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

rank of 3 tensors with 2 slices and kronecker canonical forms tensor type data are becoming important recently in various application fields we determine the rank of a 3 tensor with 2 slices in comparison with its kronecker canonical form over the complex and real number field

sufficient second order optimality conditions for semilinear control problems with pointwise state constraints second order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state in contrast to former publications on this subject the cone of critical directions is the smallest possible in the sense that the second order sufficient conditions are the closest to the associated necessary ones the theory is developed for elliptic distributed controls in domains up to dimension three moreover problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one respectively

the image of the thom map for eilenberg maclane spaces fundamental classes in bp cohomology of eilenberg mac lane spaces are defined the image of the thom map from bp cohomology to mod p cohomology is determined for arbitrary eilenberg mac lane spaces this image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of milnor primitives to the fundamental class in mod p cohomology this subalgebra in mod p cohomology is invariant under the action of the steenrod algebra and it is annihilated by all milnor primitives we also show that bp cohomology determines morava k cohomology for eilenberg mac lane spaces

tangent lines inflections and vertices of closed curves we show that every smooth closed curve immersed in euclidean space r3 satisfies the sharp inequality 2 p i v 6 which relates the numbers p of pairs of parallel tangent lines i of inflections or points of vanishing curvature and v of vertices or points of vanishing torsion of we also show that 2 p i v 4 where p is the number of pairs of concordant parallel tangent lines the proofs which employ curve shortening flow with surgery are based on corresponding inequalities for the numbers of double points singularities and inflections of closed curves in the real projective plane rp2 and the sphere s2 which intersect every closed geodesic these findings extend some classical results in curve theory from works of mobius fenchel and segre including arnold s tennis ball theorem

no skew branes on non degenerate hyperquadrics we show that non degenerate hyperquadrics in r n 2 admit no skew n branes stated more traditionally a compact codimension one immersed submanifold of a non degenerate hyperquadric of euclidean space must have parallel tangent spaces at two distinct points similar results have been proven by others but except for ellipsoids in r 3 proc amer math soc 133 3687 3690 2005 always under c 2 smoothness and genericity assumptions we use neither assumption here

accelerated solution of multivariate polynomial systems of equations we propose new las vegas randomized algorithms for the solution of a square nondegenerate system of equations with well separated roots the algorithms use oc delta csttn d 2 log d log b arithmetic operations in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal to approximate all real roots of the system as well as all roots lying in a fixed n dimensional box or disc here d is an upper bound on the number of all complex roots of the system e g bezout or bernshtein bound delta is the number of real roots or the roots lying in the box or disc and epsilon 2 b is the required upper bound on the output errors for computing the normal form modulo the ideal the efficient practical algorithms of b mourrain and p trebuchet in proceedings of the international symposium on symbolic and algebraic computation acm new york 2000 pp 231 238 or j c faugere j pure appl algebra 139 1999 pp 61 88 can be applied we also yield the bound oc csttn d 2 log d on the complexity of counting the numbers of all roots in a fixed box disc and all real roots for a large class of inputs and typically in practical computations the factor delta is much smaller than d delta o d this improves by the order of magnitude the known complexity estimates of the order of at least 3n d4 d3 log b or d4 which so far are the record estimates even for the approximation of a single root of a system and for each of the cited counting problems respectively our progress relies on proposing several novel techniques in particular we exploit the structure of matrices associated to a given polynomial system and relate it to the associated linear operators dual space of linear forms and normal forms of polynomials in the quotient algebra furthermore our techniques support the new nontrivial extension of the matrix sign and quadratic inverse power iterations to the case of multivariate polynomial systems where we emulate the recursive splitting of a univariate polynomial into factors of smaller degree

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

second order optimality conditions for semilinear elliptic control problems with finitely many state constraints this paper deals with necessary and sufficient optimality conditions for control problems governed by semilinear elliptic partial differential equations with finitely many equality and inequality state constraints some recent results on this topic for optimal control problems based upon results for abstract optimization problems are compared with some new results using methods adapted to the control problems meanwhile the lagrangian formulation is followed to provide the optimality conditions in the first case the lagrangian and hamiltonian functions are used in the second statement finally we prove the equivalence of both formulations

the yamabe invariant for non simply connected manifolds the yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature recently petean showed that the yamabe invariant is nonnegative for all closed simply connected manifolds of dimension 5 we extend this to show that yamabe invariant is nonnegative for all closed manifolds of dimension 5 with fundamental group of odd order having all sylow subgroups abelian the main new geometric input is a way of studying the yamabe invariant on toda brackets a similar method of proof shows that all closed oriented manifolds of dimension 5 with non spin universal cover with finite fundamental group having all sylow subgroups elementary abelian admit metrics of positive scalar curvature once one restricts to the complement of manifolds whose homology classes are toral the exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group so this proves important cases of the gromov lawson rosenberg conjecture once the dimension is sufficiently large

paraunitary filter banks over finite fields in real and complex fields unitary and paraunitary pu matrices have found many applications in signal processing there has been interest in extending these ideas to the case of finite fields we study the theory of pu filter banks fbs in gf q with q prime various properties of unitary and pu matrices in finite fields are studied in particular a number of factorization theorems are given we show that i all unitary matrices in gf q are factorizable in terms of householder like matrices and permutation matrices and ii the class of first order pu matrices the lapped orthogonal transform in finite fields can always be expressed as a product of degree one or degree two building blocks if q 2 we do not need degree two building blocks while many properties of pu matrices in finite fields are similar to those of pu matrices in complex field there are a number of differences for example unlike the conventional pu systems in finite fields there are pu systems that are unfactorizable in terms of smaller building blocks in fact in the special case of 2 spl times 2 systems all pu matrices that are factorizable in terms of degree one building blocks are diagonal matrices we derive results for both the cases of gf 2 and gf q with q 2 even though they share some similarities there are many differences between these two cases

conjugate points for variational problems with equality and inequality state constraints in this paper variational problems with equality and inequality state constraints are considered the theory of conjugate points for these problems is developed and necessary conditions for weak local optimality are derived in terms of this concept and the legendre condition for the case of inequality constraints the envelope like effect is taken into consideration in the accessory problem

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

no gap second order optimality conditions for optimal control problems with a single state constraint and control the paper deals with optimal control problems with only one control variable and one state constraint of arbitrary order we consider the case of finitely many boundary arcs and touch times we obtain a no gap theory of second order conditions allowing to characterize second order quadratic growth

new optimal z complementary code sets based on generalized paraunitary matrices the concept of paraunitary pu matrices arose in the early 1990 s in the study of multi rate filter banks these matrices have found wide applications in cryptography digital signal processing and wireless communications existing pu matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice motivated by this for the first time we introduce a novel concept called z paraunitary zpu matrix whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value we show that there exists an equivalence between a zpu matrix and a z complementary code set when the latter is expressed as a matrix with polynomial entries furthermore we investigate some important properties of zpu matrices which are useful for the extension of matrix sizes and sequence lengths we propose a new construction framework for optimal zpu matrices which includes existing pu matrices as special cases finally we give the enumeration formula for the constructed z complementary sequences from the optimal zpu matrices

typical ranks for 3 tensors nonsingular bilinear maps and determinantal ideals abstract let m n 3 m 1 n 1 2 p m n and u m n p the set r u n m of all real tensors with size u n m is one to one corresponding to the set of bilinear maps r m r n r u we show that r m n p has plural typical ranks p and p 1 if and only if there exists a nonsingular bilinear map r m r n r u we show that there is a dense open subset o of r u n m such that for any y o the ideal of maximal minors of a matrix defined by y in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal further we show that there is a dense open subset t of r n p m and continuous surjective open maps o r u p and t r u p where r u p is the set of u p matrices with entries in r such that if y t then rank t p if and only if the ideal of maximal minors of the matrix defined by y is a real prime ideal

second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations this paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition the case of boundary control is studied pointwise constraints on the control and certain equality and set constraints on the state are considered second order sufficient conditions for local optimality of controls are established

on highly regular embeddings a continuous map mathbb r d rightarrow mathbb r n is k regular if it maps any k pairwise distinct points to k linearly independent vectors our main result on k regular maps is the following lower bound for the existence of such maps between euclidean spaces in which k denotes the number of ones in the dyadic expansion of k

a block cipher cryptosystem using wavelet transforms over finite fields we present a new private key cryptosystem based on the finite field wavelet the encryption and decryption are performed by the synthesis and analysis banks of the nonlinear finite field wavelet transform whose filter coefficients are determined by the keys of the users we exploit the polyphase representation of the wavelets to introduce a shared key mechanism for the wavelet cryptosystem we propose to use wavelets that operate over gf 256 and a nonlinear device that performs a mapping on the field elements to their inverse in the field the introduced cryptographic system can operate in either stream cipher or block cipher modes depending on whether the filterbanks perform linear or circular convolution the block cipher system has a key length of 16 symbols 128 bits and an input block size of 30 symbols 240 bits to evaluate the efficiency of the developed two round wavelet cryptographic scheme we compare it with des and aes our results suggest that the wavelet cryptosystem has comparable computational complexity to aes and approximately half the complexity of des the security is tied to the length of the wavelet basis function and to the nonlinearity within the wavelet transform we study the security of the block cipher wavelet cryptosystem in response to classical attacks and those specific to this algorithm particularly those which use variations of the divide and conquer interpolation attack and discrete fourier transform techniques we show that chosen ciphertext attacks of the wavelet encryption system can reduce to the problem of solving a set of nonlinear equations over finite fields by considering existing classical and structure specific attacks we conclude that the lowest complexity of any of these attacks is greater than an exhaustive key search

lipschitzian stability for state constrained nonlinear optimal control for a nonlinear optimal control problem with state constraints we give conditions under which the optimal control depends lipschitz continuously in the l2 norm on a parameter these conditions involve smoothness of the problem data uniform independence of active constraint gradients and a coercivity condition for the integral functional under these same conditions we obtain a new nonoptimal stability result for the optimal control in the l infty norm and under an additional assumption concerning the regularity of the state constraints a new tight l infty estimate is obtained our approach is based on an abstract implicit function theorem in nonlinear spaces

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

homotopical localizations of spaces for a map f of spaces dror farjoun and the author have constructed an f localization functor where a space y is called f local when map f y is an equivalence this very general construction gives all known idempotent homotopy functors of spaces the main theorem of this paper shows that f localization functors always preserve fiber sequences of connected h spaces up to small error terms for instance the localization with respect to the n th morava k theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups this implies for example that a k 1 homology equivalence of h spaces must induce an isomorphism of v 1 periodic homotopy groups results are also obtained on the a nullification or a periodization functors which are just the f localization functors for the maps f from spaces a to points two spaces are said to have the same nullity when they give the same nullification functors and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds the a nullifications of nilpotent postnikov spaces are completely determined

skew flat fibrations a fibration of mathbb r n by oriented copies of mathbb r p is called skew if no two fibers intersect nor contain parallel directions conditions on p and n for the existence of such a fibration were given by ovsienko and tabachnikov a classification of smooth fibrations of mathbb r 3 by skew oriented lines was given by salvai in analogue with the classification of oriented great circle fibrations of s 3 by gluck and warner we show that salvai s classification has a topological variation which generalizes to characterize all continuous fibrations of mathbb r n by skew oriented copies of mathbb r p we show that the space of fibrations of mathbb r 3 by skew oriented lines deformation retracts to the subspace of hopf fibration and therefore has the homotopy type of a pair of disjoint copies of s 2 we discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of mathbb c n respectively mathbb h n by skew oriented copies of mathbb c p respectively mathbb h p

spectra of linear relations _ series and cohomology of eilenberg mac lane spaces on brown peterson cohomology groups of a space we introduce a natural inherent topology bp topology which is always complete hausdorff for any space we then construct a spectra map which calculates infinite bp linear sums convergent with respect to the bp topology and a spectrum which describes infinite sum bp linear relations in bp cohomology the mod p cohomology of this spectrum is a cyclic module over the steenrod algebra with relations generated by products of exactly two milnor primitives we show a close relationship between bp linear relations in bp cohomology and the action of the milnor primitives on mod p cohomology we prove main relations in the bp cohomology of eilenberg mac lane spaces these are infinite sum bp linear relations convergent with respect to the bp topology using bp fundamental classes we define v series which are v analogues of the p series finally we show that the above main relations come from the vn series

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

sufficient optimality conditions for optimal control subject to state constraints strong second order sufficient optimality conditions ssc are derived for optimal control problems of systems described by nonlinear odes subject to mixed control state and pure state constraints the obtained ssc are expressed in terms of a modified legendre clebsch condition and the associated riccati equation the role of ssc in stability analysis of solutions to parametric optimal control problems is briefly discussed

sensitivity analysis in calculus of variations some applications this paper deals with the problem of sensitivity analysis in calculus of variations a perturbation technique is applied to derive the boundary value problem and the system of equations that allow us to obtain the partial derivatives sensitivities of the objective function value and the primal and dual optimal solutions with respect to all parameters two examples of applications a simple mathematical problem and a slope stability analysis problem are used to illustrate the proposed method

the typical rank of tall three way arrays the rank of a three way array refers to the smallest number of rank one arrays outer products of three vectors that generate the array as their sum it is also the number of components required for a full decomposition of a three way array by candecomp parafac the typical rank of a three way array refers to the rank a three way array has almost surely the present paper deals with typical rank and generalizes existing results on the typical rank ofi j k arrays withk 2 to a particular class of arrays withk 2 it is shown that the typical rank isi when the array is tall in the sense thatjk j i jk in addition typical rank results are given for the case wherei equalsjk j

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

public key cryptography using paraunitary matrices in this paper we propose an algebraic approach for designing multivariate cryptosystems our method is based on formulating a general system of multivariate polynomial equations by paraunitary matrices these matrices are a special family of invertible polynomial matrices that can be completely parameterized and efficiently generated by primitive building blocks using the general formulation that involves paraunitary matrices we design a one way function that operates over the fields of characteristic two in order to include a trapdoor we make some approximations to the paraunitary matrix the result is a trapdoor one way function that is efficient to evaluate but hard to invert unless secret information about the trapdoor is known using this function we propose a paraunitary asymmetric cryptosystem pac we present an instance of the pac and show how it can be efficiently implemented this instance operates on the finite field gf 256 the message block consists of 16 to 32 symbols from gf 256 i e the block size n is an integer between 16 and 32 the ciphertext block takes its elements from the same field and has at least ten extra symbols we show that the encryption and decryption can be efficiently performed with complexities o n3 and o n2 respectively where n is the size of the message block comparing complexities of the pac to those in the hidden field equation hfe family we show that the pac is faster in public key generation and decryption we study the computational security of the pac in addition we show that the attacks developed for the hfe are not applicable on the pac

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

simplicity of core arrays in three way principal component analysis and the typical rank of p q 2 arrays abstract interpreting the solution of a principal component analysis of a three way array is greatly simplified when the core array has a large number of zero elements the possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand and by mathematical analysis on the other in the present paper it is shown that a p q 2 array with p q 2 can almost surely be transformed to have all but 2 q elements zero it is also shown that arrays of that form have three way rank p at most this has direct implications for the typical rank of p q 2 arrays also when p q when p 2q the typical rank is 2 q when q it is p and when p q the rank is typically almost surely p or p 1 these typical rank results pertain to the decomposition of real valued three way arrays in terms of real valued rank one arrays and do not apply in the complex setting where the typical rank of p q 2 arrays is also min p 2q when p q but it is p when p q

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

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

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

ranks of tensors secant varieties of segre varieties and fat points abstract a classical unsolved problem of projective geometry is that of finding the dimensions of all the higher secant varieties of the segre embeddings of an arbitrary product of projective spaces an important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t spaces fills the ambient projective space in this paper we give a new approach to these problems the crux of our method is the translation of a well known lemma of terracini into a question concerning the hilbert function of fat points in a multiprojective space our approach gives much new information on the classical problem even in the case of three factors a case also studied in the area of algebraic complexity theory

on the cohomology of elementary abelian groups the structure of the bp cohomology of elementary abelian p groups is studied obtaining a presentation expressed in terms of bp cohomology and mod p singular cohomology using the milnor derivations the arguments are based on a result on multi koszul complexes which is related to margolis s criterion for freeness of a graded module over an exterior algebra

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

second order necessary and sufficient optimality conditions for optimization problems and applications to control theory this paper deals with a class of nonlinear optimization problems in a function space where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type second order necessary and sufficient optimality conditions are established where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces the results are applied to some optimal control problems for ordinary and partial differential equations

introducing totally nonparallel immersions abstract an immersion of a smooth n dimensional manifold m r q is called totally nonparallel if for every distinct x y m the tangent spaces at f x and f y contain no parallel lines equivalently they span a 2n dimensional space given a manifold m we seek the minimum dimension tn m such that there exists a totally nonparallel immersion m r tn m in analogy with the totally skew embeddings studied by ghomi and tabachnikov we find that totally nonparallel immersions are related to the generalized vector field problem the immersion and embedding problems for real projective spaces and nonsingular symmetric bilinear maps our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space f k m of k tuples of distinct points of m for example k regular embeddings k skew embeddings k neighborly embeddings and several others typically a map satisfying one of these configuration space conditions induces some s k equivariant map on the configuration space f k m or on a bundle thereof and obstructions can be computed in the form of stiefel whitney classes however the existence problem for such conditions is relatively unstudied our main result is a whitney type theorem every smooth n manifold m admits a totally nonparallel immersion into r 4 n 1 one dimension less than given by genericity we begin by studying the local problem which requires a thorough understanding of the space of nonsingular symmetric bilinear maps after which the main theorem is established using the removal of singularities h principle technique due to gromov and eliashberg when combined with a recent non immersion theorem of davis we obtain the exact value tn r p n 4 n 1 when n is a power of 2 this is the first optimal dimension result for any closed manifold m besides s 1 for any of the recently studied configuration space conditions

on the nonimmersion of products of real projective spaces in this paper we utilize bp a generalized cohomology theory associated with the brown peterson spectrum to prove a nonimmersion theorem for products of real projective spaces 1 statement of results let v 2a 2b 1 a and let a n denote the number of l s in the binary expansion of n let pn denote the n dimensional real projective space then our main theorem is theorem 1 1 if d and k are positive integers with a m 1 a mk d k and 2v m d 2k j 1 for each i then p2mi 2d p2m2 4d x xp 2kd cannot be immersed in r4 mi mk 2d k i to our knowledge the only published works on nonimmersions of products of real projective spaces are those of suzuki 18 and kobayashi 12 their methods seem to be effective for pm x pn with m 2r1 and n 2s 1 when s r 7 their nonimmersions are in dimensions m n 2s 1 see 12 corollary 4 we can apply theorem 1 1 to the case ml 2 2 2r 3 and 2 s3 m2 2s 2 when r 3 and s 3 are greater than 14 and we have k 2 and d 2 this manifold is a submanifold of theirs and it fails to immerse indimension m 2s i 2r5 which is an improvement of 2r 5 dimensions a more detailed analysis 17 of this special case lowers the above 14 to 4 from known immersions of single real projective spaces e g adem 2 lam 1 3 cohen 7 and randall 1 5 we see that nonimmersions in theorem 1 1 received by the editors july 18 1988 1980 mathematics subject classification 1985 revision primary 57r42 55n20 kei words and phrases real projective space bp theory euler class immersion the first author was partially supported by kosef the second author was partially supported by the n s f d 1990 american mathematical society 0002 9947 90 1 00 25 per page

krull dimension and monomial orders abstract we introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence our main result shows that the krull dimension of a noetherian ring is equal to the supremum of the length of independent sequences the proof has led to other notions of independent sequences which have interesting applications for example we can show that dim r 0 j is the maximum number of analytically independent elements in an arbitrary ideal j of a local ring r and that dim b dim a if b a are not necessarily finitely generated subalgebras of a finitely generated algebra over a noetherian jacobson ring

two band wavelets and filterbanks over finite fields with connections to error control coding recently we have developed a new framework to study error control coding using finite field wavelets and filterbanks fbs this framework reveals a rich set of signal processing techniques that can be exploited to investigate new error correcting codes and to simplify encoding and decoding techniques for some existing ones the paper introduces the theory of wavelet decompositions of signals in vector spaces defined over galois fields to avoid the limitations of the number theoretic fourier transform our wavelet transform relies on a basis decomposition in the time rather than the frequency domain first by employing a symmetric nondegenerate canonical bilinear form we obtain a necessary and sufficient condition that the basis functions defined over finite fields must satisfy in order to construct an orthogonal wavelet transform then we present a design methodology to generate the mother wavelet and scaling function over finite fields by relating the wavelet transform to two channel paraunitary pu fbs finally we describe the application of this transform to the construction of error correcting codes in particular we give examples of double circulant codes that are generated by wavelets

theory of paraunitary filter banks over fields of characteristic two motivated by our wavelet framework for error control coding we proceed to develop an important family of wavelet transforms over finite fields paraunitary pu filter banks that are realizations of orthogonal wavelets by multirate filters are an important subclass of perfect reconstruction pr filter banks a parameterization of pu filter banks that covers all possible pu systems is very desirable in error control coding because it provides a framework for optimizing the free parameters to maximize coding performance this paper undertakes the problem of classifying all pu matrices with entries from a polynomial ring where the coefficients of the polynomials are taken from finite fields it constructs householder transformations that are used as elementary operations for the realization of all unitary matrices then it introduces elementary pu building blocks and a factorization technique that is specialized to obtain a complete realization for all pu filter banks over fields of characteristic two this is proved for the 2 spl times 2 case and conjectured for the m spl times m case where m spl ges 3 using these elementary building blocks we can construct all pu filter banks over fields of characteristic two these filter banks can be used to implement transforms which in turn provide a powerful new perspective on the problems of constructing and decoding arbitrary rate error correcting codes

second order sufficient conditions for optimal control problems with free final time the riccati approach second order sufficient conditions ssc for control problems with control state constraints and free final time are presented instead of deriving such ssc from first principles we transform the control problem with free final time into an augmented control problem with fixed final time for which well known ssc exist ssc are then expressed as a condition on the positive definiteness of the second variation a convenient numerical tool for verifying this condition is based on the riccati approach where one has to find a bounded solution of an associated riccati equation satisfying specific boundary conditions the augmented riccati equations for the augmented control problem are derived and their modifications on the boundary of the control state constraint are discussed two numerical examples 1 the classical earth mars orbit transfer in minimal time and 2 the rayleigh problem in electrical engineering demonstrate that the riccati equation approach provides a viable numerical test of ssc

on the generic and typical ranks of 3 tensors abstract we study the generic and typical ranks of 3 tensors of dimension l m n using results from matrices and algebraic geometry we state a conjecture about the exact values of the generic rank of 3 tensors over the complex numbers which is verified numerically for l m n 14 we also discuss the typical ranks over the real numbers and give an example of an infinite family of 3 tensors of the form l m n m 1 2 1 m 3 4 which have at least two typical ranks

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

well posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control this paper deals with the shooting algorithm for optimal control problems with a scalar control and a regular scalar state constraint additional conditions are displayed under which the so called alternative formulation is equivalent to pontryagin s minimum principle the shooting algorithm appears to be well posed invertible jacobian iff i the no gap second order sufficient optimality condition holds and ii when the constraint is of order q geq 3 there is no boundary arc stability and sensitivity results without strict complementarity at touch points are derived using robinson s strong regularity theory under a minimal second order sufficient condition the directional derivatives of the control and state are obtained as solutions of a linear quadratic problem

shadows and convexity of surfaces we study the geometry and topology of immersed surfaces in euclidean 3 space whose gauss map satisfies a certain two piece property and solve the shadow problem formulated by h wente

typical ranks for m n m 1 n tensors with m n abstract in various application fields tensor type data are used recently and then a typical rank is important there may be more than one typical ranks over the real number field it is well known that the set of 2 n n tensors has two typical ranks n n 1 for n 2 that the set of 3 4 8 tensors has two typical ranks 8 9 and that the set of 4 4 12 tensors has two typical ranks 12 13 in this paper we show that the set of m n m 1 n tensors with m n has two typical ranks m 1 n m 1 n 1 if m n where is the hurwitz radon function defined as n 2 b 8 c for nonnegative integers a b c such that n 2 a 1 2 b 4 c and 0 b 4

pontryagin s principle for local solutions of control problems with mixed control state constraints this paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state control constraints we obtain necessary optimality conditions in the form of a pontryagin s minimum principle for local solutions in the sense of l p p leq infty

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

a proof of the conner floyd conjecture conjecture in is the annihilator ideal of tn the conjecture was proved forp odd by ravenel and wilson in 4 in this paper we give a much shorter proof valid for all primes briefly the proof runs as follows first of all we may replace mu by bp and vi by vi where bp z p v1 v2 and vo p moreover it is easy to see that in annihilates tn hence it is enough to show that the resulting map bp in bp tn is injective in section 1 we describe a comparatively small spectrum m n and a map f bp b z p n bp enm n such that f tn i 0 in section 2 we use the adams spectral sequence to show that bp m n is vn vn 1 9 torsion free which completes the proof aside from the adams spectral sequence the only result of any depth required for the proof is nakaoka s calculation 3 of the cohomology of symmetric products

topological obstructions to totally skew embeddings following ghomi and tabachnikov s 2008 work we study the invariant n m n defined as the smallest dimension n such that there exists a totally skew embedding of a smooth manifold m n in r n this problem is naturally related to the question of estimating the geometric dimension of the stable normal bundle of the configuration space f 2 m n of ordered pairs of distinct points in m n we demonstrate that in a number of interesting cases the lower bounds on n m n obtained by this method are quite accurate and very close to the best known general upper bound n m n 1 n m n 4n 2 alpha n 1