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

A direct approach to co-universal algebras associated to directed graphs

We prove directly that if E is a directed graph in which every cycle has an entrance, then there exists a C*-algebra which is co-universal for Toeplitz-Cuntz-Krieger E-families. In particular, our proof does not invoke ideal-structure theory for graph algebras, nor does it involve use of the gauge action or its fixed point algebra.Comment: 9 pages; V2: the definition of a Toeplitz-Cuntz-Krieger $E$-family has been correcte

A Numerical Study of the Conjugate Conduction-Convection Heat Transfer Problem

This study investigates some of the basic aspects of conjugate, or coupled, heat transfer problems. The ultimate interest is in the improvement of an existing computational fluid dynamics (CFD) code by the inclusion of such a coupling capability. Many CFD codes in the past have treated the thermal boundary conditions of a bounding solid as the simple cases of either a surface across which there is no heat flux, or as a surface along which the temperature is a constant with respect to both space and time. These conditions are acceptable for some applications, but many real-world problems require a more-realistic treatment of the thermal wall condition. A thermal coupling may be accomplished by maintaining a continuous heat flux and temperature across the fluid-solid boundary. A heat flux is calculated on the fluid-side of the interface, and this is used as a boundary condition for a heat-conduction solver to calculate the temperature field within the solid and return an interface temperature to the fluid. This process is executed for each time-step iteration of the code, and, therefore, the temperature field of the solid and the fluid-solid interface temperature are allowed to evolve with time and space. A new heat-conduction solver is developed and coupled with an existing flow solver. For this reason, some of the study is devoted to the testing of the accuracy of the new heat-conduction solver on simple problems for which there exist analytical solutions. Additional coverage is devoted to the possibility of thermal communication between solid grid blocks. This is due to the fact that multiple grid blocking of the solid may be required for more complex geometries. For such cases, a similar procedure as that described for the fluid-solid interface is used to accomplish the solid-solid block-to-block communication. Relatively simple test cases of fluid-solid and solid-solid coupling are conducted; these cases are limited to two-dimensional grids. Other limitations include: the assumption of constant thermophysical properties for the solid, no consideration for thermal expansion of the solid, and no consideration for the radiation mode of heat transfer. The results indicate that the heat-conduction/flow solver shows potential

High-Energy theory for close Randall Sundrum branes

We obtain an effective theory for the radion dynamics of the two-brane Randall Sundrum model, correct to all orders in brane velocity in the limit of close separation, which is of interest for studying brane collisions and early Universe cosmology. Obtained via a recursive solution of the Bulk equation of motions, the resulting theory represents a simple extension of the corresponding low-energy effective theory to the high energy regime. The four-dimensional low-energy theory is indeed not valid when corrections at second order in velocity are considered. This extension has the remarkable property of including only second derivatives and powers of first order derivatives. This important feature makes the theory particularly easy to solve. We then extend the theory by introducing a potential and detuning the branes.Comment: Version published in the Physical Review

The Path Space of a Directed Graph

We construct a locally compact Hausdorff topology on the path space of a directed graph $E$, and identify its boundary-path space $\partial E$ as the spectrum of a commutative $C^*$-subalgebra $D_E$ of $C^*(E)$. We then show that $\partial E$ is homeomorphic to a subset of the infinite-path space of any desingularisation $F$ of $E$. Drinen and Tomforde showed that we can realise $C^*(E)$ as a full corner of $C^*(F)$, and we deduce that $D_E$ is isomorphic to a corner of $D_F$. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.Comment: 12 pages, all figures drawn with TikZ/PG

von Neuman algebras of strongly connected higher-rank graphs

We investigate the factor types of the extremal KMS states for the preferred dynamics on the Toeplitz algebra and the Cuntz--Krieger algebra of a strongly connected finite $k$-graph. For inverse temperatures above 1, all of the extremal KMS states are of type I$_\infty$. At inverse temperature 1, there is a dichotomy: if the $k$-graph is a simple $k$-dimensional cycle, we obtain a finite type I factor; otherwise we obtain a type III factor, whose Connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph.Comment: 16 pages; 1 picture prepared using TikZ. Version 2: this version to appear in Math. An

On some fundamental results about higher-rank graphs and their C*-algebras

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated to a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph {\Lambda} is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterisation, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.Comment: 21 pages, two pictures prepared using TiK

High-energy effective theory for matter on close Randall Sundrum branes

Extending the analysis of hep-th/0504128, we obtain a formal expression for the coupling between brane matter and the radion in a Randall-Sundrum braneworld. This effective theory is correct to all orders in derivatives of the radion in the limit of small brane separation, and, in particular, contains no higher than second derivatives. In the case of cosmological symmetry the theory can be obtained in closed form and reproduces the five-dimensional behaviour. Perturbations in the tensor and scalar sectors are then studied. When the branes are moving, the effective Newtonian constant on the brane is shown to depend both on the distance between the branes and on their velocity. In the small distance limit, we compute the exact dependence between the four-dimensional and the five-dimensional Newtonian constants.Comment: Updated version as published in PR