Although multiple global navigation satellite systems (multi-GNSS) with more visible satellites have a high success rate, they make positioning time-consuming. Partial ambiguity resolution (PAR) can improve the efficiency of multi-GNSS; however, at present PAR cannot simultaneously achieve fast and high-precision positioning with a high success rate. Therefore, PAR based on ambiguity dilution of precision- and convex-hull-based satellite selection is proposed. The experimental results of the proposed PAR, its corresponding satellite selection algorithm, the classical PAR, and the low-cutoff-elevation-angle-based multi-GNSS show that the proposed PAR outperforms the classical PAR, i.e., it achieves fast and high-precision positioning with a success rate of 100⋅0%. Furthermore, in terms of R-ratio-test-based ambiguity validation, it improves the reliability of carrier-phase-based integrity monitoring of multi-GNSS and the corresponding satellite selection algorithms. In addition, its positioning accuracy is close to that of multi-GNSS and higher than that of the classical PAR, with maximum differences of 0⋅3 and 2⋅4 cm, respectively. The proposed single (dual) frequency-based PAR improves single/dual-frequency multi-GNSS efficiency by more than 54⋅9%/80⋅4% (42⋅0%/75⋅8%) when 14⋅4 (13⋅2) out of 24⋅4 satellites are selected.

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

The convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. To approximate the distribution of the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately most of the asymptotic results are computationally intractable. To address this computational intractability, we consider consistent bootstrapping schemes for certain cases. Let $S_n=\{X_i\}_{i=1}^{n}$ be a sequence of independent and identically distributed random points uniformly distributed on an unknown convex set in $\mathbb{R}^{d}$ ($d\ge 2$ ). We suggest a bootstrapping scheme that relies on resampling uniformly from the convex hull of $S_n$ . Moreover, the resampling asymptotic consistency of certain functionals of convex hulls is derived under this bootstrapping scheme. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. For $d=2$ , we investigate the asymptotic consistency of the suggested bootstrapping scheme for the area of the symmetric difference and the perimeter difference between the actual convex set and its estimate. In all cases the consistency allows us to rely on the suggested resampling scheme to study the actual distributions, which are not computationally tractable.

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\], where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$. Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

Synthetic seed banks are a useful tool for tracking how weed populations change over time. By sowing a known number of seeds of a given species within a quadrat with defined boundaries, an investigator can measure the number remaining and thereby calculate demographic rates (e.g., mortality). The alternative is to use in situ seeds and estimate their initial population density via sampling. To make a synthetic seed bank approach useful within an agricultural system subjected to soil disturbances such as tillage, one would need a way to account for seeds leaving the initial quadrat (i.e., a way to follow how the area encompassing the sown seeds changes over time). Without accounting for the change in location/extent of the synthetic seed bank, any field operation moving soil will create additional uncertainty in population size. Depending on the “aggressiveness” of specific field operations and the initial size of the quadrat, this effect might be negligible or so large as to be intractable. Here, we describe a method for following a synthetic seed bank over time using a “living boundary” of nondormant seeds, which effectively play the role of tracers used in the study of dynamics in other scientific disciplines. Study quadrats in East Lansing, MI, and Arlington, WI, were sown with giant foxtail and velvetleaf at a rate of 2,000 seeds m−2. The study quadrats were marked on the perimeter and diagonals using nondormant seeds of three marker species: kale, radish, and rye. The areas were then subjected to tillage and planting operations. Spatial coordinates of seedling locations for the marker and weed species were obtained through digital image processing. A nonparametric comparison of the spatial displacement of marker and weed species indicated that their empirical spatial distributions did not differ. The marker species quadrats described by the 50th, 90th, and 99th quantiles of movement contained all velvetleaf seedlings in Wisconsin, all velvetleaf seedlings in Michigan, and all giant foxtail seedlings in Michigan, respectively. The results suggest a simple rule for applying the method to field demography studies: after the original quadrat is deformed and seedlings have emerged, flag the polygon containing all marker seedlings to obtain the expanded quadrat containing the study weed population.

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affine invariance—n points P1,…, Pn distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñn of points among P1,…, Pn, which are vertices of the convex hull of (0, 1), P1,…, Pn, and (1, 0). Correspondingly, D̃n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñn and var . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

Evaluation of geometric deviations for the purpose of determining compliance with specified form tolerances requires the acquisition of numerous measured data points and extensive computation to accurately characterize the inspected part’s geometry. If there are not enough data points measured or when the measured points are not distributed properly on the measured surface a high level of uncertainty in characterizing the inspected part’s geometry can be expected. However, increasing the number of data point also significantly increases the computational time and also increases computational uncertainty by adding to instability of the optimization process required to find the minimum deviation zone. Selections of number and location of the measured date points need to be performed by understanding the significance of these two sources of uncertainties. This paper discussed the effect of sampling procedure on uncertainty and precision of flatness inspection.

In Groeneboom (1988) a central limit theorem for the number of vertices Nn of the convex hull of a uniform sample from the interior of a convex polygon was derived. To be more precise, it was shown that {Nn - (2/3)rlogn} / {(10/27)rlogn}1/2 converges in law to a standard normal distribution, if r is the number of vertices of the convex polygon from which the sample is taken. In the unpublished preprint Nagaev and Khamdamov (1991) a central limit result for the joint distribution of Nn and An is given, where An is the area of the convex hull, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of the number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of Nn and An in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Cabo and Groeneboom (1994) and Nagaev (1995).

In this paper we present simple extensions of earlier works on the optimal time to exchange one basket of log Brownian assets for another. A superset and subset of the optimal stopping region in the case where both baskets consist of multiple assets are obtained. It is also shown that a conjecture of Hu and Øksendal (1998) is false except in the trivial case where all the assets in a basket are the same processes.

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

We discuss the problem of computing points of IR n whoseconvex hull contains the Euclidean ball, and is containedin a small multiple of it. Given a polytope containing the Euclidean ball, we introduce its successor obtained by intersectionwith all tangent spaces to the Euclidean ball, whose normalspoint towards the vertices of the polytope. Starting from the L ∞ ball,we discuss the computation of the two first successors, andgive a complete analysis in the case when n=6.

We study the finite projective planes with linear programmingmodels. We give a complete description of the convex hull of thefinite projective planes of order 2. We give some integer linearprogramming models whose solution are, either a finiteprojective (or affine) plane of order n, or a (n+2)-arc.

Let n points be randomly and independently placed in Rd according to a common probability law. It is known that the expected volume for the convex hull of these points, in the cases where n - d ≥ 2 and even, is related linearly to expected volumes of the convex hulls for j points, j < n. We show that similar identities for these volumes hold almost surely - and in contexts where independence and communality of law do not apply. New geometric and topological identities developed here provide a foundation for this result.

In this paper we establish large deviation results on the number of extreme points of a homogeneous Poisson point process in the unit ball of Rd. In particular, we deduce an almost-sure law of large numbers in any dimension. As an auxiliary result we prove strong localization of the extreme points in an annulus near the boundary of the ball.

The convex hull of n independent random points in ℝd, chosen according to the normal distribution, is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the variance of the ith intrinsic volume of a Gaussian polytope in ℝd, d∈ℕ, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number of i-faces and for the ith intrinsic volume of a Gaussian polytope as n→∞.

A programming tactic involving polyhedra is reported that has been widely applied in the polyhedral analysis of (constraint) logic programs. The method enables the computations of convex hulls that are required for polyhedral analysis to be coded with linear constraint solving machinery that is available in many Prolog systems.

An exact expression is determined for the asymptotic constant c2 in the limit theorem by P. Groeneboom (1988), which states that the number of vertices of the convex hull of a uniform sample of n random points from a circular disk satisfies a central limit theorem, as n → ∞, with asymptotic variance 2πc2n1/3.

The focus of our attention is the limit distribution of the sum of independent and identically distributed random vectors from which all the extreme summands are removed. The problem is rather trivial if the summands are ordered by their norms. It is of much more interest when the vertices of the convex hull generated by the vectors are taken as the extremes.