George Purdy - Academia.edu (original) (raw)

Papers by George Purdy

Research paper thumbnail of A proof of a consequence of Dirac's conjecture

Geometriae Dedicata, 1981

Research paper thumbnail of On the number of regions determined by n lines in the projective plane

Geometriae Dedicata, 1980

ABSTRACT Let d be an arrangement of n lines in the real projective plane, not all concurrent, let... more ABSTRACT Let d be an arrangement of n lines in the real projective plane, not all concurrent, letf2(~) be the number of regions into which the real projective plane is divided by the lines, and let Pk be the number of those regions having exactly k sides. Griinbaum shows in his book Arrangements and Spreads [2, p. 14] that f2 >t 2n - 2, and furthermore that if at most n - 2 lines are concurrent, then f2 >t 3n - 6. In this note, using an approach similar to [3, Theorem 4.1 ], we are able to extend this result to show THEOREM 1. lf n lines are given in the projective plane, and at most n - k go through any point, and n >>. 4k 2 + k + 1, then

Research paper thumbnail of Some external problems in geometry

Discrete Mathematics, 1974

Research paper thumbnail of Two results about points, lines and planes

Discrete Mathematics, 1986

Research paper thumbnail of E1775

The American Mathematical Monthly, 1966

Research paper thumbnail of Elementary Problems: E1845-E1854

The American Mathematical Monthly, 1966

Research paper thumbnail of Advanced Problems: 5340-5349

The American Mathematical Monthly, 1965

Research paper thumbnail of Elementary Problems: E1766-E1775

The American Mathematical Monthly, 1965

Research paper thumbnail of E1850

The American Mathematical Monthly, 1967

Research paper thumbnail of E1849

The American Mathematical Monthly, 1967

Research paper thumbnail of The minimum genus of a two-point universal graph

Journal of Combinatorial Theory, 1979

Research paper thumbnail of On Finding Ordinary or Monochromatic Intersection Points

Computing Research Repository, 2009

An algorithm is demonstrated that finds an ordinary intersection in an arrangement of nnn lines i... more An algorithm is demonstrated that finds an ordinary intersection in an arrangement of nnn lines in mathbbR2\mathbb{R}^2mathbbR2, not all parallel and not all passing through a common point, in time O(nlogn)O(n \log{n})O(nlogn). The algorithm is then extended to find an ordinary intersection among an arrangement of hyperplanes in mathbbRd\mathbb{R}^dmathbbRd, no ddd passing through a line and not all passing through

Research paper thumbnail of A Bichromatic Incidence Bound and an Application to Determined Planes

Computing Research Repository, 2010

We prove that there are O(m2/3k2/3n(d−2)/3+knd−2)O(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2})O(m2/3k2/3n(d−2)/3+knd−2) incidences between kkk red poin... more We prove that there are O(m2/3k2/3n(d−2)/3+knd−2)O(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2})O(m2/3k2/3n(d−2)/3+knd−2) incidences between kkk red points and mmm hyperplanes that are determined jointly by the red points and n−kn-kn−k blue points. This is a generalization of an incidence bound proved by Agarwal and Aronov \cite{AA92} (i.e., when k=nk=nk=n). We provide an explicit construction that attains the asymptotic result, showing that the bound is

Research paper thumbnail of Some numerical results on Fekete polynomials

Mathematics of Computation, 1975

Page 1. MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 129 JANUARY 1975, PAGES 7-23 Some Numerical... more Page 1. MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 129 JANUARY 1975, PAGES 7-23 Some Numerical Results on Fekete Polynomials By Paul T. Bateman, George B. Purdy and Samuel S. Wagstaff, Jr. Dedicated ...

Research paper thumbnail of Similar simplices in a d-dimensional point set

Symposium on Computational Geometry, 2007

We consider the problem of bounding the maximum possible number fk,d(n) of k- simplices that are ... more We consider the problem of bounding the maximum possible number fk,d(n) of k- simplices that are spanned by a set of n points in Rd and are similar to a given simplex. We first show that f2,3(n) = O(n13/6), and then tackle the general case, and show that fd 2,d(n) = O(nd 8/5) and1 fd 1,d(n) = O�(nd 72/55), for

Research paper thumbnail of Two results about points, lines and planes

Discrete Mathematics, 1986

Research paper thumbnail of Spheres Tangent to All the Faces of a Simplex

Journal of Combinatorial Theory, 1974

Research paper thumbnail of On the number of regions determined by n lines in the projective plane

Geometriae Dedicata, 1980

Let d be an arrangement of n lines in the real projective plane, not all concurrent, letf2(~) be ... more Let d be an arrangement of n lines in the real projective plane, not all concurrent, letf2(~) be the number of regions into which the real projective plane is divided by the lines, and let Pk be the number of those regions having exactly k sides. Griinbaum shows in his book Arrangements and Spreads [2, p. 14] that f2 >t

Research paper thumbnail of A binary algorithm for solving n by n systems of integer equations

Computers Mathematics With Applications, Dec 31, 1983

Research paper thumbnail of Software Protection Using ``Communal Key Cryptosystems

Research paper thumbnail of A proof of a consequence of Dirac's conjecture

Geometriae Dedicata, 1981

Research paper thumbnail of On the number of regions determined by n lines in the projective plane

Geometriae Dedicata, 1980

ABSTRACT Let d be an arrangement of n lines in the real projective plane, not all concurrent, let... more ABSTRACT Let d be an arrangement of n lines in the real projective plane, not all concurrent, letf2(~) be the number of regions into which the real projective plane is divided by the lines, and let Pk be the number of those regions having exactly k sides. Griinbaum shows in his book Arrangements and Spreads [2, p. 14] that f2 >t 2n - 2, and furthermore that if at most n - 2 lines are concurrent, then f2 >t 3n - 6. In this note, using an approach similar to [3, Theorem 4.1 ], we are able to extend this result to show THEOREM 1. lf n lines are given in the projective plane, and at most n - k go through any point, and n >>. 4k 2 + k + 1, then

Research paper thumbnail of Some external problems in geometry

Discrete Mathematics, 1974

Research paper thumbnail of Two results about points, lines and planes

Discrete Mathematics, 1986

Research paper thumbnail of E1775

The American Mathematical Monthly, 1966

Research paper thumbnail of Elementary Problems: E1845-E1854

The American Mathematical Monthly, 1966

Research paper thumbnail of Advanced Problems: 5340-5349

The American Mathematical Monthly, 1965

Research paper thumbnail of Elementary Problems: E1766-E1775

The American Mathematical Monthly, 1965

Research paper thumbnail of E1850

The American Mathematical Monthly, 1967

Research paper thumbnail of E1849

The American Mathematical Monthly, 1967

Research paper thumbnail of The minimum genus of a two-point universal graph

Journal of Combinatorial Theory, 1979

Research paper thumbnail of On Finding Ordinary or Monochromatic Intersection Points

Computing Research Repository, 2009

An algorithm is demonstrated that finds an ordinary intersection in an arrangement of nnn lines i... more An algorithm is demonstrated that finds an ordinary intersection in an arrangement of nnn lines in mathbbR2\mathbb{R}^2mathbbR2, not all parallel and not all passing through a common point, in time O(nlogn)O(n \log{n})O(nlogn). The algorithm is then extended to find an ordinary intersection among an arrangement of hyperplanes in mathbbRd\mathbb{R}^dmathbbRd, no ddd passing through a line and not all passing through

Research paper thumbnail of A Bichromatic Incidence Bound and an Application to Determined Planes

Computing Research Repository, 2010

We prove that there are O(m2/3k2/3n(d−2)/3+knd−2)O(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2})O(m2/3k2/3n(d−2)/3+knd−2) incidences between kkk red poin... more We prove that there are O(m2/3k2/3n(d−2)/3+knd−2)O(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2})O(m2/3k2/3n(d−2)/3+knd−2) incidences between kkk red points and mmm hyperplanes that are determined jointly by the red points and n−kn-kn−k blue points. This is a generalization of an incidence bound proved by Agarwal and Aronov \cite{AA92} (i.e., when k=nk=nk=n). We provide an explicit construction that attains the asymptotic result, showing that the bound is

Research paper thumbnail of Some numerical results on Fekete polynomials

Mathematics of Computation, 1975

Page 1. MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 129 JANUARY 1975, PAGES 7-23 Some Numerical... more Page 1. MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 129 JANUARY 1975, PAGES 7-23 Some Numerical Results on Fekete Polynomials By Paul T. Bateman, George B. Purdy and Samuel S. Wagstaff, Jr. Dedicated ...

Research paper thumbnail of Similar simplices in a d-dimensional point set

Symposium on Computational Geometry, 2007

We consider the problem of bounding the maximum possible number fk,d(n) of k- simplices that are ... more We consider the problem of bounding the maximum possible number fk,d(n) of k- simplices that are spanned by a set of n points in Rd and are similar to a given simplex. We first show that f2,3(n) = O(n13/6), and then tackle the general case, and show that fd 2,d(n) = O(nd 8/5) and1 fd 1,d(n) = O�(nd 72/55), for

Research paper thumbnail of Two results about points, lines and planes

Discrete Mathematics, 1986

Research paper thumbnail of Spheres Tangent to All the Faces of a Simplex

Journal of Combinatorial Theory, 1974

Research paper thumbnail of On the number of regions determined by n lines in the projective plane

Geometriae Dedicata, 1980

Let d be an arrangement of n lines in the real projective plane, not all concurrent, letf2(~) be ... more Let d be an arrangement of n lines in the real projective plane, not all concurrent, letf2(~) be the number of regions into which the real projective plane is divided by the lines, and let Pk be the number of those regions having exactly k sides. Griinbaum shows in his book Arrangements and Spreads [2, p. 14] that f2 >t

Research paper thumbnail of A binary algorithm for solving n by n systems of integer equations

Computers Mathematics With Applications, Dec 31, 1983

Research paper thumbnail of Software Protection Using ``Communal Key Cryptosystems