Projective resolutions and Poincar\'e duality complexes
D. J.
Benson;
Jon F.
Carlson
447-488
Abstract: Let k be a field lof characteristic $p > 0$ and let G be a finite group. We investigate the structure of the cohomology ring ${H^\ast}(G,k)$ in relation to certain spectral sequences determined by systems of homogeneous parameters for the cohomology ring. Each system of homogeneous parameters is associated to a complex of projective kG-modules which is homotopically equivalent to a Poincaré duality complex. The initial differentials in the hypercohomology spectral sequence of the complex are multiplications by the parameters, while the higher differentials are matric Massey products. If the cohomology ring is Cohen-Macaulay, then the duality of the complex assures that the Poincaré series for the cohomology satisfies a certain functional equation. The structure of the complex also implies the existence of cohomology classes which are in relatively large degrees but are not in the ideal generated by the parameters. We consider several other questions concerned with the minimal projective resolutions and the convergence of the spectral sequence.
A variational principle in Kre\u\i n space
Paul
Binding;
Branko
Najman
489-499
Abstract: A variational characterization, involving a max-inf of the Rayleigh quotient, is established for certain eigenvalues of a wide class of definitizable selfadjoint operators Q in a Krein space. The operator Q may have continuous spectrum and nonreal and nonsemisimple eigenvalues; in particular it may have embedded eigenvalues. Various applications are given to selfadjoint linear and quadratic eigenvalue problems with weak definiteness assumptions.
Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity
Juan R.
Esteban;
Pierangelo
Marcati
501-521
Abstract: We shall consider a model problem for the fully nonlinear parabolic equation $\displaystyle {u_t} + F(x,t,u,Du,\varepsilon {D^2}u) = 0$ and we study both the approximating degenerate second order problem and the related first order equation, obtained by the limit as $\varepsilon \to 0$. The strong convergence of the gradients is provided by semiconcavity unilateral bounds and by the supremum bounds of the gradients. In this way we find solutions in the class of viscosity solutions of Crandall and Lions.
M\"untz systems and orthogonal M\"untz-Legendre polynomials
Peter
Borwein;
Tamás
Erdélyi;
John
Zhang
523-542
Abstract: The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system $\{ {x^{{\lambda _0}}},{x^{{\lambda _1}}}, \ldots \}$ with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp ${L^2}$ Markov inequality is proved.
On the solvability of systems of inclusions involving noncompact operators
P.
Nistri;
V. V.
Obukhovskiĭ;
P.
Zecca
543-562
Abstract: We consider the solvability of a system $\displaystyle \left\{ {\begin{array}{*{20}{c}} {y \in \bar F(x,y),} {x \in \bar G(x,y)} \end{array} } \right.$ of set-valued maps in two different cases. In the first one, the map $(x,y) - \circ \bar F(x,y)$ is supposed to be closed graph with convex values and condensing in the second variable and $(x,y) - \circ \bar G(x,y)$ is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case $\bar F$ is as before with compact, not necessarily convex, values and $\bar G$ is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set $x - \circ S(x)$ of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.
Noether's theorem for Hopf orders in group algebras
David M.
Weinraub
563-574
Abstract: Let K be a local field with valuation ring R of residue characteristic p containing a primitive pth root of unity $ {\zeta _p}$. We state an analog to Noether's Theorem for modules over R-Hopf algebras and use induction techniques to deduce a criterion for this analog to hold. We then construct a family of noncommutative Hopf algebras which satisfy the criterion.
Transference for radial multipliers and dimension free estimates
P.
Auscher;
M. J.
Carro
575-593
Abstract: For a large class of radial multipliers on $ {L^p}({{\mathbf{R}}^{\mathbf{n}}})$, we obtain bounds that do not depend on the dimension n. These estimates apply to well-known multiplier operators and also give another proof of the boundedness of the Hardy-Littlewood maximal function over Euclidean balls on $ {L^p}({{\mathbf{R}}^{\mathbf{n}}})$, $p \geq 2$, with constant independent of the dimension. The proof is based on the corresponding result for the Riesz transforms and the method of rotations.
Scattering theory for semilinear wave equations with small data in two space dimensions
Kimitoshi
Tsutaya
595-618
Abstract: We study scattering theory for the semilinear wave equation ${u_{tt}} - \Delta u = \vert u{\vert^{p - 1}}u$ in two space dimensions. We show that if $p > {p_0} = (3 + \sqrt {17} )/2$, the scattering operator exists for smooth and small data. The lower bound ${p_0}$ of p is considered to be optimal (see Glassey [6, 7], Schaeffer [18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at $t = 0$ in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: $\displaystyle u(x,t) = u_0^ - (x,t) + \frac{1}{{2\pi }}\int_{ - \infty }^t {\in... ...t^{p - 1}}u)(y,s)}}{{\sqrt {{{(t - s)}^2} - \vert x - y{\vert^2}} }}dy\;ds,} }$ for $t \in R$, where $ u_0^ - (x,t)$ is a solution of $ {u_{tt}} - \Delta u = 0$ which $u(x,t)$ approaches asymptotically as $t \to - \infty$. The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of $ {u_{tt}} - \Delta u = \vert u{\vert^{p - 1}}u$ in two space dimensions.
The structure of the set of singular points of a codimension $1$ differential system on a $5$-manifold
P.
Mormul;
M. Ya.
Zhitomirskiĭ
619-629
Abstract: Generic modules V of vector fields tangent to a 5-dimensional smooth manifold M, generated locally by four not necessarily linearly independent fields $ {X_1}$, ${X_2}$, ${X_3}$, ${X_4}$, are considered. Denoting by $\omega$ the 1-form $ {X_4}\lrcorner{X_3}\lrcorner{X_2}\lrcorner{X_1}\lrcorner\mathop \Omega \limits^5$ conjugated to V ( $\mathop \Omega \limits^5 $ is a fixed local volume form on M), the loci of singular behavior of $V:{M_{\deg }}(V) = \{ p \in M\vert\omega (p) = 0\}$ and ${M_{{\text{sing}}}}(V) = \{ p \in M\vert\omega \wedge {(d\omega )^2}(p) = 0\}$ are handled. The local classification of this pair of sets is carried out (outside a curve and a discrete set in ${M_{\deg }}(V)$) up to a smooth diffeomorphism. In the most complicated case, around points of a codimension 3 submanifold of M, ${M_{{\text{sing}}}}(V)$ turns out to be diffeomorphic to the Cartesian product of ${\mathbb{R}^2}$ and the Whitney's umbrella in ${\mathbb{R}^3}$.
On the Gorenstein property of Rees and form rings of powers of ideals
M.
Herrmann;
J.
Ribbe;
S.
Zarzuela
631-643
Abstract: In this paper we determine the exponents n for which the Rees ring $R({I^n})$ and the form ring ${\text{gr}}_{A}({I^n})$ are Gorenstein rings, where I is a strongly Cohen-Macaulay ideal of linear type (including complete and almost complete intersections) or an $ \mathfrak{m}$-primary ideal in a local ring A with maximal ideal $\mathfrak{m}$.
Nonlinear stability of rarefaction waves for a viscoelastic material with memory
Harumi
Hattori
645-669
Abstract: In this paper we will discuss the stability of rarefaction waves for a viscoelastic material with memory. The rarefaction waves for which the stability is tested are not themselves solutions to the integrodifferential equations (1.1) governing the viscoelastic material. They are solutions to a related equilibrium system of conservation laws given by (1.11). We shall show that if the forcing term and the past history are small and if the initial data are close to the rarefaction waves, the solutions to (1.1) will approach the rarefaction waves in sup norm as the time goes to infinity.
Gromov's compactness theorem for pseudo holomorphic curves
Rugang
Ye
671-694
Abstract: We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.
An almost strongly minimal non-Desarguesian projective plane
John T.
Baldwin
695-711
Abstract: There is an almost strongly minimal projective plane which is not Desarguesian.
Ulam-Zahorski problem on free interpolation by smooth functions
A.
Olevskiĭ
713-727
Abstract: Let f be a function belonging to $ {C^n}[0,1]$. Is it possible to find a smoother function $g \in {C^{n + 1}}$ (or at least ${C^{n + \varepsilon }}$) which has infinitely many points of contact of maximal order n with f (or at least arbitrarily many such points with fixed norm ${\left\Vert g \right\Vert _{{C^{n + \varepsilon }}}}$)? It turns out that for n = 0 and 1 the answer is positive, but if $n \geq 2$, it is negative. This gives a complete solution to the Ulam-Zahorski question on free interpolation on perfect sets.
On the discriminant of a hyperelliptic curve
P.
Lockhart
729-752
Abstract: The minimal discriminant of a hyperelliptic curve is defined and used to generalize much of the arithmetic theory of elliptic curves. Over number fields this leads to a higher genus version of Szpiro's Conjecture. Analytically, the discriminant is shown to be related to Siegel modular forms of higher degree.
A proof of $C\sp 1$ stability conjecture for three-dimensional flows
Sen
Hu
753-772
Abstract: We give a proof of the ${C^1}$ stability conjecture for three-dimensional flows, i.e., prove that there exists a hyperbolic structure over the $\Omega$ set for the structurally stable three-dimensional flows. Mañé's proof for the discrete case motivates our proof and we find his perturbation techniques crucial. In proving this conjecture we have overcome several new difficulties, e.g., the change of period after perturbation, the ergodic closing lemma for flows, the existence of dominated splitting over $\Omega \backslash \mathcal{P}$ where $\mathcal{P}$ is the set of singularities for the flow, the discontinuity of the contracting rate function on singularities, etc. Based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems, i.e., we create unstable saddle connections if there are accumulations of periodic orbits on the singularities.
Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains
Dimitry
Ioffe;
Ross
Pinsky
773-791
Abstract: Denote points $\bar x \in {R^{d + 1}}$, $d \geq 2$, by $\bar x = (\rho ,\theta ,z)$, where $\rho > 0$, $\theta \in {S^{d - 1}}$, and $z \in R$. Let $a:[0,\infty ) \to (0,\infty )$ be a nondecreasing $ {C^2}$-function and define the "horn-shaped" domain $ \Omega = \{ \bar x = (\rho ,\theta ,z):\vert z\vert < a(\rho )\}$ and its unit "cylinder" $D = \{ \bar x = (\rho ,\theta ,z) \in \Omega :\rho < 1\}$. Under appropriate regularity conditions on a, we prove the following theorem: (i) If ${\smallint ^\infty }a(\rho )/{\rho ^2}d\rho = \infty$, then the Martin boundary at infinity for $\frac{1}{2}\Delta$ in $\Omega$ is a single point, (ii) If ${\smallint ^\infty }a(\rho )/{\rho ^2}d\rho < \infty$, then the Martin boundary at infinity for $ \frac{1}{2}\Delta$ in $ \Omega$ is homeomorphic to ${S^{d - 1}}$. More specifically, a sequence $\{ ({\rho _n},{\theta _n},{z_n})\} _{n = 1}^\infty \subset \Omega $ satisfying $ {\lim _{n \to \infty }}{\rho _n} = \infty$ is a Martin sequence if and only if $ {\lim _{n \to \infty }}{\theta _n}$ exists on $ {S^{d - 1}}$. From (i), it follows that the cone of positive harmonic functions in $\Omega$ vanishing continuously on $\partial \Omega$ is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on $\Omega$ vanishing continuously on $\partial \Omega$ is generated by a collection of minimal elements which is homeomorphic to ${S^{d - 1}}$. In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for $ \frac{1}{2}\Delta$ in $ \Omega$ in the case $ a(\rho ) = 1 + {\rho ^\gamma }$, $0 < \gamma < 1$. Our method of proof involves an analysis as $\rho \to \infty$ of the exit distribution on $\partial D$ for Brownian motion starting from $(\rho ,\theta ,z) \in \Omega$ and conditioned to hit D before exiting $ \Omega$.
The theory of Jacobi forms over the Cayley numbers
M.
Eie;
A.
Krieg
793-805
Abstract: As a generalization of the classical theory of Jacobi forms we discuss Jacobi forms on $\mathcal{H} \times {\mathbb{C}^8}$, which are related with integral Cayley numbers. Using the Selberg trace formula we give a simple explicit formula for the dimension of the space of Jacobi forms. The orthogonal complement of the space of cusp forms is shown to be spanned by certain types of Eisenstein series.
A controlled plus construction for crumpled laminations
R. J.
Daverman;
F. C.
Tinsley
807-826
Abstract: Given a closed n-manifold M $(n > 4)$ and a finitely generated perfect subgroup P of $ {\pi _1}(M)$, we previously developed a controlled version of Quillen's plus construction, namely a cobordism (W, M, N) with the inclusion $ j:N \mapsto W$ a homotopy equivalence and kernel of ${i_\char93 }:{\pi _1}(M) \mapsto {\pi _1}(W)$ equalling the smallest normal subgroup of ${\pi _1}(M)$ containing P together with a closed map $ p:W \mapsto [0,1]$ such that ${p^{ - 1}}(t)$ is a closed n-manifold for every $t \in [0,1]$ and, in particular, $M = {p^{ - 1}}(0)$ and $N = {p^{ - 1}}(1)$. We accomplished this by constructing an acyclic map of manifolds $f:M \mapsto N$ having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when ${\pi _1}(M)$ contains no finitely generated perfect subgroups. By-products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.
Transfer functions of regular linear systems. I. Characterizations of regularity
George
Weiss
827-854
Abstract: We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form $\dot x(t) = Ax(t) + Bu(t)$, $y(t) = Cx(t) + Du(t)$, like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Fourés and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.
Block Jacobi matrices and zeros of multivariate orthogonal polynomials
Yuan
Xu
855-866
Abstract: A commuting family of symmetric matrices are called the block Jacobi matrices, if they are block tridiagonal. They are related to multivariate orthogonal polynomials. We study their eigenvalues and joint eigenvectors. The joint eigenvalues of the truncated block Jacobi matrices correspond to the common zeros of the multivariate orthogonal polynomials.
Classification of rank-2 ample and spanned vector bundles on surfaces whose zero loci consist of general points
Atsushi
Noma
867-894
Abstract: Let X be an n-dimensional smooth projective variety over an algebraically closed field k of characteristic zero, and E an ample and spanned vector bundle of rank n on X. To study the geometry of (X, E) in view of the zero loci of global sections of E, Ballico introduces a numerical invariant $s(E)$. The purposes of this paper are to give a cohomological interpretation of $s(E)$, and to classify ample and spanned rank-2 bundles E on smooth complex surfaces X with $ s(E) = 2{c_2}(E)$, or $2{c_2}(E) - 1$; namely ample and spanned 2-bundles whose zero loci of global sections consist of general $ {c_2}(E)$ points or general ${c_2}(E) - 1$ points plus one. As an application of these classification, we classify rank-2 ample and spanned vector bundles E on smooth complex projective surfaces with ${c_2}(E) = 2$.
Rational homotopy of the space of self-maps of complexes with finitely many homotopy groups
Samuel B.
Smith
895-915
Abstract: For simply connected CW complexes X with finitely many, finitely generated homotopy groups,$^{1}$ the path components of the function space $ M(X,X)$ of free self-maps of X are all of the same rational homotopy type if and only if all the k-invariants of X are of finite order. In case X is rationally a two-stage Postnikov system the space ${M_0}(X,X)$ of inessential self-maps of X has the structure of rational H-space if and only if the k-invariants of X are of finite order.
Multiplier Hopf algebras
A.
Van Daele
917-932
Abstract: In this paper we generalize the notion of Hopf algebra. We consider an algebra A, with or without identity, and a homomorphism $\Delta$ from A to the multiplier algebra $M(A \otimes A)$ of $A \otimes A$. We impose certain conditions on $\Delta$ (such as coassociativity). Then we call the pair $ (A,\Delta )$ a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where $(\Delta f)(s,t) = f(st)$ with $ s,t \in G$ and $ f \in A$. We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a $ \ast$-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a $\ast$-algebra.