The world problem: on the computability of the topology of 4-manifolds

Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing machine with an arbitrary input can be encoded into the topology of a 4-manifold, such that the 4-manifold is homeomorphic to a certain other 4-manifold if and only if the corresponding Turing machine halts on the associated input. Physical implications are briefly discussed.Comment: Submitted to Class. Quant. Gra

Excising das All: Evolving Maxwell waves beyond scri

We study the numerical propagation of waves through future null infinity in a conformally compactified spacetime. We introduce an artificial cosmological constant, which allows us some control over the causal structure near null infinity. We exploit this freedom to ensure that all light cones are tilted outward in a region near null infinity, which allows us to impose excision-style boundary conditions in our finite difference code. In this preliminary study we consider electromagnetic waves propagating in a static, conformally compactified spacetime.Comment: 13 pages; incorporated material from gr-qc/051216

Universality of Swap for Qudits: A Representation Theory Approach

An open problem of quantum information theory has been to determine under what conditions universal exchange-only computation is possible for qudits encoded on d-state systems for d&gt;2. This problem can be posed in terms of representation theory by recognizing that each quantum mechanical swap, generated by exchange-interaction, can be identified with a transposition in a symmetric group, each d-state system can be identified with the fundamental representation of SU(d), and each encoded qudit can be identified with an irreducible representation of a Lie algebra generated by transpositions. Towards this end we first give a mathematical definition of exchange-only universalityin terms of a map from the special unitary algebra on the product of qudits into a representation of a Lie algebra generated by transpositions. We show that this definition is consistent with quantum computing requirements. We then proceed with the task of characterizing universal families of qudits, that is families of encoded qudits admitting exchange-only universality. This endeavor is aided by the fact that the irreducible representations corresponding to qudits are canonically labeled by partitions. In particular we derive necessary and sufficient conditions for universality on one or two such qudits, in terms of simple arithmetic conditions on the associated partitions. We also derive necessary and sufficient conditions for universality on arbitrarily many such qudits, in terms of Littlewood--Richardson coefficients. Among other results, we prove that universal families of multiple qudits are upward closed, that universality is guaranteed for sufficiently many qudits, and that any family that is not universal can be made so by simply adding at most five ancillae. We also obtain results for 2-state systems as a special case.</p

Reducing reflections from mesh refinement interfaces in numerical relativity

Full interpretation of data from gravitational wave observations will require accurate numerical simulations of source systems, particularly binary black hole mergers. A leading approach to improving accuracy in numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a manifestation of numerical interface truncation error which appears as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh refinement implementations, potentially compromising the effectiveness of mesh refinement techniques for some numerical relativity applications if left untreated. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite differencing stencil modifications, termed mesh-adapted differencing (MAD), which eliminate this pathology in both our model problem and in numerical relativity examples.Comment: 7 page

Novel chromatin regulatory activity of ESCO2 in cancer and neural development

ESCO2 has a well characterized role in the stabilization of the cohesin ring through its acetyltransferase activity. Quantitative PCR studies comparing RNA from tissue collected from ependymoma biopsies and tissue from non-tumor brain, show an increased level of ESCO2 transcription in ependymoma. High levels of ESCO2 protein expression in ependymoma tissue samples were confirmed via immunostaining and confocal microscopy. ESCO2 expression ordinarily peaks during mitosis in order to stabilize the cohesin ring, and is reported to increase the rate of replication. However, some ESCO2 expressing cells are not mitotic in these tumors. Because of the detrimental effects of deletion of ESCO2 on neural development and its high expression in ependymoma, a type of tumor that exhibits many characteristics of primitive neural cells, the current study sought to investigate the broader role of ESCO2 in primitive neural cells. Studies carried out in Zebrafish were performed to explore the possibility of an additional role for ESCO2 as a regulator of neural development. QPCR on Zebrafish embryos at developmental timepoints indicated that ESCO2 expression was highest 48 hours post-fertilization, and then declined thereafter. Confocal microscopy on transgenic NeuroD-EGFP embryos at 24-96 hpf confirmed that the proportion of ESCO2 positive brain cells decreases by 96 hpf, that non-mitotic ESCO2 expressing cells decline, and that 48 hpf appear to display a unique pattern of ESCO2 staining associated with the 4th ventricle. Using FACS and subsequent chromatin immunoprecipitation studies, Esco2- bound chromatin was isolated from NT2 neural progenitor cells and NeuroD-EGFP+ transgenic zebrafish, and utilized to identify novel sites of Esco2 regulation. The data obtained suggests an additional layer of neural regulatory activity that may be unrelated to the reported SMC3 acetyltransferase activity of ESCO2, and that could play a role in neural pathologies such as cancer. Esco2 may be a novel therapeutic target for ependymoma

Controlling Reflections from Mesh Refinement Interfaces in Numerical Relativity

A leading approach to improving the accuracy on numerical relativity simulations of black hole systems is through fixed or adaptive mesh refinement techniques. We describe a generic numerical error which manifests as slowly converging, artificial reflections from refinement boundaries in a broad class of mesh-refinement implementations, potentially limiting the effectiveness of mesh- refinement techniques for some numerical relativity applications. We elucidate this numerical effect by presenting a model problem which exhibits the phenomenon, but which is simple enough that its numerical error can be understood analytically. Our analysis shows that the effect is caused by variations in finite differencing error generated across low and high resolution regions, and that its slow convergence is caused by the presence of dramatic speed differences among propagation modes typical of 3+1 relativity. Lastly, we resolve the problem, presenting a class of finite-differencing stencil modifications which eliminate this pathology in both our model problem and in numerical relativity examples

Consistency of post-Newtonian waveforms with numerical relativity

General relativity predicts the gravitational wave signatures of coalescing binary black holes. Explicit waveform predictions for such systems, required for optimal analysis of observational data, have so far been achieved using the post-Newtonian (PN) approximation. The quality of this treatment is unclear, however, for the important late-inspiral portion. We derive late-inspiral waveforms via a complementary approach, direct numerical simulation of Einstein's equations. We compare waveform phasing from simulations of the last $\sim 14$ cycles of gravitational radiation from equal-mass, nonspinning black holes with the corresponding 2.5PN, 3PN, and 3.5PN orbital phasing. We find phasing agreement consistent with internal error estimates based on either approach, suggesting that PN waveforms for this system are effective until the last orbit prior to final merger.Comment: Replaced with published version -- one figure removed, text and other figures updated for clarity of discussio

General Relativistic Simulations of Magnetized Plasmas around Merging Supermassive Black Holes

Coalescing supermassive black hole binaries are produced by the mergers of galaxies and are the most powerful sources of gravitational waves accessible to space-based gravitational observatories. Some such mergers may occur in the presence of matter and magnetic fields and hence generate an electromagnetic counterpart. In this Letter, we present the first general relativistic simulations of magnetized plasma around merging supermassive black holes using the general relativistic magnetohydrodynamic code Whisky. By considering different magnetic field strengths, going from non-magnetically dominated to magnetically dominated regimes, we explore how magnetic fields affect the dynamics of the plasma and the possible emission of electromagnetic signals. In particular we observe a total amplification of the magnetic field of ~2 orders of magnitude which is driven by the accretion onto the binary and that leads to much stronger electromagnetic signals, more than a factor of 10^4 larger than comparable calculations done in the force-free regime where such amplifications are not possible.Comment: 7 pages, 5 figures. Minor changes to match version accepted for publication on The Astrophysical Journal Letter