Set Of Distinct Points Research Articles

Rank 2 vector bundles over $${\mathbb {P}}^{2}(\mathbb {C})$$ P 2 ( C ) whose sections vanish on points in general position

In this paper we address the classification of rank 2 vector bundles \(E,\) over \({\mathbb {P}}^{2}(\mathbb {C}),\) such that the zero locus of a general section of \(E\) is a set of distinct points in general position. We also discuss the existence of smooth threefolds \(X={\mathbb {P}}(E),\) arising from such bundles.

Nonabelian theta functions of positive genus

'. Let Cg be a smooth projective irreducible curve over C of genus g ≥ 1 and let {p 1 ,..., p s } be a set of distinct points on Cg. We fix a nonnegative integer l and denote by M g (p, λ) the moduli space of parabolic semistable vector bundles of rank r on Cg with trivial determinant and fixed parabolic structure of type A = (λ 1 ,..., λ s ) at p = (p 1 ,..., p s ), where each weight λ i is in P l (SL(r)). On M g (p, λ) there is a canonical line bundle L(λ, l), whose global sections are called generalized parabolic SL(r)-theta functions of order l. In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.

Probabilistic algorithms for extreme point identification

For a finite set of distinct points S = {pi, i ε I} , in ℝ d there exists Î ⊆ I such that all points in Ŝ = {pi , i ε Î are extreme points and conv(Ŝ) = conv(S). Since a point pk is extreme if and only if the inequality is necessary with respect to the representation of the polar dual S Δ of S, Ŝ can be found by classifying the inequalities in the representation as necessary or redundant. Thus, the problem of finding Ŝ is polynomial. This paper shows the advantage of using probabilistic hit-and-run algorithms applied to the polar dual for the quick identification of points in Ŝ and shows how, in an application to a certain cluster analysis problem, it can be used to also identify points in S/Ŝ. Further, it shows that the hit-and-run variant known as Stand-and-Hit provides approximations for the exterior solid angles at the extreme points of S.

Locally residual currents and Dolbeault cohomology on projective manifolds

Let X be a projective manifold of dimension n, and n hypersurfaces Y i ( 1 ⩽ i ⩽ n ) on X, defining ample line bundles, in complete intersection position. After introducing sheaves of locally residual currents, we enunciate the following two main theorems. First, for any positive integer i, the Dolbeault cohomology group H i ( Ω X q ) of the sheaf of holomorphic q-forms on X can be computed as the ith cohomology group of some complex of residual currents on X. We get from this the theorem of [A. Dickenstein, M. Herrera, C. Sessa, On the global liftings of meromorphic forms, Manuscripta Math. 47 (1984) 31–54] that any locally residual current on X which is ∂ ¯ -exact is globally residual. Secondly, let us assume that Y 1 ∩ ⋯ ∩ Y p ( 1 ⩽ p ⩽ n ) are reduced complete intersections. We get another exact sequence computing H i ( Ω X n ) by restricting to residual currents obtained from meromorphic forms with simple poles on the Y i . We deduce from this a reformulation of the main theorem of [B. Khesin, A. Rosly, R. Thomas, A polar De Rham theorem, Topology 43 (2004) 1231–1246], saying that we can compute the cohomology groups H i ( Ω X n ) by the cohomology of a complex of principal value currents. We also deduce from this the result from [P. Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976) 321–390] that if Y 1 ∩ ⋯ ∩ Y n is a set of distinct points { P 1 , … , P s } , then for any sequence of s complex numbers c i ( 1 ⩽ i ⩽ s ) , there is a global meromorphic n-form Ψ with simple poles on each Y i such that: ( ∀ i , 1 ⩽ i ⩽ s ) Res Y 1 , … , Y n P i Ψ = c i iff ∑ i = 1 s c i = 0 . We give proofs of the theorems by mean of several exact sequences of sheaves of locally residual currents. We conclude by giving an application to the Hodge conjecture, giving some new formulation using our theorems.

On 0-dimensional schemes with all permissible conductor degrees

IfX is a set of distinct points in ℙ2 with given graded Betti numbers, we produce a new set of pointsY with the same graded Betti numbers asX which admits all possible conductor degrees according to the graded Betti numbers. Moreover, for such schemes we can compute the conductor degree for each point.

Decompositions of the Hilbert Function of a Set of Points in ℙn

AbstractLet H be the Hilbert function of some set of distinct points in ℙn and let α = α(H) be the least degree of a hypersurface of ℙn containing these points. Write α = ds + ds–1 + … + d1 (where di > 0). We canonically decompose H into s other Hilbert functions and show how to find sets of distinct points , lying on reduced hypersurfaces of degrees ds, …, d1 (respectively) such that the Hilbert function of is and the Hilbert function of is H. Some extremal properties of this canonical decomposition are also explored.

Calculation of mean central dose in interstitial brachytherapy using Delaunay triangulation.

In 1997 the ICRU published Report 58 "Dose and Volume Specification for Reporting Interstitial Therapy" with the objective of addressing the problem of absorbed dose specification for reporting contemporary interstitial therapy. One of the concepts proposed in that report is "mean central dose." The fundamental goal of the mean central dose (MCD) calculation is to obtain a single, readily reportable and intercomparable value which is representative of dose in regions of the implant "where the dose gradient approximates a plateau." Delaunay triangulation (DT) is a method used in computational geometry to partition the space enclosed by the convex hull of a set of distinct points P into a set of nonoverlapping cells. In the three-dimensional case, each point of P becomes a vertex of a tetrahedron and the result of the DT is a set of tetrahedra. All treatment planning for interstitial brachytherapy inherently requires that the location of the radioactive sources, or dwell positions in the case of HDR, be known or digitized. These source locations may be regarded as a set of points representing the implanted volume. Delaunay triangulation of the source locations creates a set of tetrahedra without manual intervention. The geometric centers of these tetrahedra define a new set of points which lie "in between" the radioactive sources and which are distributed uniformly over the volume of the implant. The arithmetic mean of the dose at these centers is a three dimensional analog of the two-dimensional triangulation and inspection methods proposed for calculating MCD in ICRU 58. We demonstrate that DT can be successfully incorporated into a computerized treatment planning system and used to calculate the MCD.

A Matrix Problem with Application to Rapid Solution of Integral Equations

Let $\{ z_j \} _{j = 0}^{n - 1} $ be a set of distinct points in the complex plane $\mathbb{C}$, and introduce the $n \times n$ matrix $A_n = [a_{jk} ]_{j,k = 0}^{n - 1} $, $a_{jk} = (z_j - z_k )^{ - 1} ,\, j \ne k$ and $a_{jj} = 0$. Recently Golub and Trummer raised the question of whether or not, for an arbitrary vector $x \in \mathbb{C}^n $, $A{\bf x}$ can be computed in fewer than $O(n^2 )$ arithmetic operations by using the structure of $A_n $. In this paper it is assumed that there is a smooth $2\pi $-periodic bijective function $z(t)$ such that $z_j = z(2\pi (j - 1)/n),\, j = 1(1)n$, and shown that when n increases, there is a sequence of matrices of low rank $\tilde A_n ,\, n = 1,2,3, \cdots $ such that $\tilde A_n \to A_n $ as $n \to \infty $ and $\tilde A_n {\bf x}$ can be computed in $O(n\log n)$ arithmetic operations. The method to construct the matrices $\tilde A_n $ is then used in a fast solution scheme for Fredholm integral equations of the second kind with smooth periodic kernels. The integral equations are discretized by the trapezoidal rule using the nodes $z_j = z(2\pi j/n)$, $0 \leqq j < n$, and it is shown that arbitrarily accurate approximate solutions can be computed in $O(n\log n)$ arithmetic operations for large n, provided that $z(t)$ is sufficiently smooth. When the asymptotic analysis is not applicable, fast iterative $O(n^2 )$ solution methods are obtained. The scheme is applied to the solution of a Fredholm integral equation of the second kind of plane potential theory and Cauchy singular integral equations.

Separating Pairs of Points by Standard Boxes

Let A be a set of distinct points in ℝ d . A 2-subset { a , b } of A is called separated if there exists a closed box with sides parallel to the axes, containing a and b but no other points of A . Let s(A) denote the number of separated 2-sets of A and put f ( n , d ) = max { s ( A ): A ⊂ ℝ d , | A | = n }. We show that f ( n , 2) = [ n 2 /4] + n − 2 for all n ≥2 and that for each fixed dimension d f ( n , d ) = ( 1 − 1 / 2 2 d − 1 − 1 ) ⋅ n 2 / 2 + o ( n 2 ) .

Nonpolynomial interpolation

Here interpolation is meant in the following sense: given f ε C¦a, b¦. and given a set of distinct points in ¦ a, b¦ and a linearly independent set {ifu 0,...,u n} of continuous functions on ¦ a, b¦, the interpolating function L n f is the unique linear combination of u 0,..., u n that coincides with f at the given points, if such a linear combination exists. In the classical case of Lagrange interpolation. u i ( x) is a polynomial of degree i. Here we allow other choices, and prove a generalization of the mean-convergence theorem of Erdös and Turán: it is shown that if a certain condition is satisfied, then L n f converges to f, in an appropriate L 2 sense, for all continuous functions f for which E n(f) →0 where E n(f) is the error of best uniform approximation by a linear combination of u 0,..., u n . In particular, this mean convergence property is shown to hold for interpolation by the leading eigenfunctions of a regular Sturm-Liouville eigenvalue problem, if the interpolation points are taken to be the zeros of the “next” eigenfunction. (The cigenfunctions are ordered so that the eigenvalues increase.)

On a Bound Useful in the Theory of Factorial Designs and Error Correcting Codes

Consider a finite projective space $PG(r - 1, s)$ of $r - 1$ dimensions, $r \geqq 3$, based on the Galois field $GF_s$, where $s = p^h, p$ being a prime. A set of distinct points in $PG(r - 1, s)$ is said to be a non-collinear set, if no three are collinear. The maximum number of points in such a non-collinear set is denoted by $m_3(r, s)$. It is the object of this paper to find a new upper bound for $m_3(r, s)$. This bound is of importance in the theory of factorial designs and error correcting codes. The exact value of $m_3(r, s)$ is known when either $r \leqq 4$ or when $s = 2$. When $r \geqq 5, s > 3$, the best values for the upper bound on $m_3(r, s)$ are due to Tallini [10] and Barlotti [1]. Our bound improves these when $s = 3$ or when $s$ is even.

A simple triangulation method for smooth manifolds

We first triangulate a compact closed m-manifold M of differentiability class C (r>l) in a euclidean space = E. The method is simpler than earlier methods (see References) and is applicable to a wider class of spaces (see (F) and (G) below). (A) For a given rj>0, let (#i, • • • , aM) be a set of distinct points on M such that each point of M is at distance <rj from at least one point a*. Let d be the euclidean distance function in E For each * G ( 1 , • • • ,M),let

Some Metric Properties of Descriptive Planes

This paper is concerned with the study of certain properties of planes in which Axioms I-VIII of Veblen's System of Axioms for Geometry * are satisfied. Such a plane will be called a descriptive plane. A point P of a descriptive plane S will be said to be a limit point of a subset L of S if and only if every triangle of S which incloses t P also incloses a point of L distinct from P. From the results obtained in Part I, it may be concluded that a necessary and sufficient condition that a descriptive plane S be metric is that S contain a countable set of distinct points which has a limit point. In Part IV, it is shown that not every descriptive placne is metric. A conclusion which may be drawn from Part II is that a necessary and sufficient condition that a descriptive plane S be in one-to-one continutous correspondence with an everywhere den-se subset of the number-plane is that S contain a separable segment. In Part III, it is shown that not every metric descriptive plane is separable.