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The radial basis function interpolation is an effective method for approximation on scattered data. However, this interpolation method suffers from the contradiction between the accuracy and the numerical stability. With considering the distribution of the interpolation points, a bounded random shape variation scheme for the radial basis function is developed to circumvent the problem of numerical stability. In this scheme, the shape variation is bounded by the value that is determined by the maximum distance and the minimal distance which are applied to describe the average density of the interpolation centers. Within this bound, the shape of the MQ is modified through a random scheme. With applying this bounded randomly variable shape scheme, the accuracy and the stability of the MQ interpolation are balanced. Comparisons on the accuracy and the condition number of the interpolation matrix between the constant shaped MQ interpolation and this bounded randomly variable shaped MQ interpolation have been made to verify the conclusion. Furthermore, this scheme is integrated in the dual reciprocity boundary element method in the analysis of three dimensional elastic problems. Results of the numerical examples demonstrated that the developed scheme improved the accuracy of the dual reciprocity boundary element method stably.

Background: The supporting quadric method(SQM) is a versatile method for designing freeform optics for desired irradiance redistribution, but the time of solution optimization increases rapidly with the refinement of the mapping grid. Objective: As the complexity of light distribution is getting higher and higher, time consumed will also increase exponentially. This paper proposes an idea of applying the deep neural network method to optical design. Methods: In this article, we established a special corresponding relationship and prepared a dataset, which underwent deep network learning and training. Finally, a hybrid design method of deep learning and optical design was realized and verified. Results: Compared with the traditional method, this method is more efficient. Here, we used a deep neural network(DNN) to accelerate the freeform optical design. After the DNN was trained by a sample set consisting of a uniform pattern and eight different Chinese characters represented by an array with 11 × 11, it can generate a character's reflector within few milliseconds. Conclusion: As proof of this new method, a character pattern reflector was manufactured and tested, and the experimental irradiance distribution was found close to the expectation, which means that the neural network has the excellent capability to memorize all of the learned characters. SQM combined with DNN has the potential to establish a particular “optical font library” and even offers a promising path for rapid freeform optical design to realize the function of “optical typography”.

[Abstract] We analyze the characteristic polynomial associated to an ellipsoid and another quadric in the context of the contact detection problem. We obtain a necessary and sufficient condition for an efficient method to detect contact. This condition, named smallness condition, is a feature on the size and the shape of the quadrics and can be checked directly from their parameters. Under this hypothesis, contact can be noticed by means of the expressions in a discriminant system of the characteristic polynomial. Furthermore, relative positions can be classified through the sign of the coefficients of this polynomial. As an application of these results, a method to detect contact between a small ellipsoid and a combination of quadrics is given.

A quadratic set of a nonsingular quadric Q of Witt index at least three is defined as a set of points intersecting each subspace of Q in a possibly reducible quadric of that subspace. By using the theory of pseudo-embeddings and pseudo-hyperplanes, we show that if Q is one of the quadrics $$Q^+(5,2)$$ , Q(6, 2), $$Q^-(7,2)$$ , then the quadratic sets of Q are precisely the intersections of Q with the quadrics of the ambient projective space of Q. In order to achieve this goal, we will determine the universal pseudo-embedding of the geometry of the points and planes of Q.

We consider the Bipartite Boolean Quadratic Programming Problem (BQP01), which generalizes the well-known Boolean quadratic programming problem (QP01). The model has applications in graph theory, matrix factorization and bioinformatics, among others. The primary focus of this paper is on studying the structure of the Bipartite Boolean Quadric Polytope (BQP m , n ) resulting from a linearization of a quadratic integer programming formulation of BQP01. We present some basic properties and partial relaxations of BQP m , n , as well as some families of facets and valid inequalities. We find facet-defining inequalities including a family of odd-cycle inequalities. We discuss various approaches to obtain a valid inequality and facets from those of the related Boolean quadric polytope. The key strategy is based on rounding coefficients, and it is applied to the families of clique and cut inequalities in BQP m , n .

The diffusion of passing over the saddle point of a three-dimensional quadric potential energy surface was studied by analytically solving a set of coupled generalized Langevin equations. An accurate expression of the passing probability was obtained. The effect of the coupling between different degrees of freedom which is represented by the off-diagonal elements of the inertia, friction and potential-curvature tensors was analysed in detail. It is found that some of the coupling have great influence on the diffusion process, while others not. The combination of them results in an optimal injecting direction of the diffusing particles, revealing an optimal three-dimensional diffusion path.

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q ( 2 n , q ) when q is even. This characterisation is based on the number of elements of these sets through points and codimension 2 spaces and generalises [1] , [2] .