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1.
Given a collection ℬ of balls in a three-dimensional space, we wish to explore the cavities, voids, and tunnels in the complement space of ∪ℬ. We introduce the pathway axis of ℬ as a useful subset of the medial axis of the complement of ∪ℬ and prove that it satisfies several desirable geometric properties. We present an algorithm that constructs the pathway graph of ∪ℬ, a piecewise-linear approximation of the pathway axis. At the heart of our approach is an approximation scheme that constructs a collection K{\mathcal{K}} of same-size balls that approximate ℬ so that the Hausdorff distance between ∪ℬ and èK\bigcup{\mathcal{K}} is bounded by a prescribed parameter. We prove a bound on the ratio between the number of balls in K{\mathcal{K}} and the number of balls in ℬ. We employ this bound and the approximation scheme to show how to approximate the persistence diagrams for ∪ℬ, which can be used to extract major topological features such as the large voids and tunnels in the complement of ∪ℬ. We show that our approach is superior in terms of complexity to the standard point-sample approaches for the two problems that we address in this paper: approximating the pathway axis of ℬ and approximating the persistence diagrams for ∪ℬ. In a companion paper we introduce MolAxis, a tool for the identification of channels in macromolecules that demonstrates how the pathway graph and the persistence diagrams are used to identify plausible pathways in the complement of molecules.  相似文献   

2.
We define a vector representation V(u)of elliptic Ding-Iohara algebra u(q,t,p).Furthermore,we construct the tensor products of the vector representations and the Fock modulesF(u)by taking the inductive limit of certain subspaces in the finite tensor products of vector representations.  相似文献   

3.
Using inclusion-exclusion, we can write the indicator function of a union of finitely many balls as an alternating sum of indicator functions of common intersections of balls. We exhibit abstract simplicial complexes that correspond to minimal inclusion-exclusion formulas. They include the dual complex, as defined in [3], and are characterized by the independence of their simplices and by geometric realizations with the same underlying space as the dual complex.  相似文献   

4.
We introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball.  相似文献   

5.
We prove the Chern-Weil formula forSU(n + l)-singular connections over the complement of an embedded oriented surface in a smooth four-manifold. The number of representations of a positive integer n as a sum of nonvanishing squares is given in terms of the number of its representations as a sum of squares. Using this number-theoretic result, we study the irreducible SU(n +1)-representations of the fundamental group of the complement of an embedded oriented surface in smooth four-manifold.  相似文献   

6.
In this paper, we propose to study deformable necklaces—flexible chains of balls, called beads, in which only adjacent balls may intersect. Such objects can be used to model macro-molecules, muscles, ropes, and other linear objects in the physical world. We exploit this linearity to develop geometric structures associated with necklaces that are useful for collision detection in physical simulations. We show how these structures can be implemented efficiently and maintained under necklace deformation. In particular, we study a bounding volume hierarchy based on spheres which can be used for collision and self-collision detection of deforming and moving necklaces. As our theoretical and experimental results show, such a hierarchy is easy to compute and, more importantly, is also easy to maintain when the necklace deforms. Using this hierarchy, we achieve a collision detection upper bound of O(nlogn) in two dimensions and O(n2−2/d) in d-dimensions, d3. To our knowledge, this is the first subquadratic bound proved for a collision detection algorithm using predefined hierarchies. In addition, we show that the power diagram, with the help of some additional mechanisms, can be used to detect self-collisions of a necklace in a way that is complementary to the sphere hierarchy.  相似文献   

7.
By using the idea of Wakimoto's free field, we construct a class of representations for the Lie superalgebra D(2, 1; α) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter λ. Then we obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter λ satisfies (λ + m)(λ-(1+α/α)≠m)=0 for any m ∈ Z+.  相似文献   

8.
Given a set P of n points in the plane, we seek two squares such that their center points belong to P, their union contains P, and the area of the larger square is minimal. We present efficient algorithms for three variants of this problem: in the first the squares are axis parallel, in the second they are free to rotate but must remain parallel to each other, and in the third they are free to rotate independently.  相似文献   

9.
A method for robust and efficient medial axis transform (MAT) of arbitrary domains using distance solutions (or level sets) is presented. The distance field, d, is calculated by solving the hyperbolic-natured eikonal equation. The solution is obtained on Cartesian grids. Both the fast-marching method (FMM) and fast-sweeping method (FSM) are used to calculate d. Medial axis point clouds are then extracted based on the distance solution via a simple criterion: the Laplacian or the Hessian determinant of d(x). These point clouds in the pixel/voxel space are further thinned to single pixel wide so that medial axis curves or surfaces can be connected and splined. As an alternative to other methods, the current d-MAT procedure bypasses difficulties that are usually encountered by pure geometric methods (e.g. the Voronoi approach), especially in three dimensions, and provides better accuracy than pure thinning methods. It is also shown that the d-MAT approach provides the potential to sculpt/control the MAT form for specialized solution purposes. Various examples are given to demonstrate the current approach.  相似文献   

10.
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