Ideals associated to deformations of singular plane curves
Steven
Diaz;
Joe
Harris
433-468
Abstract: We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus $ES$ which parametrizes equisingular or topologically trivial deformations, the equigeneric locus $ EG$ which parametrizes deformations of constant geometric genus, and the equiclassical locus $EC$ which parametrizes deformations of constant geometric genus and class. (The class of a reduced plane curve is the degree of its dual.) It was previously known that the tangent space to $ES$ corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to $EG$ corresponds to the conductor ideal. We show that the support of the tangent cone to $EC$ corresponds to an ideal which we call the equiclassical ideal. By studying these ideals we are able to obtain information about the geometry and dimensions of $ES$, $EC$, and $EG$. This allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities.
Definable sets in ordered structures. III
Anand
Pillay;
Charles
Steinhorn
469-476
Abstract: We show that any $ o$-minimal structure has a strongly $o$-minimal theory.
Positive quadratic differential forms and foliations with singularities on surfaces
Víctor
Guíñez
477-502
Abstract: To every positive $ {C^r}$-quadratic differential form defined on an oriented two manifold is associated a pair of transversal one-dimensional $ {C^r}$-foliations with common singularities. An open set of positive $ {C^r}$-quadratic differential forms with structural stable associated foliations is characterized and it is proved that this set is dense in the space of positive $ {C^\infty }$-quadratic differential forms with ${C^2}$-topology. Also a realization theorem is established.
$v\sb 1$-periodic ${\rm Ext}$ over the Steenrod algebra
Donald M.
Davis;
Mark
Mahowald
503-516
Abstract: For a large family of modules $M$ over the $\bmod 2$ Steenrod algebra $A$, $\operatorname{Ext} _A^{s,t}(M,\,{{\mathbf{Z}}_2})$ is periodic for $t < 4s$ with respect to operators $v_1^{2n}$ of period $({2^n},\,3 \cdot {2^n})$ for varying $n$. $v_1^{ - 1}\operatorname{Ext} _A^{s,t}(M,\,{{\mathbf{Z}}_2})$ can be defined by extending this periodic behavior outside this range. We calculate this completely when $M = {H^{\ast}}(Y)$, where $Y$ is the suspension spectrum of ${\mathbf{R}}{P^2} \wedge {\mathbf{C}}{P^2}$.
Mixed norm estimates for certain means
Lennart
Börjeson
517-541
Abstract: We obtain estimates of the mean $\displaystyle F_x^\gamma (t) = {C_\gamma }\int_{\vert y\vert < 1} {{{(1 - \vert y{\vert^2})}^\gamma }f(x - ty)\,dy}$ in mixed Lebesgue and Sobolev spaces. They generalize earlier estimates of the spherical mean $F_x^{ - 1}(t) = C\;\int_{{S^{n - 1}}} {f(x - ty)\,dS(y)}$ and of solutions of the wave equation ${\Delta _x}u = {\partial ^2}u/\partial {t^2}$.
The minimal model of the complement of an arrangement of hyperplanes
Michael
Falk
543-556
Abstract: In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let ${\mathbf{A}}$ be a finite collection of hyperplanes in $ {{\mathbf{C}}^l}$, and let $M = {{\mathbf{C}}^l} - \bigcup\nolimits_{H \in {\mathbf{A}}} H$. We say ${\mathbf{A}}$ is a rational $K(\pi ,\,1)$ arrangement if the rational completion of $M$ is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of $ {\pi _1}(M)$ to the cohomology of $M$. This identity was established by group-theoretic means for the class of fiber-type arrangements in previous work. We reproduce this result by showing that the class of rational $K(\pi ,\,1)$ arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types ${A_l}$ and ${B_l}$. There is much interest in arrangements for which $M$ is a $K(\pi ,\,1)$ space. The methods developed here do not apply directly because $M$ is rarely a nilpotent space. We give examples of $K(\pi ,\,1)$ arrangements which are not rational $K(\pi ,\,1)$ for which the LCS formula fails, and $K(\pi ,\,1)$ arrangements which are not rational $K(\pi ,\,1)$ where the LCS formula holds. It remains an open question whether rational $K(\pi ,\,1)$ arrangements are necessarily $K(\pi ,\,1)$.
Bounds on the $L\sp 2$ spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality
Gregory F.
Lawler;
Alan D.
Sokal
557-580
Abstract: We prove a general version of Cheeger's inequality for discrete-time Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also prove a version of Cheeger's inequality for Markov chains and processes with killing. As an application, we prove $ {L^2}$ exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs.
Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants
Allan
Berele
581-589
Abstract: We prove Razmyslov's theorem on trace identities for ${M_{k,\,l}}$ using the invariant theory of $ \operatorname{pl} (k,\,l)$.
On the nonlinear eigenvalue problem $\Delta u+\lambda e\sp u=0$
Takashi
Suzuki;
Ken’ichi
Nagasaki
591-608
Abstract: The structure of the set $ \mathcal{C}$ of solutions of the nonlinear eigenvalue problem $\Delta u + \lambda {e^u} = 0$ under Dirichlet condition in a simply connected bounded domain $ \Omega$ is studied. Through the idea of parametrizing the solutions $(u,\,\lambda )$ in terms of $s = \lambda \,\int_\Omega {{e^u}\,dx}$, some profile of $\mathcal{C}$ is illustrated when $ \Omega$ is star-shaped. Finally, the connectivity of the branch of Weston-Moseley's large solutions to that of minimal ones is discussed.
Locally bounded sets of holomorphic mappings
José
Bonet;
Pablo
Galindo;
Domingo
García;
Manuel
Maestre
609-620
Abstract: Several results and examples about locally bounded sets of holomorphic mappings defined on certain classes of locally convex spaces (Baire spaces, $(DF)$-spaces, $C(X)$-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theory is considered.
Stochastic perturbations to conservative dynamical systems on the plane. I. Convergence of invariant distributions
G.
Wolansky
621-639
Abstract: We consider a nonlinear system on the plane, given by an oscillator with homoclinic orbits. The above system is subjected to a perturbation, composed of a deterministic part and a random (white noise) part. Assuming the existence of a finite, invariant measure to the perturbed system, we deal with the convergence of the measures to a limit measure, as the perturbation parameter tends to zero. The limit measure is constructed in terms of the action function of the unperturbed oscillator, and the strong local ${L_2}$ convergence of the associated densities is proved.
Stochastic perturbations to conservative dynamical systems on the plane. II. Recurrency conditions
G.
Wolansky
641-657
Abstract: We consider a conservative system on the plane, subjected to a perturbation. The above perturbation is composed of a deterministic part and a random (white noise) part. We discuss the conditions under which there exists a unique, finite invariant measure to the perturbed system, and the weak compactness of the above measures for small enough perturbation's parameter.
A canonical subspace of $H\sp *(B{\rm O})$ and its application to bordism
Errol
Pomerance
659-670
Abstract: A particularly nice canonical subspace of $ {H^{\ast}}(BO)$ is defined. The bordism class of a map $f:X \to Y$, where $X$ and $Y$ are compact, closed manifolds, can be determined by the characteristic numbers corresponding to elements of this subspace, and these numbers can be easily calculated. As an application, we study the "fixed-point manifold" of a parameter family of self-maps $F:M \times X \to X$, thus refining to bordism the usual homological analysis of the diagonal which is the basis of the standard Lefschetz fixed point theorem.
A Stone-type representation theorem for algebras of relations of higher rank
H.
Andréka;
R. J.
Thompson
671-682
Abstract: The Stone representation theorem for Boolean algebras gives us a finite set of equations axiomatizing the class of Boolean set algebras. Boolean set algebras can be considered to be algebras of unary relations. As a contrast here we investigate algebras of $n$-ary relations (originating with Tarski). The new algebras have more operations since there are more natural set theoretic operations on $n$-ary relations than on unary ones. E.g. the identity relation appears as a new constant. The Resek-Thompson theorem we prove here gives a finite set of equations axiomatizing the class of algebras of $ n$-ary relations (for every ordinal $n$).
Translates of exponential box splines and their related spaces
Asher
Ben-Artzi;
Amos
Ron
683-710
Abstract: Exponential box splines ($EB$-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space $ \mathcal{H}$ spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space $ {\mathbf{H}}$ of all entire functions spanned by the integer translates of an $ EB$-spline. This investigation relies on a detailed description of $\mathcal{H}$ and its discrete analog $\mathcal{S}$. The approach taken here is based on the observation that the structure of $\mathcal{H}$ is relatively simple when $\mathcal{H}$ is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex $EB$-splines. The first is when the Fourier transform of the $EB$-spline vanishes at some critical points, the second is when $ \mathcal{H}$ cannot be embedded in $ \mathcal{S}$ and the third is when $ {\mathbf{H}}$ is a proper subspace of $ \mathcal{H}$. We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.
Algebras on the disk and doubly commuting multiplication operators
Sheldon
Axler;
Pamela
Gorkin
711-723
Abstract: We prove that a bounded analytic function $f$ on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by $ {H^\infty }$ and $\overline f$ does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if $ f$ and $g$ are bounded analytic functions on the unit disk such that the commutator $ {T_f}T_g^{\ast} - T_g^{\ast}{T_f}$ (here ${T_f}$ denotes the operator of multiplication by $f$ on the Bergman space of the disk) is compact, then $(1 - \vert z{\vert^2})\min \{ \vert f' (z)\vert,\;\vert g' (z)\vert\} \to 0$ as $\vert z\vert \uparrow 1$.
Field theories in the modern calculus of variations
Andrzej
Nowakowski
725-752
Abstract: Two methods of construction of fields of extremals ("geodesic coverings") in the generalized problem of Bolza are given and, as a consequence, sufficient conditions for optimality in a form similar to Weierstrass' are formulated. The first field theory is an extension of Young's field theory-- "concourse of flights" for our problem; the other describes a nonclassical treatment of field theory which allows one to reject the "self-multiplier restriction".
Volumes of small balls on open manifolds: lower bounds and examples
Christopher B.
Croke;
Hermann
Karcher
753-762
Abstract: Question: "Under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point?" Two theorems establish such assumptions and two examples sharply limit their weakening. In particular we give an example of a metric on $ {{\mathbf{R}}^4}$ (extending to higher dimensions) of positive Ricci curvature, whose sectional curvatures decay to 0, and such that the volume of balls goes uniformly to 0 as the center goes to infinity.
Regularity of weak solutions of parabolic variational inequalities
William P.
Ziemer
763-786
Abstract: In this paper, parabolic operators of the form $\displaystyle {u_t} - \operatorname{div} A(x,\,t,\,u,\,Du) - B(x,\,t,\,u,\,Du)$ are considered where $A$ and $B$ are Borel measurable and subject to linear growth conditions. Let $\psi :\,\Omega \to {R^1}$ be a Borel function bounded above (an obstacle) where $\Omega \subset {R^{n + 1}}$. Let $u \in {W^{1,2}}(\Omega )$ be a weak solution of the variational inequality in the following sense: assume that $u \geqslant \psi $ q.e. and $\displaystyle \int_\Omega {{u_t}\varphi + A \cdot D\varphi - B\varphi \geqslant 0}$ whenever $\varphi \in W_0^{1,2}(\Omega )$ and $\varphi \geqslant u - \psi$ q.e. Here q.e. means everywhere except for a set of classical parabolic capacity. It is shown that $u$ is continuous even though the obstacle may be discontinuous. A mild condition on $\psi$ which can be expressed in terms of the fine topology is sufficient to ensure the continuity of $ u$. A modulus of continuity is obtained for $u$ in terms of the data given for $\psi$.
The box product of countably many copies of the rationals is consistently paracompact
L. Brian
Lawrence
787-796
Abstract: By proving the theorem stated in the title, we show that local compactness in the factor spaces is not necessary for paracompactness in the box product.
Fast algorithms for multiple evaluations of the Riemann zeta function
A. M.
Odlyzko;
A.
Schönhage
797-809
Abstract: The best previously known algorithm for evaluating the Riemann zeta function, $ \zeta (\sigma + it)$, with $ \sigma$ bounded and $ t$ large to moderate accuracy (within $ \pm {t^{ - c}}$ for some $ c > 0$, say) was based on the Riemann-Siegel formula and required on the order of ${t^{1/2}}$ operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of $\zeta (\sigma + it)$ with $\sigma$ fixed and $ T \leqslant t \leqslant T + {T^{1/2}}$ to within $ \pm {t^{ - c}}$ in $O({t^\varepsilon })$ operations on numbers of $ O(\log t)$ bits for any $\varepsilon > 0$, for example, provided a precomputation involving $O({T^{1/2 + \varepsilon }})$ operations and $O({T^{1/2 + \varepsilon }})$ bits of storage is carried out beforehand. These algorithms lead to methods for numerically verifying the Riemann hypothesis for the first $n$ zeros in what is expected to be $O({n^{1 + \varepsilon }})$ operations (as opposed to about ${n^{3/2}}$ operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as $\pi (x)$. The new zeta function algorithms use the fast Fourier transform and a new method for the evaluation of certain rational functions. They can also be applied to the evaluation of $L$-functions, Epstein zeta functions, and other Dirichlet series.
Hausdorff dimension in graph directed constructions
R. Daniel
Mauldin;
S. C.
Williams
811-829
Abstract: We introduce the notion of geometric constructions in ${{\mathbf{R}}^m}$ governed by a directed graph $ G$ and by similarity ratios which are labelled with the edges of this graph. For each such construction, we calculate a number $ \alpha$ which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to ${\mathcal{H}^\alpha }$ is always positive and $ \sigma$-finite. Whether the $ {\mathcal{H}^\alpha }$-measure of the object is finite depends on the order structure of the strongly connected components of $G$. Some applications are given.
On classical Clifford theory
Morton E.
Harris
831-842
Abstract: Let $k$ be a field, let $N$ be a normal subgroup of a finite group $ H$ and let $M$ be a completely reducible $ k[N]$-module. We give sufficient conditions for a finite dimensional (finite) group crossed product $k$-algebra to be a Frobenius or symmetric $ k$-algebra. These results imply that $ k[H]/(J(k[N])k[H])$ and the endomorphism $k$-algebra, ${\operatorname{End} _{k[H]}}({M^H})$, of the induced module ${M^H}$ are symmetric $k$-algebras. We also completely describe the $ k[H]$-indecomposable decomposition of ${M^H}$. It follows that the head and socle of an indecomposable component of ${M^H}$ are irreducible isomorphic $ k[H]$-modules.
Seifert matrices and $6$-knots
J. A.
Hillman;
C.
Kearton
843-855
Abstract: A new classification of simple $ {\mathbf{Z}}$-torsion-free $ 2q$-knots, $q \geqslant 3$, is given in terms of Seifert matrices modulo an equivalence relation. As a result the classification of such $2q$-knots, $q \geqslant 4$, in terms of $ F$-forms is extended to the case $q = 3$.
Correction to: ``Differential identities in prime rings with involution'' [Trans. Amer. Math. Soc. {\bf 291} (1985), no. 2, 765--787; MR0800262 (87f:16013)]
Charles
Lanski
857-859