exact solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number of integral operators that map the boundary velocity of a baffled planar structure onto the acoustic pressure radiated in far field or intermediate regions exact solutions to this problem involve prolate spheroidal wave functions that correspond to a set of independent distributions with finite spatial support and maximal energy concentration in a given bandwidth of the transform domain a stable solution to the inverse source reconstruction problem is obtained by decomposing the unknown boundary velocity into a number of efficiently radiating singular velocity patterns that correspond to the number of degrees of freedom of the radiated field it is found that the degree of ill posedness of the inverse problem is significantly reduced when considering a hemi circular observation arc with respect to a linear array of sensors by a factor scaling on the small angular aperture subtended by the observation line estimates are derived from the spatial resolution limits that can be achieved in the source reconstruction problem from the dimension of the efficiently radiating subspace

an inverse method to compute equivalent forces on a structure based on sound power measurements when modelling the dynamic response of an existing structure to predict the effects of design modifications many types of forces are difficult to characterize e g those due to unsteady flows or acoustic excitation to model the forcing function in such cases a method is presented which solves for an equivalent forcing function based on sound power measurements the method assumes that the equivalent forcing function may be represented with a superposition of force distributions on a mode by mode basis i e there is one force distribution per eigenvalue with the shape of the eigenvector the method is applied to cases where the structure is lightly damped and has low modal density experimental results show that the method accurately predicts the effect of design changes on the structure s radiated sound power the accuracy of the method is sensitive to small differences between the experimental and theoretical natural frequencies and thus it is important to match the damped natural frequencies of the physical structure as accurately as possible

on the dimension of subfield subcodes a new lower bound for the dimension of a subfield subcode is given this bound improves the well known lower bound of p delsarte 1975 and coincides in many cases with the true dimension of the code alternant codes and geometric goppa codes are discussed as special cases

comodules and landweber exact homology theories abstract we show that if e is a commutative mu algebra spectrum such that e is landweber exact over mu then the category of e e comodules is equivalent to a localization of the category of mu mu comodules this localization depends only on the heights of e at the integer primes p it follows for example that the category of e n e n comodules is equivalent to the category of v n 1 bp v n 1 bp comodules these equivalences give simple proofs and generalizations of the miller ravenel and morava change of rings theorems we also deduce structural results about the category of e e comodules we prove that every e e comodule has a primitive we give a classification of invariant prime ideals in e and we give a version of the landweber filtration theorem

an inverse aeroacoustic problem on rotor wake stator interaction an inverse aeroacoustic model on rotor wake stator interaction is proposed based on the linearized euler equations the sound field is related to the pressure distribution on the stator surface in the form of a fredholm integral equation of the first kind which is a well known ill posed problem since the sound field is fully specified numerical inversion allows the reconstruction of the pressure distribution on the stator surface for solving the discrete ill posed problem the singular value decomposition technique coupled with the tikhonov regularization method is applied to stabilize the solution the optimal regularization parameter is chosen by the generalized cross validation criterion and the discrete picard condition is employed to analyze the ill posedness of the inverse problem numerical results show that the reconstruction is fairly good when the signal to noise ratio is not very low the results become inaccurate when the noise dominates over the observer signal in addition numerical results also indicate the importance of the reduced frequency the higher the reduced frequency the better the reconstruction results

analytic solutions to the acoustic source reconstruction problem in this paper we generalize the recently developed analytical solutions of the radiation modes problem to the determination of closed form expressions for the singular value expansion of a number

a class of constacyclic codes a class of q sup m ary constacyclic codes is defined which has the property that the q ary image is equivalent to a class of shortened cyclic codes this description leads to the construction of nonlinear codes as an example one can construct nonlinear codes of length n q sup m 1 over the field gf q sup m m spl les m with a minimum distance greater than n 1 r m m where r is the rate for higher rates these codes are the best known the codes are easily encoded and decoded

estimation of acoustic source strength by inverse methods part i conditioning of the inverse problem this paper deals with the discrete inverse problem in acoustics it is assumed that a number of acoustic sources are located at known spatial positions and that the acoustic pressure is measured at a number of spatial positions in the radiated field the transfer functions relating the strengths of the acoustic sources to the radiated pressures are also assumed known in principle the strengths of the acoustic sources can be deduced from the measured acoustic pressures by inversion of this matrix of transfer functions the accuracy of source strength reconstruction in the presence of noise which contaminates the measured pressures is crucially dependent on the conditioning of the matrix to be inverted this paper examines the conditioning of this inverse problem particularly with regard to the geometry and number of sources and measurement positions and the non dimensional frequency a preliminary investigation is also presented of methods such as tikhonov regularization and singular value discarding which can improve the accuracy of source strength reconstruction in poorly conditioned cases results are also presented which enable the solution of the inverse problem when the time histories of the acoustic sources are time stationary random processes and the spectra and cross spectra are measured at a number of positions in the radiated field the paper illustrates the possibilities and limitations of the use of inverse methods in the deduction of acoustic source strength from radiated field measurements

spherical harmonics singular value decomposition and the head related transfer function abstract the singular value decomposition svd can be used to analyze the matrix of green functions relating the acoustic pressure at a number of field points to the strengths of a number of point sources on the surface of a body which radiates or scatters sound the left and right singular vectors of the resulting decomposition yield at a given frequency two sets of orthogonal basis functions describing field mode shapes and source mode shapes respectively this paper attempts to make the connection between this decomposition and the basis functions of classical acoustics in particular it is found that for a spherical co ordinate system when the source points and field points are chosen in order to sample the source and field appropriately then the matrices of left and right singular vectors are related to the sampled spherical harmonics by a unitary transformation preliminary investigations are presented of the sensitivity of this relationship to the manner in which the source and field are sampled in addition the great utility of the method is illustrated with some new results of numerical analysis of the scattering of sound by the outer ear pinna resonances and their associated mode shapes are identified at certain frequencies where high values of the dominant singular values indicate a strong coupling between source and field

inverse aeroacoustic problem for a rectangular wing an investigation is made of the feasibility of aeroacoustic inversion where the pressure on a thin flat rigid rectangular wing undergoing rigid oscillations or interacting with unsteady subsonic flow is to be predicted from the far field acoustic signal this problem is ill posed because small pressure fluctuations in the far field are larger in the near field by a factor equal to the reciprocal of the distance from the wing in the inverse model this ill posedness manifests itself in the kernel of a two dimensional fredholm integral equation of the first kind discretization of this integral equation using a physically meaningful collocation series results in an ill conditioned system of equations which is solved using the singular value decomposition svd the svd generally requires regularization techniques to discard redundant or unphysical information an algorithm is developed for optimally determining the near field pressure without relying on a user specified regularization parameter tests using numerically generated input data show the inversion is feasible and accurate for accurate input data the inversion remains feasible when errors are introduced in the far field measurements and the measurable parameters of the flow

on the rigidity theorems of witten in this paper we prove the rigidity theorems predicted by witten in 1986 about the index of certain elliptic operators on manifolds with an s1 action w witten s insight was the culmination of an interesting interchange of ideas between him and hopkins landweber ochanine and stong for the detailed history we refer the reader to la the present account is essentially a reinterpretation of the second author s taubes t original proof of the theorem the senior author s contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology in this context the final proof then appears in direct lineage of the atiyah hirzebruch theorem concerning the vanishing of the a genus of spin manifolds admitting a circle action a h and of the even older idea of lusztig concerning circle actions in the complex case there remains however the beautiful physics inspired novelty of connecting these techniques with elliptic function theory

projective elliptic genera and elliptic pseudodifferential genera abstract in this paper we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle such projective elliptic genera are rational q series that have topological definition and also have analytic interpretation via the fractional index theorem in 25 without requiring spin condition we prove the modularity properties of these projective elliptic genera as an application we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator this suggests the existence of putative s 1 equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera

tmf0 3 characteristic classes for string bundles we compute the completed tmf 0 3 cohomology of the 7 connected cover bstring of bo we use cubical structures on line bundles over elliptic curves to construct an explicit class which together with the pontryagin classes freely generates the cohomology ring

localization and periodicity in unstable homotopy theory in this paper we develop a hierarchy of natural localizations of spaces called the vn periodizations for n 0 which may be used to expose and study periodic phenomena in unstable homotopy theory these vn periodizations act less radically than the corresponding homological localizations 2 and respect fibrations to a very considerable extent a major part of this paper is devoted to developing the general theory of periodizations of spaces thereby providing a foundation for the study of the vn periodization and many others some of this general theory has been developed independently by dror farjoun 13 14 and we refer the reader to his work for an alternative approach with other interesting general results during the past decade remarkable progress was made by ravenel hopkins devinatz and smith 12 16 17 33 toward a global understanding of stable periodic phenomena and we hope that the present paper will help to prepare the way for a similar understanding of unstable periodic phenomena an excellent exposition of localization and periodicity in stable homotopy theory is now available in ravenel s book 35 major features of stable homotopy are understood chromatically as manifestations of more basic periodic phenomena these phenomena belong to a hierarchy starting with those detected rationally followed by those detected in classical k theory and in the successive morava k theories most fundamentally each finite cw spectrum has an intrinsic periodicity given by a vn self map which becomes a selfequivalence after suitable localization for simplicity we describe our results in the pointed homotopy category hoo of connected cw complexes for spaces w y e hoo we call y w periodic or w local when the pointed mapping space map w y is contractible or equivalently when xt w y for t 0 as shown more generally in 3 corollary 7 2 10 or 13 there is a natural initial example x pwx of a map from x to a w periodic space pwx in hoo and we call this the w periodization or w localization of x our notation is derived from the classical example where w sn l and pwx is the n th postnikov section

inverse aeroacoustic problem for a streamlined body ii accuracy of solutions previously established methods for performing the aeroacoustic inversion for a fiat plate airfoil are tested to determine their sensitivity to errors in both the far field input data and the mean flow input parameters the results of the sensitivity analysis for the input data show that the magnification of input error due to ill posedness can be controlled by optimal choices of the regularization parameter and the input data locations a similar sensitivity analysis of the effect of mean flow parameters mach number and reduced frequency shows strong dependence of the inverse process on these parameters in this case improvements in the reconstructions can be obtained by using input data from the midfield

subgroup subcodes subgroup subcodes and their duals which can be considered as an extension of subfield subcodes and trace codes are defined some of their basic properties are investigated examples of good subgroup subcodes are included

quadratic functions in geometry topology and m theory we describe an interpretation of the kervaire invariant of a riemannian manifold of dimension 4k 2 in terms of a holomorphic line bundle on the abelian variety h 2k 1 m otimes r z our results are inspired by work of witten on the fivebrane partition function in m theory hep th 9610234 hep th 9609122 our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics and is based on our theory of differential functions these differential functions generalize the differential characters of cheeger simons and the bulk of this paper is devoted to their study

goppa codes and trace operator we study goppa codes spl gamma l g defined by the polynomial g z a z trf sub p ms f sub p s b z it is shown that the dimension of these codes never reaches the general well known bound for goppa codes new bounds are proposed depending on the value of m and p furthermore we prove that when p 2 these codes have only even weights

lifting homotopy t algebra maps to strict maps abstract the settings for homotopical algebra categories such as simplicial groups simplicial rings a spaces e ring spectra etc are often equivalent to categories of algebras over some monad or triple t in such cases t is acting on a nice simplicial model category in such a way that t descends to a monad on the homotopy category and defines a category of homotopy t algebras in this setting there is a forgetful functor from the homotopy category of t algebras to the category of homotopy t algebras under suitable hypotheses we provide an obstruction theory in the form of a bousfield kan spectral sequence for lifting a homotopy t algebra map to a strict map of t algebras once we have a map of t algebras to serve as a basepoint the spectral sequence computes the homotopy groups of the space of t algebra maps and the edge homomorphism on 0 is the aforementioned forgetful functor we discuss a variety of settings in which the required hypotheses are satisfied including monads arising from algebraic theories and operads we also give sufficient conditions for the e 2 term to be calculable in terms of quillen cohomology groups we provide worked examples in g spaces g spectra rational e algebras and a algebras explicit calculations connected to rational unstable homotopy theory show that the forgetful functor from the homotopy category of e ring spectra to the category of h ring spectra is generally neither full nor faithful we also apply a result of the second named author and nick kuhn to compute the homotopy type of the space e coker j l k 2 r

inverse aeroacoustic problem for a streamlined body i basic formulation the inverse aeroacoustic problem associated with a flat plate airfoil in unsteady compressible flow is formulated in terms of a fredholm integral equation of the first kind the feasibility of determining the unsteady pressure along the airfoil surface from the radiated acoustic signal is demonstrated it is shown that the hadamard conditions of existence and uniqueness of the inverse solution can be satisfied the third hadamard condition for the continuous dependence of the solution on the input acoustic data shows the problem to be ill posed the singular value decomposition method with regularization is used to treat the associated ill conditioned algebraic problem discretization and collocation techniques are used to represent the unsteady pressure on the airfoil both methods give very accurate reconstructions when perfect input data are used the collocation method requires only a small amount of input data either from the far field or the near field thus making it more suitable for applications

the morava k theory of spaces related to bo abstract we calculate the p 2 morava k theory of all of the spaces in the connective omega spectra for z bo bo bso and bspin this leads to a description of the p 2 brown peterson cohomology of many of these spaces of particular interest is the space bo 8 and its relationship to bspin

products on mu modules in 2 chapter v elmendorf kriz mandell and may hereafter referred to as ekmm use their new technology of modules over highly structured ring spectra to give new constructions of mu modules such as bp k n and so on which makes it much easier to analyse product structures on these spectra unfortunately their construction only works in its simplest form for modules over mu 12 that are concentrated in degrees divisible by 4 this guarantees that various obstruction groups are trivial in the present paper we extend the ekmm results to the cases where p 2 or the homotopy groups are allowed to be nonzero in all even degrees in this context the obstruction groups are nontrivial we shall show that there are never any obstructions to associativity we prove in section 7 that the obstructions to commutativity are given by a certain power operation this was inspired by a parallel result of mironov in baas sullivan theory 6 in section 8 we shall use formal group theory to derive various formulae for this power operation in section 9 we deduce a number of results about realising 2 local mu modules as mu modules

reconstruction of the acoustic field over a limited surface area on a vibrating cylinder the acoustic field on a small region on the surface of a radiating cylindrical shell is reconstructed from measurements of pressure made on a limited nearby surface the boundary element method is applied to compute an operator relating the pressure on a small measurement surface located near the structure to the normal velocity at all points on the structure surface since the inverse problem of the reconstruction of the surface field is ill posed singular value decomposition is applied to invert the operator it is demonstrated that the surface field may be reconstructed over the portion of the structure surface located under the measurement surface

zero cycles of degree one on principal homogeneous spaces abstract let k be a field of characteristic different from 2 let g be an absolutely simple simply connected or adjoint semisimple algebraic k group of classical type we show that if a principal homogeneous space under g over k admits a zero cycle of degree 1 then it has a k rational point

quadratic algebra of square groups abstract square groups are quadratic analogues of abelian groups many properties of abelian groups are shown to hold for square groups in particular there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product

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

generalized group characters and complex oriented cohomology theories let bg be the classifying space of a finite group g given a multiplicative cohomology theory e the assignment g 7i e bg is a functor from groups to rings endowed with induction transfer maps in this paper we investigate these functors for complex oriented cohomology theories e using the theory of complex representations of finite groups as a model for what one would like to know an analogue of artin s theorem is proved for all complex oriented e the abelian subgroups of g serve as a detecting family for e bg modulo torsion dividing the order of g when e is a complete local ring with residue field of characteristic p and associated formal group of height n we construct a character ring of class functions that computes 1 e bg the domain of the characters is gn p the set of n tuples of elements in g each of which has order a power of p a formula for induction is also found the ideas we use are related to the lubin tate theory of formal groups the construction applies to many cohomology theories of current interest completed versions of elliptic cohomology e n theory etc the nth morava k theory euler characteristic for bg is computed to be the number of g orbits in gn p for various groups g including all symmetric groups we prove that k n bg concentrated in even degrees our results about e bg extend to theorems about e eg gx where x is a finite g cw complex

on the telescopic homotopy theory of spaces in telescopic homotopy theory a space or spectrum x is approximated by a tower of localizations lnx n 0 taking account of vn periodic homotopy groups for progressively higher n for each n 1 we construct a telescopic kuhn functor n carrying a space to a spectrum with the same vn periodic homotopy groups and we construct a new functor n left adjoint to n using these functors we show that the nth stable monocular homotopy category comprising the nth fibers of stable telescopic towers embeds as a retract of the nth unstable monocular homotopy category in two ways one giving infinite loop spaces and the other giving infinite ln suspension spaces we deduce that ravenel s stable telescope conjectures are equivalent to unstable telescope conjectures in particular we show that the failure of ravenel s nth stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial johnson wilson e n homology but nontrivial vn periodic homotopy groups showing a fundamental difference between the unstable chromatic and telescopic theories as a stable chromatic application we show that each spectrum is k n equivalent to a suspension spectrum as an unstable chromatic application we determine the e n localizations and k n localizations of infinite loop spaces in terms of e n localizations of spectra under suitable conditions we also determine the e n localizations and k n localizations of arbitrary postnikov h spaces

a note on lower bounds corresp a new lower bound for the parameters of nonlinear q ary codes is introduced for some q this bound improves on the varshamov gilbert bound the modular algebraic geometric bound and the very recent vl breve a duts bound

topological modular forms with level structure the cohomology theory known as mathrm tmf for topological modular forms is a universal object mapping out to elliptic cohomology theories and its coefficient ring is closely connected to the classical ring of modular forms we extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them along the way we produce a natural way to restrict to the cusps providing multiplicative maps from mathrm tmf with level structure to forms of k theory in particular this allows us to construct a connective spectrum mathrm tmf 0 3 consistent with properties suggested by mahowald and rezk this is accomplished using the machinery of logarithmic structures we construct a presheaf of locally even periodic elliptic cohomology theories equipped with highly structured multiplication on the log etale site of the moduli of elliptic curves evaluating this presheaf on modular curves produces mathrm tmf with level structure

the torsion index of the spin groups the torsion index is a positive integer associated by grothendieck to any connected compact lie group g 10 as explained in section 1 knowing the torsion index of a group has direct consequences for the abelian subgroups of g the integral cohomology of the classifying space the complex cobordism of the classifying space the chow ring of the classifying space 26 and the classification of g torsors over fields 24 demazure 5 marlin 14 and tits 25 have given upper bounds for the torsion index in this paper we compute the torsion index exactly for all the spin groups spin n in another paper we will compute the torsion index of the exceptional group e8 as discussed below this completes the calculation of the torsion index for all the simply connected simple groups the topology of the spin groups becomes more and more complicated as the dimension increases and so it was not at all clear that it would be possible to do the calculation in all dimensions indeed the answer is rather intricate and the proof in high dimensions requires some deep information from analytic number theory bauer and bennett s theorem on the binary expansion of 2 1

analytic index for a family of dirac ramond operators we derive a cohomological formula for the analytic index of a family of dirac ramond operators and exhibit its modular properties

elliptic genera and quantum field theory it is shown that in elliptic cohomology as recently formulated in the mathematical literature the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the dirac operator ink theory this leads to several insights concerning both elliptic cohomology and string theory some of the relevant calculations have been done previously by schellekens and warner in a different context

spectra of long primitive binary bch codes cannot approach the binomial distribution usually spectra weight distributions of primitive binary bch codes are supposed to approximate binomial weight distributions well for a wide range of code rates and code lengths it is shown that for any fixed code rate r 1 spectra of long n spl rarr spl infin primitive binary bch codes cannot approximate the binomial distribution at all

an e splitting of spin bordism the work determines the e structure of the k 1 local spin bordism spectrum by giving a multiplicative splitting into e cells the splitting leads to a cellular decomposition of the a map and creates new isomorphisms of conner floyd type introduction and statement of results one of the most important results on spin bordism goes back to anderson brown and peterson abp they showed that two spin manifolds are spin bordant if and only if all stiefel whitney and ko characteristic numbers coincide moreover the spin bordism groups can be computed from the additive 2 local splitting mspin

k 1 local topological modular forms we construct the witten orientation of the topological modular forms spectrum tmf in the k 1 local setting by attaching e cells to the bordism theory mo we also identify the ko homology of tmf with the ring of divided congruences

painleve formulas of the limiting distributions for nonnull complex sample covariance matrices in a recent study of large non null sample covariance matrices a new sequence of functions gener alizing the gue tracy widom distribution of random matrix theory was obtained this paper derives painleve formulas of these functions and use them to prove that they are indeed distribution functions applications of these new distribution functions to last passage percolation queues in tandem and to tally asymmetric simple exclusion process are also discussed as a part of the proof a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented

representation and character theory in 2 categories we develop a 2 categorical generalization of the theory of group representations and characters we categorify the concept of the trace of a linear transformation associating to any endofunctor of any small category a set called its categorical trace in a linear situation the categorical trace is a vector space and we associate to any two commuting self equivalences a number called their joint trace for a group acting on a linear category we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors we call this function the 2 character of such functions of commuting pairs and more generally n tuples appear in the work of hopkins kuhn and ravenel michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic on equivariant morava e theories we define the concept of induced categorical representation and show that the corresponding 2 character is given by the same formula as was obtained in michael j hopkins nicholas j kuhn douglas c ravenel generalized group characters and complex oriented cohomology theories j amer math soc 13 3 2000 553 594 electronic for the transfer map in the second equivariant morava e theory

circle equivariant classifying spaces and the rational equivariant sigma genus the circle equivariant spectrum mstring mathbb c is the equivariant analogue of the cobordism spectrum mu langle 6 rangle of stably almost complex manifolds with c1 c2 0 given a rational elliptic curve c greenlees topology 44 1213 1227 2005 constructs a ring mathbb t spectrum ec representing the associated mathbb t equivariant elliptic cohomology the core of the present paper is the construction when c is a complex elliptic curve of a map of ring mathbb t spectra mstring mathbb c to ec which is the rational equivariant analogue of the sigma orientation of ando et al invent math 146 595 687 2001 we support this by a theory of characteristic classes for calculation and a conceptual description in terms of algebraic geometry in particular we prove a conjecture the first author made in ando geom topol 7 91 153 2003

brown peterson and ordinary cohomology theories of classifying spaces for compact lie groups the steenrod algebra structures of h bg z p for compact lie groups are studied using these brown peterson cohomology and morava k theory are computed for many concrete cases all these cases have properties similar as torsion free lie groups or finite groups e g bp odd bg 0

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

localization of andre quillen goodwillie towers and the periodic homology of infinite loopspaces abstract let k n be the n th morava k theory at a prime p and let t n be the telescope of a v n self map of a finite complex of type n in this paper we study the k n homology of x the 0th space of a spectrum x and many related matters we give a sampling of our results let p x be the free commutative s algebra generated by x it is weakly equivalent to the wedge of all the extended powers of x we construct a natural map s n x l t n p x l t n x of commutative algebras over the localized sphere spectrum l t n s the induced map of commutative cocommutative k n hopf algebras s n x k n p x k n x satisfies the following properties it is always monic it is an isomorphism if x is n connected n 1 x is torsion and t i x 0 for 1 i n 1 it is an isomorphism only if k i x 0 for 1 i n 1 it is universal the domain of s n x preserves k n isomorphisms and if f is any functor preserving k n isomorphisms then any natural transformation f x k n x factors uniquely through s n x the construction of our natural transformation uses the telescopic functors constructed and studied previously by bousfield and the author and thus depends heavily on the nilpotence theorem of devanitz hopkins and smith our proof that s n x is always monic uses topological andre quillen homology and goodwillie calculus in nonconnective settings

hurewicz images of real bordism theory and real johnson wilson theories abstract we show that the hopf elements the kervaire classes and the family in the stable homotopy groups of spheres are detected by the hurewicz map from the sphere spectrum to the c 2 fixed points of the real bordism spectrum a subset of these families is detected by the c 2 fixed points of real johnson wilson theory e r n depending on n in the proof we establish an isomorphism between the slice spectral sequence and the c 2 equivariant may spectral sequence of b p r

efficient encoding for a class of subspace subcodes let s consist of all words of a code c for which each symbol is in a stipulated subalphabet possibly different for distinct positions we consider the special case where c is a linear maximum distance separable mds code and the subalphabets are linear subspaces over the ground field with equal dimensions we give an explicit algorithm for selecting the subspaces in such a way that a straightforward systematic encoding algorithm based on an encoder for c can be applied the number of information symbols that can be encoded with this algorithm equals a well known lower bound on the dimension of s

the sigma orientation is an h map in an earlier paper the authors constructed a natural map called the sigma orientation from the thom spectrum mu 6 to any elliptic spectrum mu 6 is an h ring spectrum and in this paper we show that if e c t is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p 0 then the sigma orientation is a map of h ring spectra

a bound for canonical dimension of the semi spinor groups using the theory of non negative intersections duality of the schubert varieties and pieri type formula for the varieties of maximal totally isotropic subspaces we get an upper bound for canonical dimension cd spinn of the spinor group spinn a lower bound is given by the canonical 2 dimension cd2 spinn computed in 10 if n or n 1 is a power of 2 no space is left between these two bounds therefore the precise value of cd spinn is obtained for such n we also produce an upper bound for canonical dimension of the semi spinor group giving the precise value of the canonical dimension in the case when the rank of the group is a power of 2 and show that spinor and semispinor groups are the only open cases of the question about canonical dimension of an arbitrary simple split group possessing a unique torsion prime

landweber flat real pairs and er n cohomology abstract we take advantage of the internal algebraic structure of the bockstein spectral sequence converging to e r n p t to prove that for spaces z that are part of landweber flat real pairs with respect to e n see definition 2 9 the cohomology ring e r n z can be obtained from e n z by base change in particular our results allow us to compute the real johnson wilson cohomology of the eilenberg maclane spaces z k z 2 m 1 k z 2 q 2 m k z 2 m for any natural numbers m and q as well as connective covers of bo bo bso bspin and bo 8

geometric representations of the formal affine hecke algebra abstract for any formal group law there is a formal affine hecke algebra defined by hoffnung malagon lopez savage zainoulline coming from this formal group law there is also an oriented cohomology theory we identify the formal affine hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the steinberg variety as a consequence this algebra acts on the corresponding cohomology of the springer fibers this generalizes the action of classical affine hecke algebra on the k theory of the springer fibers constructed by lusztig we also give a residue interpretation of the formal affine hecke algebra which generalizes the residue construction of ginzburg kapranov vasserot when the formal group law comes from a 1 dimensional algebraic group