The purpose of this paper is to introduce a new instance of the Mesh Adaptive Direct Search (Mads) class of algorithms, which utilizes a more uniform distribution of poll directions than do other common instances, such as OrthoMads and LtMads. Our new implementation, called QrMads, bases its poll directions on an equal area partitioning of the n-dimensional unit sphere and the QR decomposition ...

This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize...

In this paper, we consider two restricted types of dynamic orthogonal segment intersection search problems. One is the problem in which the underlying set is updated only by insertions. In the other, the set is updated only by deletions. We show that an intermixed sequence of O ( n ) queries and updates in both problems can be executed on-line in O(n1ognSK) time and O(n1ogn) space, where K is t...

We consider deeply the relation between the orthogonality and the distinguishability of a set of arbitrary states (including multi-partite states). It is shown that if a set of arbitrary states can be distinguished by local operations and classical communication (LOCC), each of the states can be written as a linear combination of product vectors such that all product vectors of one of the state...

This paper presents a proximal point approach for computing the Riemannian or intrinsic Karcher mean of, n×n, symmetric positive definite (SPD) matrices. Our method derives from proximal point algorithm with Schur decomposition developed to compute minimum points of convex functions on SPD matrices set when it is seen as a Hadamard manifold. The main idea of the original method is preserved. Ho...

Under certain constraints on the parameters a, b and c, it is known that Schrödinger’s equation −d2ψ/dx2 + (ax + bx + cx)ψ = Eψ, a > 0 with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function ψ is the generating function for a set of orthogonal polynomials {P (t) n (x)} in the energy variable E. Some of the properties of these pol...

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial order. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given order defined on the interpolating no...

A refinable function φ(x) : Rn → R or, more generally, a refinable function vector 8(x) = [φ1(x), . . . , φr (x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φj (x − α) : α ∈ Zn, 1 ≤ j ≤ r} form an orthogonal set of functions in L2(Rn)...

Let P be an orthogonal polygon with n vertices. A sliding camera travels back and forth along an orthogonal line segment s ⊆ P corresponding to its trajectory. The camera sees a point p ∈ P if there is a point q ∈ s such that pq is a line segment normal to s that is completely contained in P . In the MinimumCardinality Sliding Cameras (MCSC) problem, the objective is to find a set S of sliding ...