Early structure formation from cosmic string loops

We examine the effects of cosmic strings on structure formation and on the ionization history of the universe. While Gaussian perturbations from inflation are known to provide the dominant contribution to the large scale structure of the universe, density perturbations due to strings are highly non-Gaussian and can produce nonlinear structures at very early times. This could lead to early star formation and reionization of the universe. We improve on earlier studies of these effects by accounting for high loop velocities and for the filamentary shape of the resulting halos. We find that for string energy scales G\mu > 10^{-7} the effect of strings on the CMB temperature and polarization power spectra can be significant and is likely to be detectable by the Planck satellite. We mention shortcomings of the standard cosmological model of galaxy formation which may be remedied with the addition of cosmic strings, and comment on other possible observational implications of early structure formation by strings.Comment: 22 pages, 10 figures. References adde

Minimality and mutation-equivalence of polygons

We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine the mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface; or (ii) a surface with only singularities of type 1/3(1,1).Comment: 29 page

Time Discrete Geodesic Paths in the Space of Images

In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach, where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals over a set of image intensity maps and pairwise matching deformations. For square-integrable input images the existence of discrete, connecting geodesic paths defined as minimizers of this variational problem is shown. Furthermore, $\Gamma$-convergence of the underlying discrete path energy to the continuous path energy is proved. This includes a diffeomorphism property for the induced transport and the existence of a square-integrable weak material derivative in space and time. A spatial discretization via finite elements combined with an alternating descent scheme in the set of image intensity maps and the set of matching deformations is presented to approximate discrete geodesic paths numerically. Computational results underline the efficiency of the proposed approach and demonstrate important qualitative properties.Comment: 27 pages, 7 figure

Graph Variogram: A novel tool to measure spatial stationarity

Irregularly sampling a spatially stationary random field does not yield a graph stationary signal in general. Based on this observation, we build a definition of graph stationarity based on intrinsic stationarity, a less restrictive definition of classical stationarity. We introduce the concept of graph variogram, a novel tool for measuring spatial intrinsic stationarity at local and global scales for irregularly sampled signals by selecting subgraphs of local neighborhoods. Graph variograms are extensions of variograms used for signals defined on continuous Euclidean space. Our experiments with intrinsically stationary signals sampled on a graph, demonstrate that graph variograms yield estimates with small bias of true theoretical models, while being robust to sampling variation of the space.Comment: Submitted to IEEE Global Conference on Signal and Information Processing 2018 (IEEE GlobalSIP 2018), Nov 2018, Anaheim, CA, United States. (https://2018.ieeeglobalsip.org/

A Posteriori Error Control for the Binary Mumford-Shah Model

The binary Mumford-Shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. This paper presents robust a posteriori error estimates for a natural error quantity, namely the area of the non properly segmented region. To this end, a suitable strictly convex and non-constrained relaxation of the originally non-convex functional is investigated and Repin's functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the $L^2$-norm. In combination with a suitable cut out argument, a fully practical estimate for the area mismatch is derived. This estimate is incorporated in an adaptive meshing strategy. Two different adaptive primal-dual finite element schemes, and the most frequently used finite difference discretization are investigated and compared. Numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl