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

A comparison of soil moisture characteristics predicted by the Arya-Paris model with laboratory-measured data

Soil moisture characteristics predicted by the Arya-Paris model were compared with the laboratory measured data for 181 New Jersey soil horizons. For a number of soil horizons, the predicted and the measured moisture characteristic curves are almost coincident; for a large number of other horizons, despite some disparity, their shapes are strikingly similar. Uncertainties in the model input and laboratory measurement of the moisture characteristic are indicated, and recommendations for additional experimentation and testing are made

Elastic/plastic analyses of advanced composites investigating the use of the compliant layer concept in reducing residual stresses resulting from processing

High residual stresses within intermetallic and metal matrix composite systems can develop upon cooling from the processing temperature to room temperature due to the coefficient of thermal expansion (CTE) mismatch between the fiber and matrix. As a result, within certain composite systems, radial, circumferential, and/or longitudinal cracks have been observed to form at the fiber-matrix interface. The compliant layer concept (insertion of a compensating interface material between the fiber and matrix) was proposed to reduce or eliminate the residual stress buildup during cooling and thus minimize cracking. The viability of the proposed compliant layer concept is investigated both elastically and elastoplastically. A detailed parametric study was conducted using a unit cell model consisting of three concentric cylinders to determine the required character (i.e., thickness and material properties) of the compliant layer as well as its applicability. The unknown compliant layer mechanical properties were expressed as ratios of the corresponding temperature dependent Ti-24Al-11Nb (a/o) matrix properties. The fiber properties taken were those corresponding to SCS-6 (SiC). Results indicate that the compliant layer can be used to reduce, if not eliminate, radial and circumferential residual stresses within the fiber and matrix and therefore also reduce or eliminate the radial cracking. However, with this decrease in in-plane stresses, one obtains an increase in longitudinal stress, thus potentially initiating longitudinal cracking. Guidelines are given for the selection of a specific compliant material, given a perfectly bonded system

Vitiligo at Injection Site of PEG-IFN-α 2a in Two Patients with Chronic Hepatitis C: Case Report and Literature Review

A 72-year-old female and a 57-year-old male with chronic hepatitis C were treated with a combination therapy of pegylated interferon (PEG-IFN)-α 2a (180 μg s.c. once a week) and ribavirin (1,000 mg orally daily). This resulted in the destruction of melanocytes at the injection site in both patients. In the male patient, the depigmentation progressed to the surrounding skin area. The dermatologist concurred with vitiligo as the diagnosis in both patients. Injection and surrounding site vitiligo associated with PEG-IFN-α 2b treatment for hepatitis C was noticed in previous case studies. For the first time, the case reports below highlight the same immunological adverse event secondary to PEG IFN-α 2a/ribavirin combination therapy and explain, in part, the complex interaction between host immune response and viral genotype. In addition, we systematically review drug-induced vitiligo and autoimmune diseases associated with the depigmentation disorder

Agricultural soil moisture experiment, Colby, Kansas 1978: Measured and predicted hydrological properties of the soil

Predictive procedures for developing soil hydrologic properties (i.e., relationships of soil water pressure and hydraulic conductivity to soil water content) are presented. Three models of the soil water pressure-water content relationship and one model of the hydraulic conductivity-water content relationship are discussed. Input requirements for the models are indicated, and computational procedures are outlined. Computed hydrologic properties for Keith silt loam, a soil typer near Colby, Kansas, on which the 1978 Agricultural Soil Moisture Experiment was conducted, are presented. A comparison of computed results with experimental data in the dry range shows that analytical models utilizing a few basic hydrophysical parameters can produce satisfactory data for large-scale applications

Approximate range searching☆☆A preliminary version of this paper appeared in the Proc. of the 11th Annual ACM Symp. on Computational Geometry, 1995, pp. 172–181.

AbstractThe range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and ε>0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance εw of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in Rd can be preprocessed in O(n+logn) time and O(n) space, such that approximate queries can be answered in O(logn(1/ε)d) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O(logn+(1/ε)d−1) time. We also present a lower bound for approximate range searching based on partition trees of Ω(logn+(1/ε)d−1), which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times

Analisa Dampak Konvergensi Ifrs Ke Dalam PSAK 13, 16, Dan 30 Terhadap Aktivitas Perdagangan Saham Perusahaan Yang Terdaftar Di Bursa Efek Indonesia

The application of PSAK has been revised in reference with the IAS in the convergence of IFRS perspective. The objective of the research is to elaborate influence of this application on Trading Volume Activity (TVA) companies that implement it. This research uses quantitative method that focus to a single incident within a certain period of publication an annual financial statements of 56 companies who implemented PSAK 13, 16, and 30 revised in 2007. This study indicates that the publication of financial statements has not give a significant influence on the TVA despite there is an increase in trading activity at the time after publication.JEL Classification : M 41, L5