On the Combinatorial Complexity of Approximating Polytopes

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $\mathrm{diam}(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error parameter $\varepsilon > 0$, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from $K$ is at most $\varepsilon \cdot \mathrm{diam}(K)$. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $O(1/\varepsilon^{(d-1)/2})$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $\tilde{O}(1/\varepsilon^{(d-1)/2})$, where $\tilde{O}$ conceals a polylogarithmic factor in $1/\varepsilon$. This is a significant improvement upon the best known bound, which is roughly $O(1/\varepsilon^{d-2})$. Our result is based on a novel combination of both old and new ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering. Finally, we use a deterministic adaptation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.Comment: In Proceedings of the 32nd International Symposium Computational Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and Computational Geometr

Marrying Universal Dependencies and Universal Morphology

The Universal Dependencies (UD) and Universal Morphology (UniMorph) projects each present schemata for annotating the morphosyntactic details of language. Each project also provides corpora of annotated text in many languages - UD at the token level and UniMorph at the type level. As each corpus is built by different annotators, language-specific decisions hinder the goal of universal schemata. With compatibility of tags, each project's annotations could be used to validate the other's. Additionally, the availability of both type- and token-level resources would be a boon to tasks such as parsing and homograph disambiguation. To ease this interoperability, we present a deterministic mapping from Universal Dependencies v2 features into the UniMorph schema. We validate our approach by lookup in the UniMorph corpora and find a macro-average of 64.13% recall. We also note incompatibilities due to paucity of data on either side. Finally, we present a critical evaluation of the foundations, strengths, and weaknesses of the two annotation projects.Comment: UDW1

Implementasi Undang-Undang Republik Indonesia Nomor 26 Tahun 2007 Tentang Penataan Ruang Dalam Penyediaan Ruang Terbuka Hijau (Studi Pada Penataan Ruang Dalam Penyediaan Ruang Terbuka Hijau Di Kota Pasuruan)

The development process on each regional, particularly the urban, often put aside availability of Open Green Scape (OGS). Eventhough OGS is basic necessities a city. Constitution No.26 year 2007 about area arrangement explained that every area at least has 30% GOA from the whole area while Pasuruan City has 20.01% OGS at this time. The research intends to know how the government implements that constitution in order to reach number of OGS that determinated including the proponent and inhibitting factors. Researcher uses qualitative research method by descriptive approach. Based on research, the government publish PERDA No.31 year 2011 about RTRW Pasuruan City that existed decision about OGS that should be achieved by Pasuruan City year 2011-2013. The realizing, government do that according to priority and necessary by society, because the limits of fund and less communication. Then it need planning based on priority, good communication dan control by society

Pengaruh Umur, Pendidikan, Pekerjaan terhadap Pendapatan Rumah Tangga Miskin di Desa Bebandem

Poverty continues to be a phenomenal problem throughout Indonesia history, as well as the circumstances in Bebandem village, Karangasem Bali. With regard to the conditions that have been occured in that village, this study aims to: (1) To find out the effect of age, education and type of work simultaneously on the income of poor households in the Bebandem village, Karangasem, (2) To find out the effect of age, education, and type of work partially on the income of poor households in the Bebandem village, Karangasem, and (3) To find out the dominant effect among age, education and type of work on the income of poor households in the Bebandem village, Karangasem. By using number of population were 95 respondents with data analysis techniques of multiple linear regression. The results shows that: (1) In simultaneous the factors of age, education and type of work, have significant effect on the income of poor households in the Bebandem village, Karangasem. (2) In Partial the variable of age, education and type of work have positive and significant effect on the income of poor households in the Bebandem village, Karangasem. Age variable have negative affect on the incomes of poor households in the Bebandem village, Karangasem. (3) the dominant factor have affect on the income of poor households in the Bebandem village, Karangasem is age variable