This paper describes the theory and algorithms of distance transform for fuzzy subsets, called fuzzy distance transform (FDT). The notion of fuzzy distance is formulated by first defining the length of a path on a fuzzy subset and then finding the infimum of the lengths of all paths between two points. The length of a path π in a fuzzy subset of the n-dimensional continuous space n is defined a...

We introduce a modified dendrogram (MD) (with sub-trees to represent the feature space clusters) and display it in continuous space for multi-dimensional transfer function (TF) design and modification. Such a TF for direct volume rendering often employs a multi-dimensional feature space. In an n-dimensional (nD) feature space, each voxel is described using n attributes and represented by a vect...

Matrix descriptions of higher dimensional spherical branes are investigated. It is known that a fuzzy 2k-sphere is described by the coset space SO(2k+1)/U(k) and has some extra dimensions. It is shown that a fuzzy 2k-sphere is comprised of n k(k−1) 2 spherical D(2k − 1)-branes and has a fuzzy 2(k − 1)-sphere at each point. We can understand the relationship between these two viewpoints by the d...

Let V be an n-dimensional vector space over GF(q) and for integers k > t > 0 let m,(n, k, t) denote the maximum possible number of subspaces in a t-intersecting family 9 of k-dimensional subspaces of V, i.e.

We describe the pairs (p, n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.

We study the problem of computing the diameter of a  set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+varepsilon)$-approximation algorithm with $O(n+ 1/varepsilon^{d-1})$ time and $O(n)$ space, where $0 < varepsilonleqslant 1$. We also show that the proposed algorithm can be modified to a $(1+O(varepsilon))$-approximation algorithm with $O(n+...

This chapter reports the project MARS (Multiplicative Adaptive Refinement Search), which applies a new multiplicative adaptive algorithm for user preference retrieval to Web searches. The new algorithm uses a multiplicative query expansion strategy to adaptively improve and reformulate the query vector to learn users’ information preference. The algorithm has provable better performance than th...

• A quantum state Ψ is a linear superposition of basis states {|x〉}x and is described by the summation |Ψ〉 = ∑ x αx|x〉, with amplitudes αx ∈ C and |αx| the probability of observing the state x. • Formally, for an N dimensional system, the quantum state is represented by a "2 normalized vector in the N dimensional Hilbert with complex amplitudes. • Throughout this article we assume that our quan...

x αx|x〉, with amplitudes αx ∈ C and |α x| the probability of observing the state x. • Formally, for an N dimensional system, the quantum state is represented by a l2 normalized vector in the N dimensional Hilbert with complex amplitudes. • Throughout this article we assume that our quantum system is finite dimensional (the dimension is typically denoted by N). • A change of the state Ψ is repre...

We study 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of n-gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus 0 curves with n marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and ...