Results on the intersection of randomly located sets

1975 ◽  
Vol 12 (4) ◽  
pp. 817-823 ◽  
Author(s):  
Franz Streit
Keyword(s):  
Euclidean Space ◽  
Surface Area ◽  
Arbitrary Dimension ◽  
Expected Value ◽  
Dimensional Euclidean Space ◽  
Common Intersection ◽  
Point Set ◽  
The Common

Randomly generated subsets of a point-set A0 in the k-dimensional Euclidean space Rk are investigated. Under suitable restrictions the probability is determined that a randomly located set which hits A0. is a subset of A0. Some results on the expected value of the measure and the surface area of the common intersection-set formed by n randomly located objects and A0 are generalized and derived for arbitrary dimension k.

Results on the intersection of randomly located sets

1975 ◽  
Vol 12 (04) ◽  
pp. 817-823 ◽  
Author(s):  
Franz Streit
Keyword(s):  
Euclidean Space ◽  
Surface Area ◽  
Arbitrary Dimension ◽  
Expected Value ◽  
Dimensional Euclidean Space ◽  
Common Intersection ◽  
Point Set ◽  
The Common

Randomly generated subsets of a point-set A 0 in the k-dimensional Euclidean space Rk are investigated. Under suitable restrictions the probability is determined that a randomly located set which hits A 0. is a subset of A 0 . Some results on the expected value of the measure and the surface area of the common intersection-set formed by n randomly located objects and A 0 are generalized and derived for arbitrary dimension k.

A more general form of a theorem of Crofton

1973 ◽  
Vol 10 (2) ◽  
pp. 479-482 ◽  
Author(s):  
H. Ruben ◽  
W. J. Reed
Keyword(s):  
Euclidean Space ◽  
Expected Value ◽  
Dimensional Euclidean Space

Let Dj be a domain in nj,-dimensional Euclidean space, for j = 1, …, k. Suppose that for each j = 1,…, k, Nj points are chosen independently at random in Dj. A theorem, which is an extension of a theorem of Crofton, is proved about the expected value of functions of the points.

ALGORITHMS FOR POINT SET MATCHING WITH k-DIFFERENCES

2006 ◽  
Vol 17 (04) ◽  
pp. 903-917
Author(s):  
TATSUYA AKUTSU
Keyword(s):  
Euclidean Space ◽  
Related Problem ◽  
Time Algorithm ◽  
Two Dimensions ◽  
Efficient Algorithms ◽  
Common Point ◽  
Dimensional Euclidean Space ◽  
Maximum Cardinality ◽  
Point Set ◽  
Special Case

The largest common point set problem (LCP) is, given two point set P and Q in d-dimensional Euclidean space, to find a subset of P with the maximum cardinality that is congruent to some subset of Q. We consider a special case of LCP in which the size of the largest common point set is at least (|P| + |Q| - k)/2. We develop efficient algorithms for this special case of LCP and a related problem. In particular, we present an O(k3n1.34 + kn2 log n) time algorithm for LCP in two-dimensions, which is much better for small k than an existing O(n3.2 log n) time algorithm, where n = max {|P|,|Q|}.

The Besicovitch dimension of Cartesian product sets

1950 ◽  
Vol 46 (3) ◽  
pp. 383-386 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Euclidean Space ◽  
Hausdorff Measure ◽  
Cartesian Product ◽  
Dimensional Euclidean Space ◽  
Point Set ◽  
Cartesian Product Sets ◽  
Product Sets

A. S. Besicovitch has defined the dimension of a point-set X in n-dimensional Euclidean space in terms of its exterior Hausdorff measure as follows (2). Let (δ, X) be any enumerable class of sets whose point-set union contains X and whose members are each of diameter less than δ. Let (δ, X) denote the class of all (δ, X).

A more general form of a theorem of Crofton

1973 ◽  
Vol 10 (02) ◽  
pp. 479-482 ◽  
Author(s):  
H. Ruben ◽  
W. J. Reed
Keyword(s):  
Euclidean Space ◽  
Expected Value ◽  
Dimensional Euclidean Space ◽  
Simple Points

Let Dj be a domain in nj ,-dimensional Euclidean space, for j = 1, …, k. Suppose that for each j = 1,…, k, Nj points are chosen independently at random in Dj. A theorem, which is an extension of a theorem of Crofton, is proved about the expected value of functions of the points.

scholarly journals Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space

2019 ◽  
Vol 27 (2) ◽  
pp. 37-65 ◽  
Author(s):  
Djavvat Khadjiev ◽  
İdris Ören
Keyword(s):  
Euclidean Space ◽  
Orthogonal Group ◽  
Dimensional Euclidean Space ◽  
Plane Curves ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Special Orthogonal Group ◽  
The Common ◽  
Global Invariants ◽  
The Given

AbstractIn this paper, for the orthogonal group G = O(2) and special orthogonal group G = O+(2) global G-invariants of plane paths and plane curves in two-dimensional Euclidean space E2 are studied. Using complex numbers, a method to detect G-equivalences of plane paths in terms of the global G-invariants of a plane path is presented. General evident form of a plane path with the given G-invariants are obtained. For given two plane paths x(t) and y(t) with the common G-invariants, evident forms of all transformations g ∈ G, carrying x(t) to y(t), are obtained. Similar results have obtained for plane curves.

ON ENUMERATING AND SELECTING DISTANCES

2001 ◽  
Vol 11 (03) ◽  
pp. 291-304 ◽  
Author(s):  
TIMOTHY M. CHAN
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Randomized Algorithm ◽  
Dimensional Euclidean Space ◽  
Running Time ◽  
Selection Problems ◽  
Deterministic Algorithms ◽  
Parametric Search ◽  
Enumeration Problem ◽  
Point Set

Given an n-point set, the problems of enumerating the k closest pairs and selecting the k-th smallest distance are revisited. For the enumeration problem, we give simpler randomized and deterministic algorithms with O(n log n+k) running time in any fixed-dimensional Euclidean space. For the selection problem, we give a randomized algorithm with running time O(n log n+n2/3k1/3 log 5/3n) in the Euclidean plane. We also describe output-sensitive results for halfspace range counting that are of use in more general distance selection problems. None of our algorithms requires parametric search.

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