Search results for " set"
showing 10 items of 2095 documents
Gradient-based shape optimisation of ultra-wideband antennas parameterised using splines
2010
Methodology enabling the gradient-based optimisation of antennas parameterised using B-splines is presented. Use of the spline parametrisation allows us to obtain versatile new shapes, whereas the geometry can be represented with a small set of design variables. Moreover, good control over admissible geometries is retained. Advantages of gradient-based optimisation methods are quick convergence, and the fact that the obtained design can be guaranteed to be a local optimum. Focus of this study is to present techniques that enable the computation of exact gradients of the discrete problem, even though the complexity of the geometries does not permit establishing analytical expressions for the…
Incremental bipartite drawing problem
2001
Abstract Layout strategies that strive to preserve perspective from earlier drawings are called incremental. In this paper we study the incremental arc crossing minimization problem for bipartite graphs. We develop a greedy randomized adaptive search procedure (GRASP) for this problem. We have also developed a branch-and-bound algorithm in order to compute the relative gap to the optimal solution of the GRASP approach. Computational experiments are performed with 450 graph instances to first study the effect of changes in grasp search parameters and then to test the efficiency of the proposed procedure. Scope and purpose Many information systems require graphs to be drawn so that these syst…
Online Metric Learning Methods Using Soft Margins and Least Squares Formulations
2012
Online metric learning using margin maximization has been introduced as a way to learn appropriate dissimilarity measures in an efficient way when information as pairs of examples is given to the learning system in a progressive way. These schemes have several practical advantages with regard to global ones in which a training set needs to be processed. On the other hand, they may suffer from a poor performance depending on the quality of the examples and the particular tuning or other implementation details. This paper formulates several online metric learning alternatives using a passive-aggressive schema. A new formulation of the online problem using least squares is also introduced. The…
Solving the pentahedron problem
2015
Nowadays, all geometric modelers provide some tools for specifying geometric constraints. The 3D pentahedron problem is an example of a 3D Geometric Constraint Solving Problem (GCSP), composed of six vertices, nine edges, five faces (two triangles and three quadrilaterals), and defined by the lengths of its edges and the planarity of its quadrilateral faces. This problem seems to be the simplest non-trivial problem, as the methods used to solve the Stewart platform or octahedron problem fail to solve it. The naive algebraic formulation of the pentahedron yields an under-constrained system of twelve equations in eighteen unknowns. Even if the use of placement rules transforms the pentahedron…
Deflation-Based FastICA With Adaptive Choices of Nonlinearities
2014
Deflation-based FastICA is a popular method for independent component analysis. In the standard deflation-base d approach the row vectors of the unmixing matrix are extracted one after another always using the same nonlinearities. In prac- tice the user has to choose the nonlinearities and the efficiency and robustness of the estimation procedure then strongly depends on this choice as well as on the order in which the components are extracted. In this paper we propose a novel adaptive two- stage deflation-based FastICA algorithm that (i) allows one to use different nonlinearities for different components and (ii) optimizes the order in which the components are extracted. Based on a consist…
Additive properties of fractal sets on the parabola
2023
Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The main corollary of the paper states that if $0 0$. This information is deduced from an $L^{6}$ bound for the Fourier transforms of Frostman measures on $\mathbb{P}$. If $0 0$, then there exists $\epsilon = \epsilon(s) > 0$ such that $$ \|\hat{\mu}\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + \epsilon)} $$ for all sufficiently large $R \geq 1$. The proof is based on a reduction to a $\delta$-discretised point-circle incidence problem, and eventually to the $(s,2s)$-…
X-ray Tomography of One-forms with Partial Data
2021
If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions.
Polynomial and horizontally polynomial functions on Lie groups
2022
We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and we assume that $S$ Lie generates $\mathfrak g$. We say that a function $f:\mathbb G\to \mathbb R$ (or more generally a distribution on $\mathbb G$) is $S$-polynomial if for all $X\in S$ there exists $k\in \mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous defini…
On the shape of compact hypersurfaces with almost constant mean curvature
2015
The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capillarity problems.
Prescribing the behaviour of geodesics in negative curvature
2010
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon desc…