Causal compactification of compactly causal spaces
Frank
Betten
4699-4721
Abstract: We give a classification of causal compactifications of compactly causal spaces. Introduced by Ólafsson and Ørsted, for a compactly causal space $G/H$, these compactifications are given by $G$-orbits in the Bergman-Silov boundary of $G_1/K_1$, with $G \subset G_1$ and $(G_1, K_1, \theta)$ a Hermitian symmetric space of tube type. For the classical spaces an explicit construction is presented.
The central limit problem for convex bodies
Milla
Anttila;
Keith
Ball;
Irini
Perissinaki
4723-4735
Abstract: It is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately Gaussian. Several quite broad classes of bodies are shown to satisfy the condition.
A compactification of open varieties
Yi
Hu
4737-4753
Abstract: In this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors. This is done by showing that blowing up along an arrangement of subvarieties can be carried out. Important examples such as Ulyanov's configuration spaces and complements of arrangements of linear subspaces in projective spaces, etc., are covered. Intersection ring and (nonrecursive) Hodge polynomials are computed. Furthermore, some general structures arising from the blowup process are also described and studied.
A Baire's category method for the Dirichlet problem of quasiregular mappings
Baisheng
Yan
4755-4765
Abstract: We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any $\epsilon>0$ and any piece-wise affine map $\varphi\in W^{1,n}(\Omega;\mathbf{R}^n)$ with $\vert D\varphi(x)\vert^n\le L\det D\varphi(x)$ for almost every $x\in\Omega$ there exists a map $u\in W^{1,n}(\Omega;\mathbf{R}^n)$ such that \begin{displaymath}\begin{cases} \vert Du(x)\vert^n=L\det Du(x)\quad\text{a.e.} ... ...,\quad\Vert u-\varphi\Vert _{L^n(\Omega)}<\epsilon. \end{cases}\end{displaymath} The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.
Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers
Peter
Borwein;
Kevin
G.
Hare
4767-4779
Abstract: This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number $q$, with minimal polynomial $p$, such that $p(0) = -1$, and where $p$ has only one real root, then there exists a $C(q)$, explicitly given here, such that: (1) For all $\epsilon > 0$, all but finitely many integer quadratics $P$ satisfy \begin{displaymath}\vert P(q)\vert \geq \frac{C(q) - \epsilon}{H(P)^2}\end{displaymath} where $H$ is the height function. (2) For all $\epsilon > 0$ there exists a sequence of integer quadratics $P_n(q)$ such that \begin{displaymath}\vert P_n(q)\vert \leq \frac{C(q) + \epsilon}{H(P_n)^2}.\end{displaymath} Furthermore, $C(q) < 1$ for all $q$ in this class of cubic Pisot numbers. What is surprising about this result is how precise it is, giving exact upper and lower bounds for these approximations.
Uniqueness of the density in an inverse problem for isotropic elastodynamics
Lizabeth
V.
Rachele
4781-4806
Abstract: We consider the unique determination of the density of a nonhomogeneous, isotropic elastic object from measurements made at the surface. We model the behavior of the bounded, 3-dimensional object by the linear, hyperbolic system of operators for isotropic elastodynamics. The material properties of the object (its density and elastic properties) correspond to the smooth coefficients of these differential operators. The data for this inverse problem, in the form of the correspondence between applied surface tractions and resulting surface displacements, is modeled by the dynamic Dirichlet-to-Neumann map on a finite time interval. In an earlier paper we show that the speeds $c_{p/s}$ of (compressional and sheer) wave propagation through the object are uniquely determined by the Dirichlet-to-Neumann map. Here we extend that result by showing that the density is also determined in the interior by the Dirichlet-to-Neumann map in the case, for example, that $c_p = 2 c_s$ at only isolated points in the object. We use techniques from microlocal analysis and integral geometry to solve this fully three-dimensional problem.
A local characterization of simply-laced crystals
John
R.
Stembridge
4807-4823
Abstract: We provide a simple list of axioms that characterize the crystal graphs of integrable highest weight modules for simply-laced quantum Kac-Moody algebras.
A geometric characterization of Vassiliev invariants
Michael
Eisermann
4825-4846
Abstract: It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree $\le m$ if and only if it is a polynomial of degree $\le m$ on every geometric sequence of knots. Here a sequence $K_z$ with $z\in\mathbb{Z}$ is called geometric if the knots $K_z$ coincide outside a ball $B$, inside of which they satisfy $K_z \cap B = \tau^z$ for all $z$ and some pure braid $\tau$. As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in $\mathbb{S} ^1\times\mathbb{S} ^2$that can be distinguished by $\mathbb{Z} {/}{2}$-invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over $\mathbb{Z}$ a universal Vassiliev invariant of degree $1$ for knots in $\mathbb{S} ^1\times\mathbb{S} ^2$.
Tight surfaces in three-dimensional compact Euclidean space forms
Marc-Oliver
Otto
4847-4863
Abstract: In this paper we define and discuss tight surfaces -- smooth or polyhedral -- in three-dimensional compact Euclidean space forms and prove existence and non-existence results. It will be shown that all orientable and most of the non-orientable surfaces can be tightly immersed in any of these space forms.
The limiting curve of Jarník's polygons
Greg
Martin
4865-4880
Abstract: In 1925, Jarník defined a sequence of convex polygons for use in constructing curves containing many lattice points relative to their curvatures. Properly scaled, these polygons converge to a certain limiting curve. In this paper we identify this limiting curve precisely, showing that it consists piecewise of arcs of parabolas, and we discuss the analogous problem for sequences of polygons arising from generalizations of Jarník's construction.
Stratified transversality by isotopy
C.
Murolo;
D.
J. A.
Trotman;
A.
A.
Du Plessis
4881-4900
Abstract: For $\mathcal{X}$ an abstract stratified set or a $(w)$-regular stratification, hence for any $(b)$-, $(c)$- or $(L)$-regular stratification, we prove that after stratified isotopy of $\mathcal{X}$, a stratified subspace $\mathcal{W}$ of $\mathcal{X}$, or a stratified map $h : \mathcal{Z} \to \mathcal{X}$, can be made transverse to a fixed stratified map $g: \mathcal{Y} \to \mathcal{X}$.
Families of nodal curves on projective threefolds and their regularity via postulation of nodes
Flaminio
Flamini
4901-4932
Abstract: The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given a smooth projective threefold $X$, a rank-two vector bundle $\mathcal{E}$ on $X$, and integers $k\geq 0$, $\delta >0$, denote by ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ the subscheme of ${\mathbb{P}}(H^0({\mathcal{E}}(k)))$ parametrizing global sections of ${\mathcal{E}}(k)$ whose zero-loci are irreducible $\delta$-nodal curves on $X$. We present a new cohomological description of the tangent space $T_{[s]}({\mathcal{V}}_{\delta} ({\mathcal{E}} (k)))$ at a point $[s]\in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$. This description enables us to determine effective and uniform upper bounds for $\delta$, which are linear polynomials in $k$, such that the family ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ is smooth and of the expected dimension (regular, for short). The almost sharpness of our bounds is shown by some interesting examples. Furthermore, when $X$ is assumed to be a Fano or a Calabi-Yau threefold, we study in detail the regularity property of a point $[s] \in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$related to the postulation of the nodes of its zero-locus $C = V(s) \subset X$. Roughly speaking, when the nodes of $C$ are assumed to be in general position either on $X$, or on an irreducible divisor of $X$ having at worst log-terminal singularities or to lie on a l.c.i. and subcanonical curve in $X$, we find upper bounds on $\delta$ which are, respectively, cubic, quadratic and linear polynomials in $k$ ensuring the regularity of ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ at $[s]$. Finally, when $X= \mathbb{P}^3$, we also discuss some interesting geometric properties of the curves given by sections parametrized by ${\mathcal{V}}_{\delta} ({\mathcal{E}} \otimes \mathcal{O}_X(k))$.
Spines and topology of thin Riemannian manifolds
Stephanie
B.
Alexander;
Richard
L.
Bishop
4933-4954
Abstract: Consider Riemannian manifolds $M$ for which the sectional curvature of $M$ and second fundamental form of the boundary $B$ are bounded above by one in absolute value. Previously we proved that if $M$ has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of $B$ exhibits canonical branching behavior of arbitrarily low branching number. In particular, if $M$is thin in the sense that its inradius is less than a certain universal constant (known to lie between $.108$ and $.203$), then $M$collapses to a triply branched simple polyhedral spine. We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of $M$ when $B$ is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When $M$ is $3$-dimensional and compact, $M$ has complexity $0$ in the sense of Matveev, and is a connected sum of $p$ copies of the real projective space $P^3$, $t$ copies chosen from the lens spaces $L(3,\pm1)$, and $\ell$ handles chosen from $S^2\times S^1$ or $S^2\tilde\times S^1$, with $\beta$ 3-balls removed, where $p+t+\ell +\beta \ge 2$. Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.
On the equations defining toric l.c.i.-singularities
Dimitrios
I.
Dais;
Martin
Henk
4955-4984
Abstract: Based on Nakajima's Classification Theorem we describe the precise form of the binomial equations which determine toric locally complete intersection (``l.c.i.'') singularities.
A pair of difference differential equations of Euler-Cauchy type
David
M.
Bradley
4985-5002
Abstract: We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.
Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations
Rabi
N.
Bhattacharya;
Larry
Chen;
Scott
Dobson;
Ronald
B.
Guenther;
Chris
Orum;
Mina
Ossiander;
Enrique
Thomann;
Edward
C.
Waymire
5003-5040
Abstract: A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.
Regularity of isoperimetric hypersurfaces in Riemannian manifolds
Frank
Morgan
5041-5052
Abstract: We add to the literature the well-known fact that an isoperimetric hypersurface $S$ of dimension at most six in a smooth Riemannian manifold $M$ is a smooth submanifold. If the metric is merely Lipschitz, then $S$ is still Hölder differentiable.
The free entropy dimension of hyperfinite von Neumann algebras
Kenley
Jung
5053-5089
Abstract: Suppose $M$ is a hyperfinite von Neumann algebra with a normal, tracial state $\varphi$ and $\{a_1,\ldots,a_n\}$ is a set of selfadjoint generators for $M$. We calculate $\delta_0(a_1,\ldots,a_n)$, the modified free entropy dimension of $\{a_1,\ldots,a_n\}$. Moreover, we show that $\delta_0(a_1,\ldots,a_n)$ depends only on $M$ and $\varphi$. Consequently, $\delta_0(a_1,\ldots,a_n)$ is independent of the choice of generators for $M$. In the course of the argument we show that if $\{b_1,\ldots,b_n\}$ is a set of selfadjoint generators for a von Neumann algebra $\mathcal R$ with a normal, tracial state and $\{b_1,\ldots,b_n\}$has finite-dimensional approximants, then $\delta_0(N) \leq \delta_0(b_1,\ldots,b_n)$ for any hyperfinite von Neumann subalgebra $N$of $\mathcal R.$ Combined with a result by Voiculescu, this implies that if $\mathcal R$ has a regular diffuse hyperfinite von Neumann subalgebra, then $\delta_0(b_1,\ldots,b_n)=1$.
Codimension growth and minimal superalgebras
A.
Giambruno;
M.
Zaicev
5091-5117
Abstract: A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope $G(A)$ of a finite dimensional superalgebra $A$. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: $A$ is a minimal superalgebra if and only if the ideal of identities of $G(A)$ is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties $\mathcal{V}$ such that $\exp({\mathcal{V}})=d\ge 2$ and $\exp(\mathcal{U})<d$ for all proper subvarieties ${\mathcal{U}}$ of ${\mathcal{V}}$. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003). As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.
Erratum to ``A Berger-Green type inequality for compact Lorentzian manifolds"
Manuel
Gutiérrez;
Francisco
J.
Palomo;
Alfonso
Romero
5119-5120