported consequences are discussed for exponential perturbation theory construction of numerical methods control theory or in general for problems where time ordered integrals are involved

the equivariant conner floyd isomorphism this paper proves two equivariant generalizations of the conner floyd isomorphism relating unitary cobordism and k theory it extends a previous result of okonek for abelian groups to all compact lie groups we also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism 1 introduction in 6 conner and floyd established a relation between cobordism and rf theory they proved that mu x mu k k x where mu is unitary cobordism and k is complex a theory a generalization of this result to the equivariant context was proved by okonek 15 this paper improves upon that generalization in two ways first it expands the class of groups considered from abelian groups to all compact lie groups second it shows that both of the usual generalizations of cobordism to the equivariant context can be used the rf theory that we use is kg the equivariant k theory of segal 17 which is the obvious generalization of k theory to the equivariant world atiyah 2 proved that this theory has bott periodicity it is periodic with respect to spinc representations of the group there are two common ways of generalizing cobordism we will define these carefully in 2 one gives geometric cobordism which we denote by ug and the other homotopical cobordism mug both of these are multiplicative cohomology theories there are multiplicative maps p u g k g and p mug kg writing u g for the ring ug and similarly for the other theories the two main results we will show are theorem a p ug x w kg kg x is an isomorphism when g is a finite group theorem b p mug x mu kg kg x is an isomorphism for every compact lie group g it should be pointed out that these results use integer grading and not ro g grading we will use integer grading throughout this paper except on those occa sions when it is necessary or particularly convenient to use ro g we will make it clear at these points that we are using the larger grading theorem b has already been shown for g abelian in 15 and we will use that result as a starting point theorem a follows from theorem b as shown in 3 by

freedom in small parameter expansion for nonlinear perturbations the freedom of choice of the zero order term in the perturbative analysis of harmonic oscillators that are perturbed by a nonlinear perturbation is investigated in detail within the framework of the method of normal forms in the case of the unforced duffing oscillator it is demonstrated that the choice leading to minimal normal forms mnf is by far the best indicating that mnf may be a way to significantly improve the convergence properties of the perturbation series relative to the traditional expansions

on the homotopy type of higher orbifolds and haefliger classifying spaces abstract we describe various equivalent ways of associating to an orbifold or more generally a higher etale differentiable stack a weak homotopy type some of these ways extend to arbitrary higher stacks on the site of smooth manifolds and we show that for a differentiable stack x arising from a lie groupoid g the weak homotopy type of x agrees with that of b g using this machinery we are able to find new presentations for the weak homotopy type of certain classifying spaces in particular we give a new presentation for the borel construction m g e g of an almost free action of a lie group g on a smooth manifold m as the classifying space of a category whose objects consist of smooth maps r n m which are transverse to all the g orbits where n dim m dim g we also prove a generalization of segal s theorem which presents the weak homotopy type of haefliger s groupoid q as the classifying space of the monoid of self embeddings of r q b emb r q and our generalization gives analogous presentations for the weak homotopy type of the lie groupoids 2 q s p and r q which are related to the classification of foliations with transverse symplectic forms and transverse metrics respectively we also give a short and simple proof of segal s original theorem using our machinery

linear time dependent hamiltonian systems beyond the adiabatic limit a classical version of the magnus expansion well suited to studying adiabatic time evolution is built up the method improves the adiabatic approximation while being symplectic in character it is shown that the first order approximation is already accurate enough even far from the adiabatic limit an analysis of the changes suffered by the adiabatic invariant of a linear hamiltonian system along its time evolution illustrates part of the above results asymptotic formulae for such changes are also obtained with explicit computation of pre exponential factors

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

geometric versus homotopy theoretic equivariant bordism by results of loffler and comezana the pontrjagin thom map from geometric g equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian g if g s1 s1 we prove that the associated fixed point square is a pull back square thus confirming a recent conjecture of sinha 22 this is used in order to determine the image of the pontrjagin thom map for toralg

the magnus expansion for classical hamiltonian systems lie operator techniques and the magnus expansion are developed in the framework of classical mechanics this leads to an exponential perturbation theory that preserves the canonical character at each order of approximation the treatment is kept as close as possible to the quantum mechanical case in order take full advantage of the properties of the expansion the explicit relationship with secular perturbation theory is established and a recursive procedure for obtaining higher order approximants is provided finally the formalism is applied to two problems of physical interest

the group theoretical treatment of aberrating systems iii the classification of asymmetric aberrations a lie theoretical classification of the aberrations of systems modeled by asymmetric optical devices is given the classification is done on the basis of aberration order and axial symmetry of the first order part this leads to finite dimensional nonunitary representations of sp 4 r reduced with respect to its sp 2 r subalgebra with helicity and symplectic spin labels based on pure magnifier systems a weight label reproduces and completes seidel s traditional classification of axis symmetric aberrations based on other first order systems such as optical fibers other classification schemes are indicated

decomposition of time ordered products and path ordered exponentials we present a decomposition formula for un an integral of time ordered products of operators in terms of sums of products of the more primitive quantities cm which are the integrals of time ordered commutators of the same operators the resulting factorization enables a summation over n to be carried out to yield an explicit expression for the time ordered exponential an expression which turns out to be an exponential function of cm the campbell baker hausdorff formula and the non abelian eikonal formula obtained previously are both special cases of this result

stochastic expansions in an overcomplete wavelet dictionary we consider random functions defined in terms of members of an overcomplete wavelet dictionary the function is modelled as a sum of wavelet components at arbitrary positions and scales where the locations of the wavelet components and the magnitudes of their coefficients are chosen with respect to a marked poisson process model the relationships between the parameters of the model and the parameters of those besov spaces within which realizations will fall are investigated the models allow functions with specified regularity properties to be generated they can potentially be used as priors in a bayesian approach to curve estimation extending current standard wavelet methods to be free from the dyadic positions and scales of the basis functions

computation of error effects in nonlinear hamiltonian systems using lie algebraic methods there exist lie algebraic methods for obtaining transfer maps around any given trajectory of a hamiltonian system this paper describes an iterative procedure for finding transfer maps around the same trajectory when the hamiltonian is perturbed by small linear terms such terms often result when an actual system deviates from an ideal one due to errors two examples from accelerator physics are worked out comparisons with numerical computations and in simple cases exact analytical calculations demonstrate the validity of the procedure

equivariant bp cohomology for finite groups the brown peterson cohomology rings of classifying spaces of finite groups are studied considering relations to the other generalized cohomology theories in particular bp af are computed for minimal nonabelian pgroups m as an application we give a necessary condition for the existence of nonabelian p subgroups of compact lie groups introduction the topology of classifying space bg for a finite group g is important in algebraic topology given generalized cohomology theory h h bg plays the central role e g cohomology of a group completion of the representation ring and the burnside ring when h is the ordinary cohomology the complex ktheory and the stable cohomotopy theory respectively recently the morava a theory of bg has been studied by hopkins kuhn and ravenel 20 for simplicity let us denote k bg by k g in this paper we study the brown peterson cohomology bp g for a prime p and the related cohomology k g with the coefficient k bp ideal s where s is a set of generators in bp landweber showed 3 that bp z r is a flat bp module and for an abelian group a bp is given by the tensor product of bp z for nonabelian p groups when g p bp is determined by tezuka yagita 11 and some relations to the other cohomology theories are given by bp c7 bp z p even c7 and k nf g k nf bp bp g consider the map induced from restrictions r rc t7 liminv k a a c g conjugacy classes of abelian groups ravenel conjectured that for k bp r is an isomorphism 8 unfortunately this does not hold however we show that for k bp z p r is an f isomorphism by using quillen s argument which showed that f isomorphy for k hz p the ordinary mod 7 cohomology 6 moreover we show that p bp g bp z p h g z p is f epic received by the editors december 28 1987 1980 mathematics subject classification 1985 revision primary 55n22 secondary 57r77

multiplicativity of the idempotent splittings of the burnside ring and the g sphere spectrum abstract we provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the g equivariant sphere spectrum including their hill hopkins ravenel norms where g is any finite group our results describe explicitly how these structures depend on the subgroup lattice and conjugation in g algebraically our analysis characterizes the multiplicative transfers on the localization of the burnside ring of g at any idempotent element which is of independent interest to group theorists as an application we obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category

new identities in dendriform algebras abstract dendriform structures arise naturally in algebraic combinatorics where they allow for example the splitting of the shuffle product into two pieces and through rota baxter algebra structures the latter appear among others in differential systems and in the renormalization process of pqft we prove new combinatorial identities in dendriform algebras that appear to be strongly related to classical phenomena such as the combinatorics of lyndon words rewriting rules in lie algebras or the fine structure of the malvenuto reutenauer algebra one of these identities is an abstract noncommutative dendriform generalization of the bohnenblust spitzer identity and of an identity involving iterated chen integrals due to c s lam

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

quasi elliptic cohomology i abstract quasi elliptic cohomology is a variant of elliptic cohomology theories it is the orbifold k theory of a space of constant loops for global quotient orbifolds it can be expressed in terms of equivariant k theories thus the constructions on it can be made in a neat way this theory reflects the geometric nature of the tate curve in this paper we provide a systematic introduction of its construction and definition

jolt factorization of pendulum map in this paper we apply the symplectic integration method using jolt factorization described in an earlier paper to the symplectic map describing the nonlinear pendulum hamiltonian we compare results obtained with this method with those obtained using nonsymplecticmethods and demonstrate that our results are much better

explicit near symplectic mappings of hamiltonian systems with lie generating functions the construction of explicit near symplectic mappings for generic hamiltonian systems with the utilization of lie transforms is presented the method is mathematically rigorous and systematically extended to high order with respect to a perturbation parameter the explicit mappings are compared to their implicit counterparts which use mixed variable generating functions in terms of conservation of invariant quantities calculation speed and accurate construction of poincare surfaces of sections the comparative study considers a wide range of parameters and initial conditions for which different time scales are involved due to large differences between internal and external frequencies of the system

from time ordered products to magnus expansion recent work by lam on the decomposition of time ordered products appearing in the time evolution operator in terms of sums of products of nested commutators is endowed with a recursive algorithm a sufficient condition for absolute convergence is subsequently obtained connection with magnus expansion is established

the complex bordism of groups with periodic cohomology is is proved that if bg is the classifying space of a group g with periodic cohomology then the complex bordism groups mu bg are obtained from the connective k theory groups ku bg by just tensoring up with the generators of mu as a polynomial algebra over ku the explicit abelian group structure is also given the bulk of the work is the verification when g is a generalized quaternionic group 1 statement of results it is well known ce that a finite group has periodic cohomology if and only if its sylow subgroups are all cyclic or generalized quaternionic another characterization sw is that these are precisely the finite groups which can act freely on a finite simplicial homotopy sphere in wo it was shown exactly which of these the spherical space form groups admit a free orthogonal action on a standard sphere let mu denote reduced complex bordism and bu connective ktheory homology it is well known cf1 that if bg denotes the classifying space of a finite group g then mun bg is isomorphic to the group of bordism classes of stably almost complex n manifolds with free g action here and elsewhere x is the space obtained from x by adjoining a disjoint basepoint the coefficient rings are mu mu s z x2 i 1 and bu bu s0 z x2 where x21 is a generator of degree 2i in a polynomial algebra our main result proves an extension of a conjecture of gilkey g bd theorem 1 1 if g is any finite group with periodic cohomology then there is an isomorphism of graded abelian groups mu bg bu bg 0 z x2i i 2 received by the editors may 1 1988 1980 mathematics subject classification 1985 revision primary 57r85 secondary 55r35 55n22

sixth order lie group integrators abstract in this paper we present the coefficients of several sixth order symplectic integrators of the type developed by r ruth to get these results we fully exploit the connection with lie groups these integrators as well as all the explicit integrators of ruth may be used in any equation where some sort of lie bracket is preserved in fact if the lie operator governing the equation of motion is separable into two solvable parts the ruth integrators can be used

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

an exact formulation of the time ordered exponential using path sums we present the path sum formulation for the time ordered exponential of a time dependent matrix the path sum formulation gives the time ordered exponential as a branched continued fraction of finite depth and breadth the terms of the path sum have an elementary interpretation as self avoiding walks and self avoiding polygons on a graph our result is based on a representation of the time ordered exponential as the inverse of an operator the mapping of this inverse to sums of walks on a graphs and the algebraic structure of sets of walks we give examples demonstrating our approach we establish a super exponential decay bound for the magnitude of the entries of the time ordered exponential of sparse matrices we give explicit results for matrices with commonly encountered sparse structures

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

computation of nonlinear behavior of hamiltonian systems using lie algebraic methods lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general hamiltonian systems a procedure is presented for the computation of nonlinear effects of arbitrarily high degree and explicit formulas are given through effects of degree 5 expected applications include accelerator design charged particle beam and light optics other problems in the general area of nonlinear dynamics and perhaps with suitable modification the area of s matrix expansions in quantum field theory

equivariant properties of symmetric products sp n x x n n we denote by sp n fsp n s m gm 0 the spectrum whose terms are the n th symmetric products of spheres a celebrated theorem of dold and thom 7 asserts that sp 1 s m is an eilenberg mac lane space of type z m for m 1 so sp 1 is an eilenberg mac lane spectrum for the group z the resulting ltration

lie algebraic approach to fer s expansion for classical hamiltonian systems the so called fer s expansion is proposed as a solution for the time evolution operator of classical time dependent hamiltonian systems the quadratic hamiltonians treated as examples show that under very different regimes the second order approximation already gives extremely good results

bayesian tree structured image modeling using wavelet domain hidden markov models wavelet domain hidden markov models have proven to be useful tools for statistical signal and image processing the hidden markov tree hmt model captures the key features of the joint probability density of the wavelet coefficients of real world data one potential drawback to the hmt framework is the need for computationally expensive iterative training to fit an hmt model to a given data set e g using the expectation maximization algorithm we greatly simplify the hmt model by exploiting the inherent self similarity of real world images the simplified model specifies the hmt parameters with just nine meta parameters independent of the size of the image and the number of wavelet scales we also introduce a bayesian universal hmt uhmt that fixes these nine parameters the uhmt requires no training of any kind while extremely simple we show using a series of image estimation denoising experiments that these new models retain nearly all of the key image structure modeled by the full hmt finally we propose a fast shift invariant hmt estimation algorithm that outperforms other wavelet based estimators in the current literature both visually and in mean square error

orbifold genera product formulas and power operations we generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h using h tuples rather than pairs of commuting elements we show that our genera are in fact orbifold invariants and we prove integrality results for them if the genus arises from an h8 map into the morava lubin tate theory eh then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold m in terms of the genus of m itself our formula is the p typical analogue of the dijkgraaf moore verlinde verlinde formula for the orbifold elliptic genus r dijkgraaf et al elliptic genera of symmetric products and second quantized strings comm math phys 185 1 1997 197 209 it depends only on h and not on the genus

virtual vector bundles and graded thom spectra we introduce a convenient framework for constructing and analyzing orthogonal thom spectra arising from virtual vector bundles this framework enables us to set up a theory of orientations and graded thom isomorphisms with good multiplicative properties the theory is applied to the analysis of logarithmic structures on commutative ring spectra

on the farrell jones conjecture for higher algebraic k theory here b is the classifying space of the group and we denote by k r the non connective algebraic k theory spectrum of the ring r the homotopy groups of this spectrum are denoted kn r and coincide with quillen s algebraick groups of r qui73 in positive dimensions and with the negative k groups of bass bas68 in negative dimensions the homotopy groups of the spectrum x k r are denotedhn x k r they yield a generalized homology theory and in particular standard computational tools such as the atiyah hirzebruch spectral sequence apply to the left hand side of the assembly map above as a corollary of the main result of this paper we prove conjecture 1 1 in the case where is the fundamental group of a closed riemannian manifold with strictly negative sectional curvature in fact our result is more general and applies to group rings r where r is a completely arbitrary coefficient ring note that if one replaces in conjecture 1 1 the coefficient ring z by an arbitrary coefficient ringr the corresponding conjecture would be false already in the simplest non trivial case if c is the infinite cyclic group the bass heller swan formula bhs64 gra76 p 236 for kn rc kn r t 1 yields that kn rc kn 1 r kn r nkn r nkn r where nkn r is defined as the cokernel of the split inclusion kn r kn r t and does not vanish in general but since s is a model for bc one obtains on the left hand side of the assembly map only hn bc k r kn r kn 1 r

bordism of semi free s1 actions we calculate geometric and homotopical bordism rings associated to semi free s1 actions on complex manifolds giving explicit generators for the geometric theory the classification of semi free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry

computation of lie transfer maps for perturbed hamiltonian systems time evolution of a hamiltonian system can be viewed as a canonical transformation therefore perturbations giving rise to near identity deviations from an unperturbed solution can be represented by products of lie transformations or together with the unperturbed solution lie transfer maps in this paper i broaden the applicability to all perturbed hamiltonian systems the method of dragt and finn and subsequent co workers who developed a representation using a product of lie transformations factored by phase space variable order in the present paper perturbation parameters are no longer necessarily associated with the phase space variables this method treats both internal and external perturbations on an equal footing and a rank is assigned to each variable to reflect the degree of perturbation it represents with the companion program pglt analytic development of the lie transfer maps is relatively easy for many systems

localization and completion theorems for mu module spectra let g be a finite extension of a torus working with highly structured ring and module spectra let m be any module over mu examples include all of the standard homotopical mu modules such as the brown peterson and morava k theory spectra we shall prove localization and completion theorems for the computation of m bg and m bg the g spectrum mug that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum sg and there is an mug module mg whose underlying mu module is m this allows the use of topological analogues of constructions in commutative algebra the computation ofm bg andm bg is expressed in terms of spectral sequences whose respective e2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring mu and its module m the central feature of the proof is a new norm map in equivariant stable homotopy theory the construction of which involves the new concept of a global i functor with smash product

any order propagation of the nonlinear schrodinger equation we derive an exact propagation scheme for nonlinear schrodinger equations this scheme is entirely analogous to the propagation of linear schrodinger equations we accomplish this by defining a special operator whose algebraic properties ensure the correct propagation as applications we provide a simple proof of a recent conjecture regarding higher order integrators for the gross pitaevskii equation extend it to multicomponent equations and to a new class of integrators

on the nonexistence of elements of kervaire invariant one we show that the kervaire invariant one elements j 2j 1 2s exist only for j 6 by browder s theorem this means that smooth framed manifolds of kervaire invariant one exist only in dimensions 2 6 14 30 62 and possibly 126 except for dimension 126 this resolves a longstanding problem in algebraic topology