pulsating flows of the 2d euler poisson equations abstract it is well known that solutions of gaseous or fluid dynamical systems can easily blowup or develop shock waves in finite time in this paper we show the existence of a class of radially symmetric rotational solutions to the two dimensional pressureless euler poisson equations the flows are global i e exist for all t 0 have compact support at all times and pulsate periodically the method of construction is novel it comprises the piecing together of suitable shells of moving particles in a delicate manner with careful choice of initial data each shell is the solution of a member of a continuum of ordinary differential equations detailed analysis of the equations is carried out to ensure that neighboring shells can be chosen to pulsate with the same period we also show that any such solution can be extended to a larger flow by adding annuli of pulsating flows our result exhibits another example of rotation preventing the blowup of solutions

correction to the stable geometric dimension of vector bundles over real projective spaces trans amer math soc 268 1981 no 1 39 61 mr0628445 83c 55006

blowup solutions for a class of fluid dynamical equations in rn abstract in this paper we consider the blowup solutions for the euler poisson type equations such as isentropic euler poisson navier stokes poisson euler and navier stokes equations and the systems with frictional damping it is shown that these systems have similar blowup phenomenon which generalize the previous results of the blowup solution for the euler poisson equations for gaseous stars besides the blowup rate is given furthermore under the assumption r 0 some radial symmetric blowup solutions for the euler poisson type equations are given

totally skew embeddings of manifolds we study a version of whitney s embedding problem in projective geometry what is the smallest dimension of an affine space that can contain an n dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points this problem is related to the generalized vector field problem existence of non singular bilinear maps and the immersion problem for real projective spaces we use these connections and other methods to obtain several specific and general bounds for the desired dimension

new method for blowup of the euler poisson system in this paper we provide a new method for establishing the blowup of c2 solutions for the pressureless euler poisson system with attractive forces for rn n 2 with 0 x0 0 and 0ij x0 12 iuj 0 x0 jui 0 x0 0 at some point x0 rn by applying the generalized hubble transformation div u t x0 t na t a t to a reduced riccati differential inequality derived from the system we simplify the inequality into the emden equation a t a t n 1 a 0 1 a 0 div u 0 x0 n known results on its blowup set allow us to easily obtain the blowup conditions of the euler poisson system

the periodic homotopy groups of an unstable sphere at odd primes the mod p v periodic homotopy groups of a space x are defined by considering the homotopy classes of maps of a moore space into x and then inverting the adams self map in this paper we compute the mod p v1 periodic homotopy groups of an odd dimensional sphere localized at an odd prime this is done by showing that these groups are isomorphic to the stable mod p v1 periodic homotopy groups of b 2 p1l n the 2 p l n skeleton of the classifying space for the symmetric group y there is a map 2n 1s2n 1 q c co j a b e2 p on where j is a spectrum constructed from connective k theory and the image in homotopy is studied

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

root invariants in the adams spectral sequence let e be a ring spectrum for which the e adams spectral sequence converges we define a variant of mahowald s root invariant called the filtered root invariant which takes values in the e 1 term of the e adams spectral sequence the main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant and explain a relationship between filtered root invariants and differentials and compositions in the e adams spectral sequence these theorems are compared to some known computations of root invariants at the prime 2 we use the filtered root invariants to compute some low dimensional root invariants of v 1 periodic elements at the prime 3 we also compute the root invariants of some infinite v 1 periodic families of elements at the prime 3

improved blowup results for the euler and euler poisson equations with repulsive forces abstract some results are presented on the formation of singularities in the solutions of the radially symmetric n dimensional euler or euler poisson equations with repulsive forces based on the integration method of m w yuen we generalize the blowup results with constant compact radius r of solutions to the case with general compact radius r t and to the case with no compact support restriction

stabilities for euler poisson equations in some special dimensions abstract we study the stabilities and classical solutions of euler poisson equations describing the evolution of the gaseous star in astrophysics in fact we extend the study of the stabilities of euler poisson equations with or without frictional damping term to some special dimensional spaces besides by using the second inertia function in 2 dimension of euler poisson equations we prove the non global existence of classical solutions with 2 u 2 2 p d x g m 2 for any

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

morava modules and brown comenetz duality fix a prime number p and let x be a finite spectrum whose n 1 st morava k theory is trivial but whose n th morava k theory is nontrivial n 0 we prove following a method outlined to us by m j hopkins that if 2 p n 2 n 2 the morava module of the brown comenetz dual of the e n localization of x is isomorphic to a suspension of the pontryagin dual of the morava module of x to complete this proof we found it necessary to develop a more canonical construction of certain modified adams spectral sequences this construction should be of independent interest

the goodwillie tower of the identity functor and the unstable periodic homotopy of spheres we investigate goodwillie s taylor tower of the identity functor from spaces to spaces more specifically we reformulate johnson s description of the goodwillie derivatives of the identity and prove that in the case of an odd dimensional sphere the only layers in the tower that are not contractible are those indexed by powers of a prime moreover in the case of a sphere the tower is finite in v k periodic homotopy

analytical blowup solutions to the 2 dimensional isothermal euler poisson equations of gaseous stars ii in this article we construct analytical blowup solutions with non radial symmetry for the 2 dimensional euler poisson equations based on the previous solutions with radial symmetry for the 2 dimensional isothermal euler poisson equations some special blowup solutions with non radial symmetry are constructed by the separation method

blowup for the c1 solutions of the euler poisson equations of gaseous stars in rn abstract the newtonian euler poisson equations with attractive forces are the classical models for the evolution of gaseous stars and galaxies in astrophysics in this paper we use the integration method to study the blowup problem of the n dimensional system with adiabatic exponent 1 in radial symmetry we could show that the c 1 non trivial classical solutions v with compact support in 0 r where r 0 is a positive constant with t r 0 and v t r 0 for r r under the initial condition 1 h 0 0 r r n v 0 d r 2 r 2 n n 4 m n n 1 n n 2 with an arbitrary constant n max n 2 0 and the total mass m blow up before a finite time t for pressureless fluids or 1 our results could fill some gaps about the blowup phenomena to the classical c 1 solutions of that attractive system with pressure under the first boundary condition in addition the corresponding result for the repulsive systems is also provided here our result fully covers the previous case for n 1 in m w yuen blowup for the euler and euler poisson equations with repulsive forces nonlinear anal 74 2011 1465 1470

the stable geometric dimension of vector bundles over real projective spaces an elementary argument shows that the geometric dimension of any vector bundle of order 2e over rp depends only on e and the residue of n mod 8 for n sufficiently large in this paper we calculate this geometric dimension which is approximately 2e the nonlifting results are easily obtained using the spectrum bj the lifting results require bo resolutions half of the paper is devoted to proving mahowald s theorem that beginning with the second stage bo resolutions act almost like k z2 resolutions

stable geometric dimension of vector bundles over even dimensional real projective spaces in 6 the geometric dimension of all stable vector bundles over real projective spacep n was determined ifn is even and sufficiently large with respect to the order 2 e of the bundle in f

solutions of euler poisson equations for gaseous stars in this paper we study the euler poisson equations governing gas motion under self gravitational force we are interested in the evolution of the gaseous stars for which the density function has compact support we establish existence theory for the stationary solutions and describe the behavior of the solutions near the vacuum boundary the boundary behavior thus obtained agrees with the physical boundary condition proposed and studied in l ly for both euler equations with damping and the euler poisson equations existence non existence uniqueness and instability of the stationary solutions with vacuum are also discussed in terms of the adiabatic exponent and the entropy function and the phenomena of the blowup that is the collapsing of the star to a single point with finite mass as well as the drifting of part of the star to infinity in space are also studied and shown to agree with the conjecture from the physical considerations

a class of 2 local finite spectra which admit a v21 self map abstract at the prime 2 behrens hill hopkins and mahowald showed that m 2 1 4 admits a 32 periodic v 2 self map more recently in joint work with mahowald we showed that a 1 also admits a 32 periodic v 2 self map this leads to the question of whether there exists a finite 2 local complex with periodicity less than 32 we answer this question in the affirmative by producing a class of finite 2 local spectra z all of which admit a 1 periodic v 2 self map

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

the fiber of the secondary suspension map there are several different choices for the indexing this one works nicely for both odd primes and the prime 2 in this paper we will only consider the prime 2 the groups ek are an associated graded group to lrk qs0 filtered by the sphere of origin the ehp sequence from a quite general point of view is discussed in 13 the chain complex to calculate the e2 term is given by dl elknkn ek 1 n 1 if we invert the prime 2 then d1 is an isomorphism if k n 1 and n is odd the spectral sequence collapses at e2 this gives serre s theorem that rationally s2nis just k q 2n1

unstable towers in the odd primary homotopy groups of spheres the unstable elements in filtration 2 of the unstable novikov spectral sequence are computed these elements are shown to survive to elements in the homotopy groups of spheres which are related to im j the computation is applied to determine the hopf invariants of compositions of im and the exponent of certain sphere bundles over spheres

tangential thickness of manifolds a notion of tangential thickness of a manifold is introduced an extensive calculation within the class of lens and fake lens spaces leads to complete classication of such manifolds with thickness 1 3 or 2k for k 1 on the other hand calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants

correction to the stable geometric dimension of vector bundles over real projective spaces the theory of ho resolutions as utilized in the stable geometricc dimension of vector bundles over real projective spacces did not give adequate care to the k z2 s occurring at each stage of the resolution this restricts somewhat the set of integers e for which we can prove that the geometric dimension of vector bundles of order 2e on large real projective spaces is precisely 2 e 8 1 introduction the theory of bo resolutions played a central role in obtaining the liftings in the main theorem of 2 w lellman pointed out that the analysis of bo resolutions presented in 3 and 2 did not give adequate care to k z2 s occurring at each stage of the resolution in 4 mahowald discussed a modification in the theorem of bo resolutions which takes these k z2 s into account and showed that the application to v periodic homotopy in 5 remains valid in this note we discuss the changes in 2 required by these considerations

the bo adams spectral sequence due to its relation to the image of the j homomorphism and first order periodicity bott periodicity connective real k theory is well suited for problems in 2 local stable homotopy that arise geometrically on the other hand the use of generalized homology theories in the construction of adams type spectral sequences has proved to be quite fruitful provided one is able to get a hold on the respective e2 terms in this paper we make a first attempt to construct an algebraic and computational theory of the e2 term of the bo adams spectral sequence this allows for some concrete computations which are then used to give a proof of the bounded torsion theorem of 8 as used in the geometric application of 2 the final table of the e2 term for qts in dim o where b n denotes an integral brown gitler spectrum 8 received by the editors april 5 1985 and in revised form january 31 1986 and april 7 1986 1980 mclthemclties subject clclssificcltion 1985 revision primary 55t15 55n15 55s25 55q45

formation of singularities in three dimensional compressible fluids presented are several results on the formation of singularities in solutions to the three dimensional euler equations for a polytropic ideal fluid under various assumptions on the initial data in particular it is shown that a localized fluid which is initially compressed and outgoing on average will develop singularities regardless of the size of the initial disturbance

periodic homotopy groups of exceptional lie groups torsion free cases the v 1 periodic homotopy groups v 1 751 x p are computed explicitly for all pairs x p where x is an exceptional lie group whose integral homology has no p torsion this yields new lower bounds for p exponents of actual homotopy groups of these spaces delicate calculations with the unstable novikov spectral sequence are required in the proof

odd primary periodic phenomena in the classical adams spectral sequence we study certain periodic phenomena in the cohomology of the mod p steenrod algebra which are related to the polynomial generators v e ir bp a chromatic resolution of the e2 term of the classical adams spectral sequence is constructed one of the major goals of homotopy theory is the understanding of 1r s the stable homotopy groups of spheres a technique for studying these groups is by the construction of certain systematic families of classes first due to m g barratt 3 one way to express this idea is as follows let x be a finite complex all spaces and complexes are objects in the stable category localized at a prime p a self map of degree i v ixx is nonnilpotent if the k fold composition vk vovo ov 3kix x is essential for all k 0 definition 1 for a given nonnilpotent map v a class a e 7rj so is vperiodic if a can be decomposed as st x x t 1 ayst i where x k denotes the k skeleton of x and the composite kix v kx p a st i is essential for all k 0 4 a v periodic class a e xr so determines an infinite systematic family in the following manner for each k 0 there exists an integer e with 0 0 determines a class or classes in lr ki e t i so here are several well known examples of this sort of phenomenon example 2 let mp denote the mod p moore space p 3 then adams 1 has constructed a nonnilpotent map a eymp mp where q 2 p1 this map determines a family of nontrivial classes at t 1 with at e lrqt so given by the following diagram eqtm at mp 1 1 sqt at s1 received by the editors august 28 1986 1980 mathematics subject classification 1985 revision primary 55t15 55s10 supported in part by the n s f 1988 american mathematical society 0002 9947 88 1 00 25 per page

compressible euler equations with vacuum abstract in this paper we first show that the regular solutions of compressible euler equations in r 3 with damping will not be global if the initial density function has compact support this implies that c 2 cannot be smooth across the boundary separating the gas and the vacuum after a finite time where c is the speed of sound then we study the local existence of solutions for isentropic gas flow in r when c 0 1 is smooth across using the energy method and the characteristics method

v1 and v2 periodicity in stable homotopy theory this paper we construct some self maps related to the elements v1 e 7r2 bp and v2 e 76 bp and use them to obtain families in the 2 primary stable homotopy of spheres in particular we obtain nonzero elements in the 48k 10 stem for all k 0 here bp denotes the brown peterson spectrum associated to p 2 although our results have convenient interpretations in the bp adamsnovikov spectral sequence 251 we make hardly any use of this spectral sequence except in the final step of one argument the only knowledge of bp required here is its homotopy ring our methods are those of the classical k z2 adams spectral sequence ass as we shall be working entirely in the stable category we will denote by x y the abelian group of stable homotopy classes of maps x to y when x is a finite cw complex and y a spectrum we will identify x y with eix y for any integer i and we will not distinguish between a map and its stable homotopy class if v e e x x vk denotes

the bousfield kuhn functor and topological andre quillen cohomology we construct a natural transformation from the bousfield kuhn functor evaluated on a space to the topological andre quillen cohomology of the k n local spanier whitehead dual of the space and show that the map is an equivalence in the case where the space is a sphere this results in a method for computing unstable v n periodic homotopy groups of spheres from their morava e cohomology as modules over the dyer lashof algebra of morava e theory we relate the resulting algebraic computations to the algebraic geometry of isogenies between lubin tate formal groups

the image of j in the ehp sequence the ehp sequence is based on a result of james j who showed that there is h a map h such that s q sn s2n 1 is a fibration when localized at 2 the map sn sn is usually labeled e the boundary homomorphism in the homotopy sequence is usually labeled p for our purposes it will be most convenient to combine all the ehp sequences into one system this gives the following filtration of o201 s q s

blowup for the euler and euler poisson equations with repulsive forces abstract in this paper we study the blowup of the n dim euler or euler poisson equations with repulsive forces in radial symmetry we provide a novel integration method to show that the non trivial classical solutions v with compact support in 0 r where r 0 is a positive constant and in the sense which t r 0 and v t r 0 for r r under the initial condition 1 h 0 0 r r v 0 d r 0 blow up on or before the finite time t r 3 h 0 for pressureless fluids or 1 the main contribution of this article provides the blowup results of the euler 0 or euler poisson 1 equations with repulsive forces and with pressure 1 as the previous blowup papers makino et al 1987 18 makino and perthame 1990 19 perthame 1990 20 and chae and tadmor 2008 24 cannot handle the systems with the pressure term for c 1 solutions

the energy density in the planar ising model we study the critical ising model on the square lattice in bounded simply connected domains with and free boundary conditions we relate the energy density of the model to a discrete fermionic spinor and compute its scaling limit by discrete complex analysis methods as a consequence we obtain a simple exact formula for the scaling limit of the energy field one point function in terms of the hyperbolic metric this confirms the predictions originating in physics but also provides a higher precision

the homotopy groups of the localized toda smith spectrum 1 at the prime 3 in this paper we try to compute the homotopy groups of the l2 localized toda smith spectrum v 1 at the prime 3 by using the adamsnovikov spectral sequence and have almost done so this computation involves non trivial differentials d5 and dg of the adams novikov spectral sequence different from the case p 3 we also determine the homotopy groups of some l2 localized finite spectra relating to v 1 we further show some of the non trivial differentials on elements relating so called 3 elements in the adams novikov spectral sequence for 7r s0

on the kummer congruences and the stable homotopy of u we study the torsion free part of the stable homotopy groups of the space bu by considering upper and lower bounds the upper bound is furnished by the ring pk bu of coaction primitives into which nrs bu is mapped by the complex k theoretic hurewicz homomorphism xs bu pk bu we characterize pk b u in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical kummer congruences among the bernoulli numbers for a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in bu from among these we define families of classes constructed from regular neighborhoods of embeddings of iterated thom complexes in euclidean space employing techniques of duality theory we deduce that these two families correspond except possibly in the lowest dimensions under the hurewicz homomorphism which thus provides a link between the algebra and the geometry in the course of this work we greatly extend certain e invariant calculations of j f adams

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

immersing projective spaces in euclidean space using the ko 2 theoretic obstruction theory developed in 4 and 5 necessary and sufficient conditions are derived for quaternionic projective spaces p k and odd dimensional complex projective spaces p 2k 1 of real dimension m say to immerse in euclidean space 2m 1 in the range l 14 the results refine those obtained by davis and mahowald 10 11 and earlier authors

rate of convergence for cardy s formula we show that crossing probabilities in 2d critical site percolation on the triangular lattice in a piecewise analytic jordan domain converge with power law rate in the mesh size to their limit given by the cardy smirnov formula we use this result to obtain new upper and lower bounds of e o sqrt rm log rm log r r 1 3 for the probability that the cluster at the origin in the half plane has diameter r improving the previously known estimate of r 1 3 o 1