The path space of a higher-rank graph

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\Lambda$. We identify the boundary-path space $\partial\Lambda$ as the spectrum of a commutative $C^*$-subalgebra $D_\Lambda$ of $C^*(\Lambda)$. Then, using a construction similar to that of Farthing, we construct a finitely aligned $k$-graph \wt\Lambda with no sources in which $\Lambda$ is embedded, and show that $\partial\Lambda$ is homeomorphic to a subset of \partial\wt\Lambda . We show that when $\Lambda$ is row-finite, we can identify $C^*(\Lambda)$ with a full corner of C^*(\wt\Lambda), and deduce that $D_\Lambda$ is isomorphic to a corner of D_{\wt\Lambda}. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.Comment: 30 pages, all figures drawn with TikZ/PGF. Updated numbering and minor corrections to coincide with published version. Updated 29-Feb-2012 to fix a compiling error which resulted in the arXiv PDF output containing two copies of the articl

Laboratory tests in patients treated with isotretinoin: occurrence of liver and muscle abnormalities and failure of AST and ALT to predict liver abnormality.

Current laboratory monitoring may not be optimal. A retrospective chart review was performed on thelaboratory results of 246 patients who were treated with isotretinoin for acne over a 9-year period. Tests obtained were CBC, lipid panel, AST, ALT, CK, GGT,and C-reactive protein. Thirty-five patients had an elevated AST and 35 of these had an elevated CK; 32 had an elevated ALT and 11 of these had an elevated CK. Thirteen patients had an elevated GGT; in 5 this was the only abnormality, whereas 8 had a GGT elevation accompanied by an elevated AST or ALT. Two had an elevated GGT and an elevated CK with normal AST and ALT. Fifty-two patients had a single episode of elevated CK, of which 22 were female. However, 57 had multiple CK elevations and only one was female. Thirty-five patients had CK elevationsnormal; 38 had levels between 2 and 3 times normal, 18 had levels between 3 and 4 times normal, and 18 had levels greater than 4 times normal. We suggest that ALT and AST are not useful for monitoring isotretinoin therapy and that GGT and CK may be of greater value in managing patients

Semigroup Well-posedness of A Linearized, Compressible Fluid with An Elastic Boundary

We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile $\mathbf{U}$. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically \cite {Chu2013-comp}---the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a $C_{0}$-semigroup $\left\{ e^{ \mathcal{A}t}\right\} _{t\geq 0}$ on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator $\mathcal{A}$ which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field $\mathbf{U}\in \mathbf{H}^{3}(\mathcal{O})$ has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.Comment: 1 figur

Urban Regeneration of Industrial Areas: Affordable Housing for Low Income Populations in Cities

The UK-China Sustainable Development Dialogue (SDD) is a partnership between the UK and Chinese Governments to promote collaboration and good practice on sustainable development. It is framed by a 2004 joint Prime Ministerial declaration and was formally established in 2005 by an agreement signed by UK Deputy Prime Minister John Prescott and State Councillor Tang. The SDD was founded on the recognition that, in an interdependent world, international co-operation is needed to ensure that learning is shared and efforts are made collectively towards achieving common goals. This report is the 13th and Final Paper in the SDD (Urban Strand. It draws together the core conclusions from the previous twelve papers and offers recommendations for taking the dialogue forward

Polynomial approximation via compressed sensing of high-dimensional functions on lower sets

This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. Of particular interest is the setting where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we present an innovative weighted $\ell_1$ minimization procedure with a precise choice of weights, and a new iterative hard thresholding method, for imposing the downward closed preference. Theoretical results reveal that our computational approaches possess a provably reduced sample complexity compared to existing compressed sensing techniques presented in the literature. In addition, the recovery of the corresponding best approximation using these methods is established through an improved bound for the restricted isometry property. Our analysis represents an extension of the approach for Hadamard matrices in [5] to the general case of continuous bounded orthonormal systems, quantifies the dependence of sample complexity on the successful recovery probability, and provides an estimate on the number of measurements with explicit constants. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the novel weighted $\ell_1$ minimization strategy.Comment: 33 pages, 3 figure