Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

Science Conference Presenters’ Images of Inquiry

Inquiry‐focused professional development and conceptions of inquiry held by eight professional development leaders were investigated within the context of a state science teacher conference. The prominent session format involved session leaders modeling classroom experiences. In all sessions, classroom inquiry was portrayed as a teacher‐guided activity with the primary goal being to increase motivation for engaging students in classroom inquiry. The leaders’ conceptualized inquiry primarily as a teaching approach with various goals, characteristics, and potential barriers. The findings of this study provide evidence of how inquiry, a prominent feature of science education reform, was portrayed in sessions at a conference sponsored by a state affiliate of the National Science Teachers Association and thought about by persons who led these sessions. The findings have implications for teacher learning from conference‐based professional development and its potential influence on science teacher thinking and practice.     

Injectivity relative to closed submodules

Let R be a ring. An R-module X is called c-injective if, for every closed submodule L of every R-module M, every homomorphism from L to X lifts to M. It is proved that if R is a Dedekind domain then an R-module X is c-injective if and only if X is isomorphic to a direct product of homogeneous semisimple R-modules and injective R-modules. It is also proved that a commutative Noetherian domain R is Dedekind if and only if every simple R-module is c-injective.     

Covering the crosspolytope by equal balls

Summary We determine the minimal radius of <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>n=2$, $d$ or $2d$ congruent balls, which cover the $d$-dimensional crosspolytope.     

L 1 factorizations and invariant subspaces for weighted composition operators

For a contraction operator T with spectral radius less than one on a Banach space , it is shown that the factorization of certain L¹ functions by vectors x in and x*. in , in the sense that for n ≧ 0, implies the existence of invariant subspaces for T. Explicit formulae for such factorizations are given in the case of weighted composition operators on reproducing kernel Hilbert spaces. An interpolation result of McPhail is applied to show how this can be used to construct invariant subspaces of hyperbolic weighted composition operators on H². Received: 1 November 2005     

The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains

This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used.     

The Roles Played by Order of Convexity or Starlikeness and the Bloch Condition in the Extension of Mappings from the Disk to the Ball

Let ${G: \mathbb {C}^{n-1} \rightarrow \mathbb {C}}$ be holomorphic such that G(0)?=?0 and DG(0)?=?0. When f is a convex (resp. starlike) normalized (f(0)?=?0, f??(0)?=?1) univalent mapping of the unit disk ${\mathbb {D}}$ in ${\mathbb {C}}$ , then the extension of f to the Euclidean unit ball ${\mathbb {B}}$ in ${\mathbb {C}^n}$ given by ${\Phi_G(f)(z)=(f(z_1)+G(\sqrt{f^{\prime}(z_1)} \, \hat{z}),\sqrt{f^{\prime}(z_1)}\, \hat{z})}$ , ${\hat{z}=(z_2,\dots,z_n) \in \mathbb {C}^{n-1}}$ , is known to be convex (resp. starlike) if G is a homogeneous polynomial of degree 2 with sufficiently small norm. Conversely, it is known that G cannot have terms of degree greater than 2 in its expansion about 0 in order for ${\Phi_G(f)}$ to be convex (resp. starlike), in general. We examine whether the restriction that f be either convex or starlike of a certain order ${\alpha \in (0,1]}$ allows, in general, for G to contain terms of degree greater than 2 and still have ${\Phi_G(f)}$ maintain the respective geometric property. Related extension results for convex and starlike Bloch mappings are also given.     

Wavelet-based dynamic time warping

Dynamic Time Warping (DTW), a pattern matching technique traditionally used for restricted vocabulary speech recognition, is based on a temporal alignment of the input signal with the template models. The principal drawback of DTW is its high computational cost as the lengths of the signals increase. This paper shows extended results over our previously published conference paper, which introduces an optimized version of the DTW that is based on the Discrete Wavelet Transform (DWT).     

Multi-objective evacuation routing in transportation networks

In this paper, the optimal design and analysis of evacuation routes in transportation networks is examined. An methodology for optimal egress route assignment is suggested. An integer programming (IP) formulation for optimal route assignment is presented, which utilizes M/G/c/c state dependent queueing models to cope with congestion and time delays on road links. M/G/c/c simulation software is used to evaluate performance measures of the evacuation plan: clearance time, total travelled distance and blocking probabilities. Extensive experimental results are included.     

Hereditarily indecomposable subcontinua of the product of two Souslin arcs are metric

We prove that if each of X and Y is a Souslin arc (a Hausdorff arc that is the compactification of a connected Souslin line), then every hereditarily indecomposable subcontinuum of X×Y is metric.