Z4-Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets

When m is odd, spreads in an orthogonal vector space of type⁺(2m + 2,2) are related to binary Kerdock codes and extremalline-sets in 2^{m + 1} with prescribed angles. Spreads in a 2m-dimensionalbinary symplectic vector space are related to Kerdock codesover Z₄ and extremal line-sets in with prescribed angles. These connections involve binary, realand complex geometry associated with extraspecial 2-groups.A geometric map from symplectic to orthogonal spreads is shownto induce the Gray map from a corresponding Z₄-Kerdock codeto its binary image. These geometric considerations lead tothe construction, for any odd composite m, of large numbersof Z₄-Kerdock codes. They also produce new Z₄-linear Kerdockand Preparata codes. 1991 Mathematics Subject Classification:primary 94B60; secondary 51M15, 20C99.     

K-Theory for Algebras of Operators on Banach Spaces

It is proved that, for each pair (m, n) of non-negative integers,there is a Banach space X for which K₀(B(X))Z^m and K₁(B(X))Zⁿ.The K-groups of all closed ideals of operators contained inthe ideal of strictly singular operators are computed, and someresults about the existence of splittings of certain short exactsequences are derived.     

Piecewise Linear Non-Representability of Out {pi}1 M

For every 6, examples are given of closed negatively curvedriemannian n-manifolds for which the natural map PLM Out ₁Mis not an epimorphism. PLM denotes the group of piecewise linearhomeomorphisms of M. 2000 Mathematics Subject Classification53C20, 57Q25, 57R55.     

Linear Automorphisms that are Symplectomorphisms

Let K be the field of real or complex numbers. Let (X K²ⁿ,) be a symplectic vector space and take 0 < k < n,N =. Let L₁,...,L_N X be 2k-dimensionallinear subspaces which are in a sufficiently general position.It is shown that if F : X X is a linear automorphism whichpreserves the form ^k on all subspaces L₁,...,L_N, then F is an_k-symplectomorphism (that is, F^* = _k, where ). In particular, if K = R and k is odd then F mustbe a symplectomorphism. The unitary version of this theoremis proved as well. It is also observed that the set A_l,2r ofall l-dimensional linear subspaces on which the form has rank 2r is linear in the Grassmannian G(l,2n), that is, there isa linear subspace L such that A_l,2r = L G(l, 2n). In particular,the set A_l,2r can be computed effectively. Finally, the notionof symplectic volume is introduced and it is proved that itis another strong invariant.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity

This paper is concerned with the global existence, exponentialstability of solutions and associated nonlinear C₀-semigroupas well as the existence of maximal attractors in Hⁱ (i = 1,2, 4) for a nonlinear one-dimensional thermoviscoelasticitydescribing a kind of solid-like material. Some new ideas andmore delicated estimates are employed to prove the global existenceand exponential stability of solutions. The important featurefor the existence of maximal attractors in Hⁱ₊ (i = 1, 2, 4)is that the metric spaces H¹₊, H²₊ and H⁴₊ we work with arethree incomplete metric spaces, as can be seen from the physicalconstraints, i.e. > 0 and u > 0, with and u being absolutetemperature and deformation gradient (strain). For any positiveparameters ₁, ₂, ..., ₅ verifying some conditions, a sequenceof closed subspaces Hⁱ Hⁱ+ (i = 1, 2, 4) is found, and theexistence of maximal attractors in Hⁱ (i = 1, 2, 4) is established.     

Determination of a Convex Body from Minkowski Sums of its Projections

For a convex body K in R^d and 1 K d – 1, let P_K (K)be the Minkowski sum (average) of all orthogonal projectionsof K onto k-dimensional subspaces of R^d. It is Known that theoperator P_k is injective if kd/2, k=3 for all d, and if k =2, d 14. It is shown that P_2k (K) determines a convex body K among allcentrally symmetric convex bodies and P_2k+1(K) determines aconvex body K among all bodies of constant width. Correspondingstability results are also given. Furthermore, it is shown thatany convex body K is determined by the two sets P_k (K) and P_k'(K) if 1 < k < k'. Concerning the range of P_k , 1 k d–2, it is shown that its closure (in the Hausdorff-metric)does not contain any polytopes other than singletons.     

The Natural Morphisms between Toeplitz Algebras on Discrete Groups

Let G be a discrete group and (G, G₊) be a quasi-ordered group.Set G₊(G₊)^–1 and G₁= (G₊\){e}. Let F^G₁(G) andF^G₊(G) be the corresponding Toeplitz algebras. In the paper,a necessary and sufficient condition for a representation ofF^G₊(G) to be faithful is given. It is proved that when G isabelian, there exists a natural C*-algebra morphism from F^G₁(G)to F^G₊(G). As an application, it is shown that when G = Z² andG₊ = Z₊ x Z, the K-groups K₀(F^G₁(G)) Z², K₁(F^G₁(G)) Z andall Fredholm operators in F^G₁(G) are of index zero.     

The homotopy invariance of the string topology loop product and string bracket

Let Mⁿ be a closed, oriented, n-manifold, and LM its free loopspace. In [Chas and Sullivan, ‘String topology’,Ann. of Math., to appear] a commutative algebra structure inhomology, H_*(LM), and a Lie algebra structure in equivarianthomology , were defined.In this paper, we prove that these structures are homotopy invariantsin the following sense. Let f:M₁ M₂ be a homotopy equivalenceof closed, oriented n-manifolds. Then the induced equivalence,Lf:LM₁ LM₂ induces a ring isomorphism in homology, and an isomorphismof Lie algebras in equivariant homology. The analogous statementalso holds true for any generalized homology theory h_* thatsupports an orientation of the M_i. Received February 5, 2007.     

Construction of p-1 Irreducible Modules with Fundamental Highest Weight for the Symplectic Group in Characteristic p

Let K be a field and let V be a vector space of dimension 2mover K. Let V denote the exterior algebra of V and ^kV its kthexterior power for 0k2m. Let f be a non-degenerate alternatingbilinear form defined on VxV. The symplectic group Sp_2m(K) isthe group of all isometries of f and it acts as a group of vectorspace automorphisms on ^kV. In the case that K is algebraicallyclosed and 1km, it is known that ^kV contains a composition factorcorresponding to the fundamental weight _k of a root system oftype C_m. We shall refer to the irreducible module for Sp_2m(K)given by this composition factor as a fundamental module.     

Connected Sums of 4-Manifolds as Codimension-k Fibrators

The main result provides mild conditions under which a closed,orientable, PL 4-manifold N = N₁#N₂ with ₁(N_e) residually finite(e=1,2) is a codimension-5 PL fibrator. The paper also presentsa rich variety of conditions on a closed 4-manifold N⁴ underwhich every PL map between manifolds, where the domain is orientableand all point inverses are copies of N⁴, must be an approximatefibration.     

Convex Bodies with Homothetic Sections

We prove that if K is a convex body in Eⁿ⁺¹, n2, and p₀ is apoint of K with the property that all n-sections of K throughp₀ are homothetic, then K is a Euclidean ball.     

On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E/K overa number field K satisfying the property that there exist P₁,P₂ K[t] such that the twists and are of positiverank over K(t). As a consequence of this result on twists, weshow that for those elliptic curves E/K, and for each , the rank of E over the fixed field (K^ab) under is infinite, where K^ab is the maximal abelian extension ofK.     

ON THE INTEGRAL GALOIS MODULE STRUCTURE OF CYCLIC EXTENSIONS OF p-ADIC FIELDS

We strengthen results of Miyata on the integral Galois modulestructure of totally ramified cyclic Kummer extensions K ofdegree pⁿ of a p-adic field k. Let c₁(K/k) be the first ramificationnumber of K/k, and let c(K/k) be the least non-negative residueof c₁(K/k) modulo pⁿ. Suppose that K is of the form k() with^pn k and val _K(–1)>0, (val _K(–1), p)= 1. Thenthe valuation ring of K is free over its associated order ifc(K/k) divides p^m–1 for some m with 1mn; the converseholds if n= 2; and is a Hopf order (or a Gorenstein order)if and only if c(K/k) = pⁿ–1.     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

Rank of Elliptic Curves over almost Separably Closed Fields

Let E be an elliptic curve over a finitely generated infinitefield K. Let K^s denote a separable closure of K, an elementof the Galois group G_K=Gal(K^s/K), and K^s() the invariant subfieldof K^s. If the characteristic of K is not 2 and belongs to asuitable open subgroup of G_K, then E(K^s()) has infinite rank.2000 Mathematics Subject Classification 11G05.     

Small volume closed hyperbolic 4-manifolds

Further examples of non-orientable compact hyperbolic 4-manifoldsof volume 32²/3 arising from torsion-free subgroups of the [5,3, 3, 3] Coxeter group are given. These are the smallest knownclosed hyperbolic 4-manifolds and arise by consideration ofmaps from the [5, 3, 3, 3] Coxeter group onto the simple simplecticgroup S₄(4).     

Overconvergent Real Closed Quantifier Elimination

Let K be the (real closed) field of Puiseux series in t overR endowed with the natural linear order. Then the elements ofthe formal power series rings [[₁,...,_n]] converge t-adicallyon [–t,t]ⁿ, and hence define functions [–t,t]ⁿ K. Let L be the language of ordered fields, enriched with symbolsfor these functions. By Corollary 3.15, K is o-minimal in L.This result is obtained from a quantifier elimination theorem.The proofs use methods from non-Archimedean analysis. 2000 MathematicsSubject Classification 03C64, 32P05, 32B05, 32B20, 03C10 (primary),03C98, 03C60, 14P15 (secondary).     

Decay rates of solutions to thermoviscoelastic plates with memory

We consider the thermoelastic plate under the presence of along range memory. We find uniform rates of decay (in time)of the energy, provided that suitable assumptions on the relaxationfunctions are given. Namely, if the relaxation decays exponentiallythen the first order energy also decays exponentially. Whenthe relaxation g satisfies -c₁g(t)^1+1/_p g'(t) -c_og(t)^1+1/_p; and g,g^1-1/_p L¹ (R) withp > 2 then the energy decays as 1/(1+t)^p. A new Liapunov functionalis built for this problem.     

Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture

Smale's mean value conjecture asserts that for every polynomial P of degree d satisfying P(0)=0,where K = (d–1)/d and the minimum is taken over all criticalpoints of P. A stronger conjecture due to Tischler assertsthat with . Tischler's conjecture is known to be true: (i) for local perturbations of the extremumP₀(z)=z^d – dz, and (ii) for all polynomials of degreed 4. In this paper, Tischler's conjecture is verified for alllocal perturbations of the extremum P₁(z)=(z – 1)^d –(–1)^d, but counterexamples to the conjecture are givenin each degree d 5. In addition, estimates for certain weightedL¹- and L²-averages of the quantities are established, which lead to the best currentlyknown value for K₁ in the case d=5. 2000 Mathematics SubjectClassification 30C15.