Minimum Number Of Points Research Articles

Interval Packing and Covering in the Boolean Lattice

Let be the hypergraph whose points are the subsets X of [n] := {1,…,n} with l≤ |X| ≤ u, l < u, and whose edges are intervals in the Boolean lattice of the form I = {C ⊆[n] : X⊆C⊆Y} where |X| = l, |Y| = u, X ⊆ Y.We study the matching number i.e. the the maximum number of pairwise disjoint edges, and the covering number i.e. the minimum number of points which cover all edges. We prove that max and that for every ε > 0 the inequalities hold, where for the lower bounds we suppose that n is not too small. The corresponding fractional numbers can be determined exactly. Moreover, we show by construction that

A computer aided tolerancing tool II: Tolerance analysis

A computer aided tolerance analysis tool is presented that assists the designer in evaluating worst case quality of assembly after tolerances have been specified. In tolerance analysis calculations, sets of equations are generated. The number of equations can be restricted by using a minimum number of points in which quality of assembly is calculated. The number of points needed depends on the type of surface association. The number of parameters in the set of equations can be reduced by considering the most critical direction for the assembly condition. The latter direction, called virtual plan fragment direction, is determined using a virtual plan fragment table, based on an analogy to the plan fragment table used in degrees of freedom (DOF) analysis. This reduced set of equations is then solved and optimized in order to find the maximum/minimum values for the assembly condition using simulated annealing. This method for tolerance analysis has been implemented in a feature based (re-)design support system called FROOM, as part of the functional tolerancing module.

Histogram regression estimation using data-dependent partitions

We establish general sufficient conditions for the $L_2$-consistency of multivariate histogram regression estimates based on data-dependent partitions. These same conditions insure the consistency of partitioning regression estimates based on local polynomial fits, and, with an additional regularity assumption, the consistency of histogram estimates for conditional medians. Our conditions require shrinking cells, subexponential growth of a combinatorial complexity measure and sublinear growth of restricted cell counts. It is not assumed that the cells of every partition be rectangles with sides parallel to the coordinate axis or that each cell contain a minimum number of points. Response variables are assumed to be bounded throughout. Our results may be applied to a variety of partitioning schemes. We established the consistency of histograms regression estimates based on cubic partitions with data-dependent offsets, k-thresholding in one dimension and empirically optimal nearest-neighbor clustering schemes. In addition, it is shown that empirically optimal regression trees are consistent when the size of the trees grows with the number of samples at an appropriate rate.

Consistency of data-driven histogram methods for density estimation and classification

We present general sufficient conditions for the almost sure $L_1$-consistency of histogram density estimates based on data-dependent partitions. Analogous conditions guarantee the almost-sure risk consistency of histogram classification schemes based on data-dependent partitions. Multivariate data are considered throughout. In each case, the desired consistency requires shrinking cells, subexponential growth of a combinatorial complexity measure and sublinear growth of the number of cells. It is not required that the cells of every partition be rectangles with sides parallel to the coordinate axis or that each cell contain a minimum number of points. No assumptions are made concerning the common distribution of the training vectors. We apply the results to establish the consistency of several known partitioning estimates, including the $k_n$-spacing density estimate, classifiers based on statistically equivalent blocks and classifiers based on multivariate clustering schemes.

The Number of Points of an Empirical or Poisson Process Covered by Unions of Sets

For thed-dimensional empirical process and thed-dimensional homogeneous Poisson process we give weak laws for the maximal and minimal number of points in unions of sets from certain families.

A unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids

Given a set of n points in R 2 bounded within a rectangular floor F, and a rectangular plate P of specified size, we consider the following two problems: find an isothetic position of P such that it encloses (i) maximum and (ii) minimum number of points, keeping P totally contained within F. For both of these problems, a new algorithm based on interval tree data structure is presented, which runs in O( nlog n) time and consumes O( n) space. If polygonal objects of arbitrary size and shape are distributed in R 2, the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity. Finally, the algorithm is extended for identifying a cuboid, i.e., a rectangular parallelepiped that encloses maximum number of polyhedral objects in R 3. Thus, the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition.

Algebraic functions for recognition

In the general case, a trilinear relationship between three perspective views is shown to exist. The trilinearity result is shown to be of much practical use in visual recognition by alignment-yielding a direct reprojection method that cuts through the computations of camera transformation, scene structure and epipolar geometry. Moreover, the direct method is linear and sets a new lower theoretical bound on the minimal number of points that are required for a linear solution for the task of reprojection. The proof of the central result may be of further interest as it demonstrates certain regularities across homographics of the plane and introduces new view invariants. Experiments on simulated and real image data were conducted, including a comparative analysis with epipolar intersection and the linear combination methods, with results indicating a greater degree of robustness in practice and a higher level of performance in reprojection tasks. >

On the number of points of a homogeneous poisson process

We investigate the dense and sparse regions of a d -dimensional Poisson process and establish strong laws for both the maximal and the minimal number of points in families of sets of a certain volume.

Rejection power of a horizontal RPC telescope for left and right coming cosmic muons

The possibility of performing neutrino astronomy by means of a detector above the ground depends critically on the feasibility of a rejection power on the order of 10 11 required to discriminate the enormous background of cosmic downward going muons from the signal of upward going muons produced by neutrinos. In order to check whether and how this rejection is obtainable, we have built in the Physics Department of the University of Bari a horizontal cosmic muon telescope (MINI) instrumented with resistive plate counters. By performing time-of-flight measurements, we have estimated the rejection power of our telescope for left and right coming cosmic muons. The rejection dependence on a few fundamental parameters like minimum number of points per track, telescope length, RPC time resolution and on trigger configuration has been investigated.

A generalization of Menger's theorem for certain unicyclic graphs

LetG be a unicyclic graph such that at most two points on its cycle have degrees ?3. Then the minimum number of points separating any independent setS of points inG is the maximum number of disjoint paths between the points of S.

Finding minimum areak-gons

Given a setP ofn points in the plane and a numberk, we want to find a polygon[Figure not available: see fulltext.] with vertices inP of minimum area that satisfies one of the following properties: (1)[Figure not available: see fulltext.] is a convexk-gon, (2)[Figure not available: see fulltext.] is an empty convexk-gon, or (3)[Figure not available: see fulltext.] is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn3). The space complexity isO(n) fork=4 andO(kn2) fork?5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.

A lower bound for structuring element decompositions

A theoretical lower bound on the number of points required in the decomposition of morphological structuring elements is described. It is shown that the decomposition of an arbitrary N-point structuring element will require at least (3 ln N/ln 3)points. Using this lower bound it is possible to find the optimal decompositions (in terms of the minimum number of unions or the minimum number of points) for all one-dimensional connected line segments. L-dimensional rectangles may be decomposed by optimally decomposing the L one-dimensional line segments that describe the rectangle. >

Effects of Harvest on Antlers of Simulated Populations of Elk

Because the long-term effects of harvest on elk (Cervus elaphus) antlers are largely unkown, I used computer simulations to evaluate a variety of harvest plans. Parameters were systematically altered in the simulations to determine the effect on harvest yields and frequencies of genes influencing the number of antler points. Although the yield and the amount of change in gene frequencies varied as a result of the alterations, the pattern for each harvest plan was consistent. Some plan, like those restricting harvest of branch antlered bulls to certain years, always increased the frequency of alleles promoting more points, whereas plans based on a minimum number of points always decreased the frequency of these alleles

Covering and guarding polygons using L k -sets

The Art Gallery Problem is the problem of finding a minimum number of points (called guards) in a given polygon such that every point in the polygon is visible to at least one of the guards. Chvatal [5] was the first to show that, in the worst case, [n/3] such points will suffice for any polygon of n sides. O'Rourke [15] later showed that only [n/4] guards were needed if line segments, rather than points, were allowed as guards. In this paper, we unify these results, and extend them to many other classes of guards, while using a generalization of visibility known as link-visibility. In particular, we present the following theorems: The results of Chvatal and O'Rourke are special cases of a corollary of these theorems. Other such special cases are bounds on the cardinality of guard sets for star-shaped, convex, L k -convex, and segment-visible guards. We also obtain bounds on the maximum number of pieces in a minimum cover of a polygon by such sets.

Model robust extrapolation designs

We seek designs which are optimal in some sense for extrapolation when the true regression function is in a certain class of regression functions. More precisely, the class is defined to be the collection of regression functions such that its ( h + 1)-th derivative is bounded. The class can be viewed as representing possible departures from an ‘ideal’ model and thus describes a model robust setting. The estimates are restricted to be linear and the designs are restricted to be with minimal number of points. The design and estimate sought is minimax for mean square error. The optimal designs for cases X = [0, ∞] and X = [-1, 1], where X is the place where observations can be taken, are discussed.

Quadrature formulae with the weight functionX1/2

The method of undetermined coefficients is used to establish the third, fourth and fifth degree formulae with respect to the weight function x 1/2 where the region of integration is the space Ω = [(x y): 0≦ x≦ l and −l≦ y≦ l]. Since the minimum number of points in numerical integrations are very important, we note that the fourth degree formulae does not attain the minimum points while the fifth degree formula does. We prove that the minimum points for the fifth degree formula is attained by constructing the same fifth degree formula using Radon's method [4]. However, in future we shall investigate an alternative approach for the possibility of constructing a fourth degree formula with least points, see [5].

Minimal linear spaces

A decent linear space DLS( k) is a linear space with minimal line size at least three and with maximal line size exactly k. Denote by v k (resp. b k ) the minimum number of points (resp. lines) in a DLS( k). We determine the numbers v k and b k for all k and prove that each DLS( k) with b k lines has v k points. Thus the DLS( k)'s with b k lines are the minimal linear spaces.

Conditional connectivity

AbstractFor a noncomplete graph G, the traditional definition of its connectivity is the minimum number of points whose removal results in a disconnected subgraph with components H1,…,Hk. The conditional connectivity of G with respect to some graph‐theoretic property P is the smallest cardinality of a set S of points, if any, such that every component Hi of the disconnected graph G ‐ S has property P. A survey of promising properties P is presented. Questions for various P‐connectivities are listed in analogy with known results on connectivity and line‐connectivity.

On a measure of communication network vulnerability

AbstractA measure of communication network vulnerability was studied in a recent article by Boesch, Harary, and Kabell. The persistence (line persistence) is the minimum number of points (lines) whose removal from a graph increases its diameter. Their approach differed from earlier work on this topic in that they examined the analogs of Menger's theorem for these invariants. A result of Lovász, Neumann‐Lara, and Plummer is used to prove modified versions of their theorems (the original proofs are incorrect). Some related problems are also solved, including one posed by Hartman and Rubin.

Application of the theory of symmetry of nonrigid molecules to the calculation of the conformational dependences of the torsional potential and dipole moment of acetone‐related molecules

AbstractWe have investigated the conformational dependences of the torsional potential and dipole moment of double‐rotor molecules related to acetone, using semiempirical and ab initio calculations and expressing the results in terms of limited Fourier‐series expansions. The use of the isodynamic operations of nonrigid molecules to obtain symmetry‐adapted quasianalytic forms for the various properties helps compute the representative surfaces with a minimum number of points. Potential surfaces have been calculated for planar ground‐state acetone (CNDO/2, STO/3G, and STO/4‐31G) and both pyramidal excited‐triplet acetone and ground‐state dimethylamine (CNDO/2). For groundstate acetone STO/4‐31G brings the results obtained with STO/3G closer to those from CNDO/2 and from experiment. The potential surface of excited‐triplet acetone appears intermediate between those of ground‐state acetone and dimethylamine. For dipole moments the convergence of the harmonic expansions of the vector components is slower than that of the torsional potential whereas that of the vector magnitude is faster.