
	Where results make sense

Topic: Halfspace

	Robust Estimators of Location (Site not responding. Last check: 2007-10-13)
		Note: Although Tukey is credited with the notion of the halfspace median, it was Hotelling in 1929 who introduced this interpretation for the univariate median.
		Take the minimum number found over all halfspaces to be the depth of that point.
		The halfspace median is generally not a unique point.
	cgm.cs.mcgill.ca /~athens/Geometric-Estimators/halfspace.html (295 words)

	Another Simple Earth Model: Low-Velocity Layer Over a Halfspace (Site not responding. Last check: 2007-10-13)
		This is true with expection to the relative curvature and the wavelength differences of the refracted wavefield compared to the direct and the reflected wavefield.
		As the refracted arrival propagates through the halfspace, because it travels faster than the direct arrival in the layer, it begins to move across the layer boundary before the direct arrival.
		The refracted arrival is propogating horizontally at the velocity of the halfspace, the direct and the reflected arrivals propagate horizontally at the speed of the layer.
	gretchen.geo.rpi.edu /roecker/AppGeo96/lectures/seismic/simp1.html (581 words)

	qhalf -- halfspace intersection about a point
		Halfspace intersection by the convex hull of 6 points in 3-d: Number of halfspaces: 6 Number of non-redundant halfspaces: 6 Number of intersection points: 8 Statistics for: RBOX c
		The dual polytope is the convex hull of the vertices dual to the original faces in regard to the unit sphere (i.e., halfspaces at distance d from the origin are dual to vertices at distance 1/d).
		vertex - a dual vertex in the convex hull corresponding to a non-redundant halfspace
	www.qhull.org /html/qhalf.htm (1531 words)

	Two-Layered Media: Another Example
		Let's now fix the resistivities of the two layers and the halfspace and vary the thickness of the middle layer.
		For example, if we were to make the thickness of the middle layer very large, you would expect the apparent resistivity to approach the resistivity of the middle layer, as electrode spacing is increased, rather than approaching the resistivity of the halfspace.
		As thickness is increased, it is apparent that this hump represents a flattening in the apparent resistivity curve at the resistivity of the middle layer, 250 ohm.m Thus, if the middle layer is thick enough, it can be distinguished in the apparent resistivity curve.
	www.geo.ucalgary.ca /~maillol/goph365/RES/mlayer2.html (327 words)

	Apparent Resistivity Curves in Two-Layered Media (Site not responding. Last check: 2007-10-13)
		A suite of resistivity curves, each generated assuming a different resistivity for the underlying halfspace, is shown below (resistivity next to each curve indicates the resistivity of the halfspace).
		For this particular model notice that if the resistivity of the halfspace is larger than the resistivity of the lower layer that the affect of all three media can be discerned in the apparent resistivity curve.
		As electrode spacing increases the apparent resistivity decreases monotonically, eventually approaching the resistivity of the halfspace.
	gretchen.geo.rpi.edu /roecker/AppGeo96/lectures/res/mlayer1.html (375 words)

	qhull -- convex hull and related structures
		The Delaunay triangulation and furthest-site Delaunay triangulation are equivalent to a convex hull in one higher dimension.
		Halfspace intersection about a point is equivalent to a convex hull by polar duality.
		For halfspace intersection, an interior point may be prepended (see qhalf input).
	www.math.sunysb.edu /~sorin/online-docs/qhull/qhull.htm (709 words)

	Department of Computer Science
		Learning a halfspace is one of the oldest and most fundamental tasks in machine learning.
		Intersections of halfspaces are a powerful concept class; any convex body can be expressed as an intersection of halfspaces, and little is known about the complexity of learning the intersection of even two halfspaces.
		As a result, we give a perceptron-like algorithm for learning the intersection of a constant number of halfspaces which runs in polynomial time as long as the margin is at least $1/2^{\sqrt{\log n}}$.
	www.cs.uchicago.edu /events/267 (315 words)

	Resources at Fugro Airborne Surveys
		In this case the "halfspace" to which the system will respond is that of the conductive clay, and the response of the sand will have very little effect on the EM system.
		If the upper layer is more conductive than the halfspace, but thin enough to have only a small effect on the total signal received, the apparent resistivity measured will be closer to that of the halfspace.
		The effect of the upper layer, however, will be to boost the signal measured, making the system appear to be closer to the halfspace than it actually is, and the algorithm will produce a negative apparent depth.
	www.fugroairborne.com /resources/technical_notes/heli_em/res_algorithm.html (813 words)

	Geophysical Observations and the Crust-Mantle Transition
		An 80 layer transition zone is constructed with slownesses evenly distributed between 1/V2 and 1/V1, where V1 and V2 are the velocities in the layer and in the halfspace, respectively.
		The layer thickness is varied between 0 km and 0.5128 km with the result that the transition zone is centered at a 40 km depth and at its extreme range consists of a linear gradient between 19.4872 and 80.5228 km.
		The short period fundamental mode dispersion is controlled by the surface velocity and the long period by the halfspace velocity.
	www.eas.slu.edu /People/RBHerrmann/RFTN/rftn.html (1299 words)

	Dynamic stiffness of foundation embedded in layered halfspace based on wave propagation in cones (Site not responding. Last check: 2007-10-13)
		The dynamic stiffness of a foundation embedded in a multiple-layered halfspace is calculated postulating one-dimensional wave propagation in cone segments.
		This method postulating one-dimensional wave propagation in cone segments with reflections and refractions at layer interfaces is evaluated, calculating the dynamic stiffness of a foundation embedded in a multiple-layered halfspace.
		For sites resting on a flexible halfspace and fixed at the base, engineering accuracy (deviation of ±20%) is achieved for all degrees of freedom with a vast parameter variation.
	www.igt.ethz.ch /dynDBpages/PublicationDisplay640.htm (235 words)

	ECCC Report TR06-061 and related Papers (Site not responding. Last check: 2007-10-13)
		In the noise-free case, when a halfspace consistent with all the training examples exists, the problem can be solved in polynomial time using linear programming.
		However, under the promise that a halfspace consistent with a fraction (1-eps) of the examples exists (for some small constant eps > 0), it was not known how to efficiently find a halfspace that is correct on even 51% of the examples.
		This settles a question that was raised by Blum et al in their work on learning halfspaces in the presence of random classification noise, and in some more recent works as well.
	eccc.hpi-web.de /eccc-reports/2006/TR06-061 (296 words)

	Graphics Archive - Halfspace intersection with Qhull by Brad Barber (Site not responding. Last check: 2007-10-13)
		A halfspace is all points to one side of a plane.
		The cone with the small square facets is the convex hull of two cospherical polygons and an apex.
		For halfspace intersection, this is needed when more than three planes meet at the same point.
	www.geom.uiuc.edu /graphics/pix/Special_Topics/Computational_Geometry/half.html (266 words)

	flipcode - Octrees For Visibility (Site not responding. Last check: 2007-10-13)
		If we are testing for a node to be completely within the positive halfspace, then we must choose the point within the node that has the least potential of going positive.
		Note that the latter two cases (node is completely within the postive halfspace of a single plane, or bisected by that plane) is not our complete answer.
		Since there are multiple planes involved, a node may be within the positive halfspace of one plane, but within the negative halfspace of another.
	www.flipcode.com /cgi-bin/fcarticles.cgi?show=64466 (2516 words)

	Optimization Online - Optimal distance separating halfspace
		Abstract: One recently proposed criterion to separate two datasets in discriminant analysis, is to use a hyperplane which minimises the sum of distances to it from all the misclassified data points.
		Here all distances are supposed to be measured by way of some fixed norm,while misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace.
		It also follows that in any dimension any optimal separating halfspace always balances the misclassified points, where the balancing criterion now takes the shape of the used gauges into account.
	www.optimization-online.org /DB_HTML/2004/10/970.html (245 words)

	Figure Captions
		Data are shown as a bar centered on the nominal back azimuth and apparent velocity, with the bar's orientation parallel to the azimuth of the the fast direction, and its length proportional to the delay.
		The symmetry axis is horizontal in both layers, and has an azimuth of N30E in the top layer and N60E in the bottom layer.
		The anisotropic tensor is the arithmetic mean of the tensors in the two-layer case.
	www.ldeo.columbia.edu /users/menke/ApUr/node11.html (1167 words)

	Head Waves From a Dipping Layer: Shooting Down Dip (Site not responding. Last check: 2007-10-13)
		As was the case in the other examples where velocity increases with depth, in this case, head waves will be generated along the top of the halfspace that will propagate back up through the layer and be observed on the surface of the Earth.
		Notice, if we were able to put geophones inside the Earth along a line that passes through the source and parallels the top of the halfspace (fl dashed line), we would observe the head wave as if it had been generated on a flat boundary.
		Thus, the times that it takes the head wave to travel from the source back up to the fl dashed line are identical to the times we've discussed for flat boundaries.
	www.geo.ucalgary.ca /~maillol/goph365/SEIS/dip1.html (680 words)

	Plain Waves HowTo
		Both halfspaces are homogenious and of infinite extension, i.e.
		The seismic traces for the source signals implemented are calculated by multiplying the corresponding Fourier transforms by the response functions in the frequency domain before transforming back to the time domain.
		in the upper halfspace or at the surface,
	www.ifg.tu-clausthal.de /java/pwav/pwav-e.html (2041 words)

	Failure to Cut box with halfspace that doesn't intersect box.
		Failure to Cut box with halfspace that doesn't intersect box.
		The matter of the halfspace is entirely above the box, therefore the "cut" operation should do nothing and just return the box.
		And here is the call stack showing exactly where the exception was thrown.
	www.opencascade.org /org/forum/thread_9598 (248 words)

	Article in IWGGCR--6: Dynamic-stiffness matrix of surface foundation layered halfspace based on stiffness-matrix ...
		The soil consists of horizontal layers of visco-elastic material with hysteretic damping which rest on a halfspace with the same material characteristics.
		It requires the evaluation of the flexibility-influence functions (Green functions) of the layered site which are calculated as integrals in the wave-number domain.
		The dynamic-stiffness coefficients are given for a rigid disc as a function of frequency for a visco-elastic halfspace, a single layer resting on a halfspace and a single layer built-in at its base.
	www.iaea.org /inis/aws/htgr/abstracts_c/abst_iwggcr6_9.html (264 words)

	DIMACS Workshop on Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications
		The halfspace depth of a d-dimensional point ø relative to a data set X1; : : : ;Xn ε IRd is defined as the smallest number of observations in any closed halfspace with boundary through ø.
		For bivariate data, the halfspace depth of a point can be computed in O(n log n) time (Ma tousek 1991, Rousseeuw and Ruts 1996).
		When the depth function is halfspace or simplicial depth and F1; : : : ;FM are elliptically contoured, the previous rule assigns x to the distribution with minimum Mahalanobis' distance which suggests that the results should be comparable with those produced by the linear or quadratic discriminant functions.
	dimacs.rutgers.edu /Workshops/Depth/abstracts.html (5135 words)

	Qhull functions, macros, and data structures
		Geometrically, a vertex is a point with d coordinates and a facet is a halfspace.
		The roundoff error in halfspace computation is accounted for by computing the distance from vertices to the halfspace.
		A facet's hyperplane may define a halfspace that does not include the interior point.This is called a flipped facet.
	www.msri.org /about/computing/docs/qhull/qh-c.htm (4658 words)

	Geothermics : 1-D Model HowTo
		The applet simulates the conductive cooling / heating of simple 1-d model structures with simplified material properties and for simplified boundary conditions :
		(halfspace X <&0 : initial temperature difference D_TEMP to halfspace X > 0 at T = 0),
		harmonic temperature variaton of amplitude D_TEMP at the surface Z = 0 of a halfspace Z >= 0.
	www.ifg.tu-clausthal.de /java/thrm/thrm_how-e.html (450 words)

	Random Sampling, Halfspace Range Reporting, and Construction of (<=k)-Levels in Three Dimensions (ResearchIndex) (Site not responding. Last check: 2007-10-13)
		This document uses CoBlitz to cache paper downloads.
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		Abstract: Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(log n+k) expected time for an output size k.
	citeseer.ist.psu.edu /207082.html (836 words)

	New lower bounds for halfspace emptiness (Site not responding. Last check: 2007-10-13)
		The author derives a lower bound of /spl Omega/(n/sup 4/3/) for the halfspace emptiness problem: given a set of n points and n hyperplanes in R/sup 5/, is every point above every hyperplane?
		This matches the best known upper bound to within polylogarithmic factors, and improves the previous best lower bound of /spl Omega/(nlogn).
		The lower bound applies to partitioning algorithms in which every query region is a polyhedron with a constant number of facets.
	csdl.computer.org /comp/proceedings/focs/1996/7594/00/75940472abs.htm (155 words)

	Dr. Musharraf Zaman - Publications
		"Analysis of Circular Plate-Elastic Halfspace Interaction Using an Energy Approach" (M.M. Zaman, A.R. Kukreti and A. Issa).
		"Analysis of Foundation-Elastic Halfspace Interaction Using a Mixed-Variational Approach" (M.M. Zaman, N. Uddin and M.O. Faruque).
		"Analysis of the Interaction Between a Moderately Thick Plate and an Isotropic Elastic Halfspace by Using a Mixed-Variational Principle" (M.O. Faruque and M.M. Zaman).
	faculty-staff.ou.edu /Z/Musharraf.Zaman-1/publications.html (7600 words)

	1.3 Polyhedral Geometry
		The set of all nonzero elements of a recession cone are called its rays.
		Observation 1.14A polyhedron is the intersection of finitely many halfspaces.
		Sketch of proof: Choose two points of P which lie in the same halfspace H; show that any convex combination of them also lies in H. Thus any halfspace is convex.
	www.ms.uky.edu /~sills/webprelim/sec013.html (1309 words)

	Erik D. Demaine, Jeff Erickson, and Stefan Langerman: On the Complexity of Halfspace Volume Queries
		Erik D. Demaine, Jeff Erickson, and Stefan Langerman, ``On the Complexity of Halfspace Volume Queries,'' in Proceedings of the 15th Canadian Conference on Computational Geometry (CCCG 2003), Halifax, Nova Scotia, Canada, August 11-13, 2003, pages 159-160.
		with n vertices, a halfspace volume query asks for the volume of P intersect H for a given halfspace H.
		We show that, for d ≥ 3, such queries can require Ω(n) operations even if the polyhedron P is convex and can be preprocessed arbitrarily.
	theory.lcs.mit.edu /~edemaine/papers/VolumeQueries_CCCG2003 (126 words)

	New Lower Bounds for Halfspace Emptiness - Erickson (ResearchIndex)
		Abstract: We derive a lower bound of\Omega n 4=3) for the halfspace emptiness problem: Given a set of n points and n hyperplanes in IR 5, is every point above every hyperplane ?
		This matches the best known upper bound to within polylogarithmic factors, and improves the previous best lower bound of \Omega n log n).
		21 How hard is halfspace range searching (context) - Bronnimann, Chazelle et al.
	citeseer.ist.psu.edu /246603.html (712 words)

	[No title]
		-- --/T A domain is a set of convexes that are bounded by a set of halfspaces --/T defined by planes.
		------------------------------------------------------------- begin declare @convexID int select @convexID = max(convexID) + 1 from HalfSpace where domainID = @domainID set @convexID = coalesce(@convexID, 0) return @convexID end go ------------------------------------------------------------------------ -- add a half-space constraint to a convex, returns the id of the new halfspace.
		declare @myConvex int select @myConvex = min(convexID) from HalfSpace where domainID = @domain and convexID > @convex --===================================================== -- the ME table is a list of all the halfPlanes of this convex.
	research.microsoft.com /~Gray/papers/Polygon.doc (2288 words)

	Qhull code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection about a Point
		Qhull computes the convex hull, Delaunay triangulation, Voronoi diagram, halfspace intersection about a point, furthest-site Delaunay triangulation, and furthest-site Voronoi diagram.
		Clarkson's hull program with exact arithmetic for convex hulls, Delaunay triangulations, Voronoi volumes, and alpha shapes.
		Fukuda's cdd program for halfspace intersection and convex hulls
	www.qhull.org (545 words)

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