Curves in Grassmannians
David
Perkinson
3179-3246
Abstract: Curves in Grassmannians are analyzed using the special structure of the tangent bundle of a Grassmannian, resulting in a theory of inflections or Weierstrass behavior. A duality theorem is established, generalizing the classical duality theorem for projective plane curves. The appendices summarize basic information about principal parts bundles and their application to studying the inflections of curves in projective space.
Hybrid spaces with interesting cohomology
Kathryn
Lesh
3247-3262
Abstract: Let $p$ be an odd prime, and let $ R$ be a polynomial algebra over the Steenrod algebra with generators in dimensions prime to $p$. To such an algebra is associated a $p$-adic pseudoreflection group $ W$, and we assume that $ W$ is of order prime to $ p$ and irreducible. Adjoin to $R$ a one-dimensional element $z$, and give $R[z]$ an action of the Steenrod algebra by $\beta z = 0$ and $\beta x = (\left\vert x \right\vert/2)zx$ for an even dimensional element $x$. We show that the subalgebra of elements of $ R[z]$ consisting of elements of degree greater than one is realized uniquely, up to homotopy, as the cohomology of a $ p$-complete space. This space can be thought of as a cross between spaces studied by Aguade, Broto, and Notbohm, and the Clark-Ewing examples, further studied by Dwyer, Miller, and Wilkerson.
Continuous functions on extremally disconnected spaces
J.
Vermeer
3263-3285
Abstract: Using results and techniques due to Abramovich, Arenson and Kitover it is shown that each fixed-point set of a selfmap of a compact extremally disconnected space is a retract of that space, and that the retraction can be constructed from the particular selfmap itself. Also, the closure of the set of periodic points turns out to be a retract of the space. Several decomposition theorems for arbitrary selfmaps on extremally disconnected spaces are obtained similar to the theorem of Frolík on embeddings. Conditions are obtained under which the set of fixed points is clopen.
On Euler characteristics associated to exceptional divisors
Willem
Veys
3287-3300
Abstract: Let $k$ be an algebraically closed field and $f \in k[{x_1}, \ldots ,{x_{n + 1}}]$. Fix an embedded resolution $h:X \to {\mathbb{A}^{n + 1}}\quad {\text{of}}\quad {f^{ - 1}}\{ 0\}$ and denote by ${E_i}$, $i \in S$, the irreducible components of ${h^{ - 1}}({f^{ - 1}}\{ 0\} )$ with multiplicity ${N_i}$ in the divisor of $ f{\text{o}}h$. Put also $ {\mathop E\limits^{\text{o}} _i}: = {E_i}\backslash { \cup _{j \ne i}}{E_j}$, and denote by $\chi ({E_i})$ its Euler characteristic. Several conjectures concerning Igusa's local zeta function and the topological zeta function of $ f$ motivate the study of Euler characteristics associated to subsets ${ \cup _{i \in T}}{E_i}$ of ${ \cup _{i \in S}}{E_i}$, which are maximal connected with respect to the property that $ d\vert{N_i}$ for all $ i \in T$. Here $d \in \mathbb{N},d > 1$. We prove that if $ h$ maps ${ \cup _{i \in T}}{E_i}$ to a point, then $\displaystyle {( - 1)^n}\sum\limits_{i \in T} {\chi ({{\mathop E\limits^{\text{o}} }_i}) \geqslant 0}$ This generalizes a well-known result for curves. We also prove some vanishing results concerning the $\chi ({\mathop E\limits^{\text{o}} _i})$ for such a maximal connected subset ${ \cup _{i \in T}}{E_i}$ and give an application on the above-mentioned zeta functions, yielding some confirmation of the holomorphy conjecture for those zeta functions.
Stability of optimal-order approximation by bivariate splines over arbitrary triangulations
C. K.
Chui;
D.
Hong;
R. Q.
Jia
3301-3318
Abstract: Let $\Delta$ be a triangulation of some polygonal domain in $ {\mathbb{R}^2}$ and $S_k^r(\Delta )$, the space of all bivariate $ {C^r}$ piecewise polynomials of total degree $ \leqslant k$ on $ \Delta$. In this paper, we construct a local basis of some subspace of the space $S_k^r(\Delta )$, where $k \geqslant 3r + 2$, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of $\Delta$ with the exception of the smallest angle in the partition. This result is obtained by means of a careful choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimal-order approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their ${\text{B}}$-net representations is derived for this purpose.
The approximation theorem and the $K$-theory of generalized free products
Roland
Schwänzl;
Ross E.
Staffeldt
3319-3345
Abstract: We use methods of abstract algebraic $K$-theory as developed by Friedhelm Waldhausen to give a new derivation of the decomposition theorem for the algebraic $K$-theory of a generalized free product ring. The result takes the form of a fibration sequence which relates the algebraic $K$-theory of such a ring with the algebraic $ K$-theory of its factors, plus a Nil-term.
Intertwining operators associated to the group $S\sb 3$
Charles F.
Dunkl
3347-3374
Abstract: For any finite reflection group $G$ on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in $G$. There exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a singular set of parameter values (containing only certain negative rational numbers). This paper constructs an integral transform implementing the intertwining operator for the group ${S_3}$, the symmetric group on three objects, for parameter value $ \geqslant \frac{1} {2}$. The transform is realized as an absolutely continuous measure on a compact subset of ${M_2}({\mathbf{R}})$, which contains the group as a subset of its boundary. The construction of the integral formula involves integration over the unitary group $U(3)$.
Global uniqueness for a two-dimensional semilinear elliptic inverse problem
Victor
Isakov;
Adrian I.
Nachman
3375-3390
Abstract: For a general class of nonlinear Schrödinger equations $- \Delta u + a(x,u) = 0$ in a bounded planar domain $\Omega$ we show that the function $a(x,u)$ can be recovered from knowledge of the corresponding Dirichlet-to-Neumann map on the boundary $ \partial \Omega$.
Automorphisms of spaces with finite fundamental group
Georgia
Triantafillou
3391-3403
Abstract: Let $X$ be a finite CW-complex with finite fundamental group. We show that the group ${\text{aut}}(X)$ of homotopy classes of self-homotopy equivalences of $X$ is commensurable to an arithmetic group. If in addition $X$ is an oriented manifold then the subgroup $ {\text{au}}{{\text{t}}_t}(X)$ of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group. Moreover if $X$ is a smooth manifold of dimension $\geqslant 5$ then the subgroup ${\text{diff}}(X)$ of ${\text{aut}}(X)$ the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group.
Global surjectivity of submersions via contractibility of the fibers
Patrick J.
Rabier
3405-3422
Abstract: We give a sufficient condition for a ${C^1}$ submersion $F:X \to Y$, $X$ and $Y$ real Banach spaces, to be surjective with contractible fibers ${F^{ - 1}}(y)$. Roughly speaking, this condition "interpolates" two well-known but unrelated hypotheses corresponding to the two extreme cases: Hadamard's criterion when $Y \simeq X$ and $F$ is a local diffeomorphism, and the Palais-Smale condition when $Y = \mathbb{R}$. These results may be viewed as a global variant of the implicit function theorem, which unlike the local one does not require split kernels. They are derived from a deformation theorem tailored to fit functionals with a norm-like nondifferentiability.
The structure of MFD shock waves for rectilinear motion in some models of plasma
Mahmoud
Hesaaraki
3423-3452
Abstract: The mathematical question of the existence of structure for "fast", "slow" and "intermediate" MFD shock waves in the case of rectilinear motion in some model of plasma is stated in terms of a six-dimensional system of ordinary differential equations, which depends on five viscosity parameters. In this article we shall show that this system is gradient-like. Then by using the Conley theory we prove that the fast and the slow shocks always possess structure. Moreover, the intermediate shocks do not admit structure. Some limiting cases for singular viscosities are investigated. In particular, we show how the general results in the classical one fluid MHD theory are obtained when "the plasma viscosities" $\beta$ and $\chi$ tend to zero.
Test ideals in local rings
Karen E.
Smith
3453-3472
Abstract: It is shown that certain aspects of the theory of tight closure are well behaved under localization. Let $J$ be the parameter test ideal for $ R$, a complete local Cohen-Macaulay ring of positive prime characteristic. For any multiplicative system $U \subset R$, it is shown that $J{U^{ - 1}}R$ is the parameter test ideal for ${U^{ - 1}}R$. This is proved by proving more general localization results for the here-introduced classes of " $ {\text{F}}$-ideals" of $ R$ and " ${\text{F}}$-submodules of the canonical module" of $ R$, which are annihilators of $R$ modules with an action of Frobenius. It also follows that the parameter test ideal cannot be contained in any parameter ideal of $R$.
The $7$-connected cobordism ring at $p=3$
Mark A.
Hovey;
Douglas C.
Ravenel
3473-3502
Abstract: In this paper, we study the cobordism spectrum $MO\left\langle 8 \right\rangle$ at the prime $3$. This spectrum is important because it is conjectured to play the role for elliptic cohomology that Spin cobordism plays for real $K$-theory. We show that the torsion is all killed by $ 3$, and that the Adams-Novikov spectral sequence collapses after only $ 2$ differentials. Many of our methods apply more generally.
Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case
José M.
Arrieta
3503-3531
Abstract: We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in $ {\mathbb{R}^2}$. This domain consists of two disjoint domains ${\Omega ^L}$, $ {\Omega ^R}$ joined by a channel $ {R_\varepsilon }$ of height of the order of the parameter $\varepsilon$. When an eigenvalue approaches an eigenvalue of the Laplacian in ${\Omega ^L} \cup {\Omega ^R}$, the order of convergence is $ \varepsilon$, while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: $\varepsilon \left\vert {{\text{ln}}\varepsilon } \right\vert$. We also obtain estimates on the behavior of the eigenfunctions.
Bounded geodesics of Riemann surfaces and hyperbolic manifolds
J. L.
Fernández;
M. V.
Melián
3533-3549
Abstract: We study the set of bounded geodesics of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.
Integrally closed modules over two-dimensional regular local rings
Vijay
Kodiyalam
3551-3573
Abstract: This paper is based on work of Rees on integral closures of modules and initiates the study of integrally closed modules over two-dimensional regular local rings in analogy with the classical theory of complete ideals of Zariski. The main results can be regarded as generalizations of Zariski's product theorem. They assert that the tensor product mod torsion of integrally closed modules is integrally closed, that the symmetric algebra mod torsion of an integrally closed module is a normal domain and that the first Fitting ideal of an integrally closed module is an integrally closed ideal. A construction of indecomposable integrally closed modules is also given. The primary technical tool is a study of the Buchsbaum-Rim multiplicity.
Elliptic equations of order $2m$ in annular domains
Robert
Dalmasso
3575-3585
Abstract: In this paper we study the existence of positive radial solutions for some semilinear elliptic problems of order $ 2m$ in an annulus with Dirichlet boundary conditions. We consider a nonlinearity which is either sublinear or the sum of a sublinear and a superlinear term.
Circle bundles and the Kreck-Stolz invariant
Xianzhe
Dai;
Wei Ping
Zhang
3587-3593
Abstract: We present a direct analytic calculation of the $s$-invariant of Kreck-Stolz for circle bundles, by evaluating the adiabatic limits of $ \eta$ invariants. We believe that this method should have wider applications.
A note on singularities in semilinear problems
Mohammed
Guedda;
Mokhtar
Kirane
3595-3603
Abstract: We consider the equation $ \Delta u - \frac{1} {2}x.\Delta u - \frac{u} {{q - 1}} + {u^q} = 0,{\text{for}}q > 1$. We study the isolated singularities and present a nonlinear technique, and give a complete classification.
The zero-sets of the radial-limit functions of inner functions
Charles L.
Belna;
Robert D.
Berman;
Peter
Colwell;
George
Piranian
3605-3612
Abstract: A set $ E$ on the unit circle is the zero-set of the radial-limit function of some inner function if and only if $E$ is a countable intersection of ${F_\sigma }$-sets of measure 0.
Finite generalized triangle groups
J.
Howie;
V.
Metaftsis;
R. M.
Thomas
3613-3623
Abstract: We give an almost complete classification of those generalized triangle groups that are finite, building on previous results of Baumslag, Morgan and Shalen [1], Conder [4], Rosenberger [12] and Levin and Rosenberger [11]. There are precisely two groups for which we cannot decide whether or not they are finite.
All finite generalized triangle groups
L.
Lévai;
G.
Rosenberger;
B.
Souvignier
3625-3627
Abstract: We complete the classification of those generalized triangle groups that are finite.
Sobolev orthogonal polynomials and spectral differential equations
I. H.
Jung;
K. H.
Kwon;
D. W.
Lee;
L. L.
Littlejohn
3629-3643
Abstract: We find necessary and sufficient conditions for a spectral differential equation $\displaystyle {L_N}[y](x) = \sum\limits_{i = 1}^N {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)}$ to have Sobolev orthogonal polynomials of solutions, which are orthogonal relative to the Sobolev (pseudo-) inner product $\displaystyle \phi (p,q) = \int_\mathbb{R}^{} {pqd\mu + \int_\mathbb{R}^{} {p'q'dv,} }$ where $d\mu$ and $dv$ are signed Borel measures having finite moments. This result generalizes a result by H. L. Krall, which handles the case when $dv = 0$.
Harmonic diffeomorphisms between Hadamard manifolds
Peter
Li;
Luen-Fai
Tam;
Jiaping
Wang
3645-3658
Abstract: In this paper, we study the Dirichlet problem at infinity for harmonic maps between complete hyperbolic Hadamard surfaces. We will address the existence and uniqueness questions relating to the problem. In particular, we generalize results in the work of Li-Tam and Wan.
On the cohomology of $\Gamma\sb p$
Yining
Xia
3659-3670
Abstract: Let ${\Gamma _g}$ denote the mapping class group of genus $ g$. In this paper, we calculate $p$-torsion of Farrell cohomology $ {\widehat{H}^*}({\Gamma_p})$ for any odd prime $p$.