An Emergent Space for Distributed Data with Hidden Internal Order through Manifold Learning

Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here on the case of time series data that can ultimately be modelled as a spatially distributed system (e.g. a partial differential equation, PDE), but where we do not know the space in which this PDE should be formulated. Hence, even the spatial coordinates for the distributed system themselves need to be identified - to emerge from - the data mining process. We will first validate this emergent space reconstruction for time series sampled without space labels in known PDEs; this brings up the issue of observability of physical space from temporal observation data, and the transition from spatially resolved to lumped (order-parameter-based) representations by tuning the scale of the data mining kernels. We will then present actual emergent space discovery illustrations. Our illustrative examples include chimera states (states of coexisting coherent and incoherent dynamics), and chaotic as well as quasiperiodic spatiotemporal dynamics, arising in partial differential equations and/or in heterogeneous networks. We also discuss how data-driven spatial coordinates can be extracted in ways invariant to the nature of the measuring instrument. Such gauge-invariant data mining can go beyond the fusion of heterogeneous observations of the same system, to the possible matching of apparently different systems

Embedding variables in finite dimensional models

Global problems associated with the transformation from the Arnowitt, Deser and Misner (ADM) to the Kucha\v{r} variables are studied. Two models are considered: The Friedmann cosmology with scalar matter and the torus sector of the 2+1 gravity. For the Friedmann model, the transformations to the Kucha\v{r} description corresponding to three different popular time coordinates are shown to exist on the whole ADM phase space, which becomes a proper subset of the Kucha\v{r} phase spaces. The 2+1 gravity model is shown to admit a description by embedding variables everywhere, even at the points with additional symmetry. The transformation from the Kucha\v{r} to the ADM description is, however, many-to-one there, and so the two descriptions are inequivalent for this model, too. The most interesting result is that the new constraint surface is free from the conical singularity and the new dynamical equations are linearization stable. However, some residual pathology persists in the Kucha\v{r} description.Comment: Latex 2e, 29 pages, no figure

Constant mean curvature slicings of Kantowski-Sachs spacetimes

We investigate existence, uniqueness, and the asymptotic properties of constant mean curvature (CMC) slicings in vacuum Kantowski-Sachs spacetimes with positive cosmological constant. Since these spacetimes violate the strong energy condition, most of the general theorems on CMC slicings do not apply. Although there are in fact Kantowski-Sachs spacetimes with a unique CMC foliation or CMC time function, we prove that there also exist Kantowski-Sachs spacetimes with an arbitrary number of (families of) CMC slicings. The properties of these slicings are analyzed in some detail

A Framework for Spatio-Temporal Data Analysis and Hypothesis Exploration

We present a general framework for pattern discovery and hypothesis exploration in spatio-temporal data sets that is based on delay-embedding. This is a remarkable method of nonlinear time-series analysis that allows the full phase-space behaviour of a system to be reconstructed from only a single observable (accessible variable). Recent extensions to the theory that focus on a probabilistic interpretation extend its scope and allow practical application to noisy, uncertain and high-dimensional systems. The framework uses these extensions to aid alignment of spatio-temporal sub-models (hypotheses) to empirical data - for example satellite images plus remote-sensing - and to explore modifications consistent with this alignment. The novel aspect of the work is a mechanism for linking global and local dynamics using a holistic spatio-temporal feedback loop. An example framework is devised for an urban based application, transit centric developments, and its utility is demonstrated with real data

Dynamical Meson Melting in Holography

We discuss mesons in thermalizing gluon backgrounds in the N=2 supersymmetric QCD using the gravity dual. We numerically compute the dynamics of a probe D7-brane in the Vaidya-AdS geometry that corresponds to a D3-brane background thermalizing from zero to finite temperatures by energy injection. In static backgrounds, it has been known that there are two kinds of brane embeddings where the brane intersects the black hole or not. They correspond to the phases with melted or stable mesons. In our dynamical setup, we obtain three cases depending on final temperatures and injection time scales. The brane stays outside of the black hole horizon when the final temperature is low, while it intersects the horizon and settles down to the static equilibrium state when the final temperature is high. Between these two cases, we find the overeager case where the brane dynamically intersects the horizon although the final temperature is not high enough for a static brane to intersect the horizon. The interpretation of this phenomenon in the dual field theory is meson melting due to non-thermal effects caused by rapid energy injection. In addition, we comment on the late time evolution of the brane and a possibility of its reconnection.Comment: 42 pages, 23 figures, v2: references adde

Holographic charge localization at brane intersections

Using gauge/gravity duality, we investigate charge localization near an interface in a strongly coupled system. For this purpose we consider a top-down holographic model and determine its conductivities. Our model corresponds to a holographic interface which localizes charge around a (1+1)-dimensional defect in a (2+1)-dimensional system. The setup consists of a D3/D5 intersection at finite temperature and charge density. We work in the probe limit, and consider massive embeddings of a D5-brane where the mass depends on one of the field theory spatial directions, with a profile interpolating between a negative and a positive value. We compute the conductivity in the direction parallel and perpendicular to the interface. For the latter case we are able to express the DC conductivity as a function of background horizon data. At the interface, the DC conductivity in the parallel direction is enhanced up to five times with respect to that in the orthogonal one. We study the implications of broken translation invariance for the AC and DC conductivities.Comment: 36 pages, 12 figures. v2: typos corrected, JHEP versio

Axially symmetric Einstein-Straus models

The existence of static and axially symmetric regions in a Friedman-Lemaitre cosmology is investigated under the only assumption that the cosmic time and the static time match properly on the boundary hypersurface. It turns out that the most general form for the static region is a two-sphere with arbitrarily changing radius which moves along the axis of symmetry in a determined way. The geometry of the interior region is completely determined in terms of background objects. When any of the most widely used energy-momentum contents for the interior region is imposed, both the interior geometry and the shape of the static region must become exactly spherically symmetric. This shows that the Einstein-Straus model, which is the generally accepted answer for the null influence of the cosmic expansion on the local physics, is not a robust model and it is rather an exceptional and isolated situation. Hence, its suitability for solving the interplay between cosmic expansion and local physics is doubtful and more adequate models should be investigated.Comment: Latex, no figure