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

Electrochemical immunosensor for tumor necrosis factor-alpha detection in undiluted serum

An immunosensor for the sensitive detection and estimation of tumor necrosis factor-alpha (TNF-α) in undiluted serum has been developed via an electrochemical enzyme-linked immunosorbent assay (ELISA) process. Electrochemical sensing was performed using a TNF-α specific monoclonal antibody modified self-assembled monolayer of dithiobis(succinimidyl propionate) on a comb-shaped gold electrode microarray. After anti-TNF-α antibody binding, unreacted active groups of DTSP were blocked using ethanol amine (EA) and nonspecific binding was prevented using phosphate buffer based starting block T20 (SB). Sensitive and disposable SB–EA–anti-TNF-α/DTSP/Au electrodes were exposed to solutions with different TNF-α concentrations for 20 min in undiluted serum. Conversion of 4-aminophenyl phosphate to 4-aminophenol and its electrochemical oxidation was utilized for indirect estimation of TNF-α. Results for SB–anti-TNF-α/DTSP/Au electrodes indicate that the sensors can be used for the sensitive estimation of TNF-α in undiluted serum in the range 500 pg/ml to 100 ng/ml with a detection limit of 60 pg/ml and sensitivity of 0.46 (ng/ml)−1. Negligible interference from serum and other biomarker proteins was observed. The described electrochemical ELISA is much faster than conventional ELISA and can be applied for sensing of a range of analytes in real patient samples

Electrochemical ELISA Protein Biosensing in Undiluted Serum Using a Polypyrrole-Based Platform

An electrochemical enzyme-linked immunosorbent assay (ELISA) biosensor platform using electrochemically prepared ~11 nm thick carboxylic functionalized popypyrrole film has been developed for bio-analyte measurement in undiluted serum. Carboxyl polypyrrole (PPy-COOH) film using 3-carboxy-pyrrol monomer onto comb-shaped gold electrode microarray (Au) was prepared via cyclic voltammetry (CV). The prepared Au/PPy-COOH was then utilized for electrochemical ELISA platform development by immobilizing analyte-specific antibodies. Tumor necrosis factor-alpha (TNF-α) was selected as a model analyte and detected in undiluted serum. For enhanced performance, the use of a polymeric alkaline phosphatase tag was investigated for the electrochemical ELISA. The developed platform was characterized at each step of fabrication using CV, electrochemical impedance spectroscopy and atomic force microscopy. The bioelectrodes exhibited linearity for TNF-α in the 100 pg/mL–100 ng/mL range when measured in spiked serum, with limit of detection of 78 pg/mL. The sensor showed insignificant signal disturbance from serum proteins and other biologically important proteins. The developed platform was found to be fast and specific and can be applicable for testing and measuring various biologically important protein markers in real samples

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

Approximate Geometric MST Range Queries

Range searching is a widely-used method in computational geometry for efficiently accessing local regions of a large data set. Typically, range searching involves either counting or reporting the points lying within a given query region, but it is often desirable to compute statistics that better describe the structure of the point set lying within the region, not just the count. In this paper we consider the geometric minimum spanning tree (MST) problem in the context of range searching where approximation is allowed. We are given a set P of n points in R^d. The objective is to preprocess P so that given an admissible query region Q, it is possible to efficiently approximate the weight of the minimum spanning tree of the subset of P lying within Q. There are two natural sources of approximation error, first by treating Q as a fuzzy object and second by approximating the MST weight itself. To model this, we assume that we are given two positive real approximation parameters eps_q and eps_w. Following the typical practice in approximate range searching, the range is expressed as two shapes Q^- and Q^+, where Q^- is contained in Q which is contained in Q^+, and their boundaries are separated by a distance of at least eps_q diam(Q). Points within Q^- must be included and points external to Q^+ cannot be included. A weight W is a valid answer to the query if there exist subsets P\u27 and P\u27\u27 of P, such that Q^- is contained in P\u27 which is contained in P\u27\u27 which is contained in Q^+ and wt(MST(P\u27)) <= W <= (1+eps_w) wt(MST(P\u27\u27)). In this paper, we present an efficient data structure for answering such queries. Our approach uses simple data structures based on quadtrees, and it can be applied whenever Q^- and Q^+ are compact sets of constant combinatorial complexity. It uses space O(n), and it answers queries in time O(log n + 1/(eps_q eps_w)^{d + O(1)}). The O(1) term is a small constant independent of dimension, and the hidden constant factor in the overall running time depends on d, but not on eps_q or eps_w. Preprocessing requires knowledge of eps_w, but not eps_q

Methods and compositions related to viral inhibition

Disclosed herein are compounds, compositions and methods related to viral inhibition. In some forms, the compounds, compositions and methods are related to binding RNA

Prevalence and severity of dental fluorosis in some endemically afflicted villages of district Doda, Jammu and Kashmir, India

A cross-sectional study was conducted among the residents of three villages of Doda district, J&amp;K. A total of 391 individuals (202 males and 189 females) were examined from 60 households by house to house survey for recording the prevalence and severity of dental fluorosis by using Dean’s Index (1942) along with WHO health assessment form (1997b) and Community fluorosis Index. Of the total studied population 299 (76.47%) have found to be affected with various grades of dental fluorosis with moderate (33.5%) level of fluorosis to be the most frequent category observed. Prevalence of dental fluorosis was more in males (86.61%) than females (63.49%). No significant relation (P&gt;0.05) between prevalence of dental fluorosis to the socioeconomic status was found. Community fluorosis Index was found to be 2.05 which denotes “marked” category of public health significance. High prevalence and high community fluorosis Index suggest that fluorosis is a major public health problem in the area

Methods and Compositions Related to Viral Inhibition

Disclosed herein are compounds, compositions and methods related to viral inhibition. In some forms, the compounds, compositions and methods are related to binding RNA

Congenital constriction band syndrome: a case report

Congenital constriction bands are anomalous bands formed during antenatal period that encircle a digit or an extremity in a fetus leading to a wide spectrum of manifestations, ranging from shallow grooves to acrosyndactyly and amputations. It is also associated with skeletal and other birth defects like craniofacial abnormalities, spinal dysraphism, porencephaly, hydrancephaly and visceral body wall malformations. We hereby present one such case which presented to us with below knee amputation of the lower limb and syndactyly