Modeling the Laying Material Trajectory on the Technological Mandrel Surface Specified by an Irregular Spatial Point Set

The increasing use of unstructured grids for reservoir modeling motivates the development of geostatistical techniques to populate them with properties such as facies proportions, porosity and permeability. Unstructured grids are often populated by upscaling high-resolution regular grid models, but the size of the regular grid becomes unreasonably large to ensure that there is sufficient resolution for small unstructured grid elements. The properties could be modeled directly on the unstructured grid, which leads to an irregular configuration of points in the three-dimensional reservoir volume. Current implementations of Gaussian simulation for geostatistics are for regular grids. This paper addresses important implementation details involved in adapting sequential Gaussian simulation to populate irregular point configurations including general storage and computation issues, generating random paths for improved long range variogram reproduction, and search strategies including the superblock search and the k-dimensional tree. An efficient algorithm for computing the variogram of very large irregular point sets is developed for model checking.

By Birkhoff's ergodic theorem, the set of points Xϕ for which the Birkhoff averages of a continuous function ϕ diverge has zero measure with respect to any finite invariant measure.Thus, at least from the point of view of ergodic theory, this set could not be smaller.Nevertheless, it can be large from other points of view.For example, for subshifts with the weak specification property, we showed recently that Xϕ is residual whenever it is nonempty (it is a simple exercise to show that Xϕ is dense whenever it is nonempty).The main purpose of this note is to convey in the simplest possible manner the proof of our result in the particular case of the full shift on a finite number of symbols.This has the advantage of avoiding some accessory technicalities that are necessary in the general case.In fact, we consider also the more general case when the set of accumulation points of the Birkhoff averages of a continuous function is a prescribed closed interval and we show that it is residual whenever it is nonempty.

For shifts with weak specification, we show that the set of points for which the Birkhoff averages of a continuous function diverge is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with specification. In addition, we show that the set of points for which the Birkhoff averages of a continuous function have a prescribed set of accumulation points is also residual. The proof consists of bridging together strings of sufficiently large length corresponding to a dense set of limits of Birkhoff averages. Finally, we consider intersections of finitely many irregular sets and show that they are again residual. As an application, we show that the set of points for which the Lyapunov exponents on a conformal repeller are not limits is residual.

Even though all presented components of our multilevel PUM, the PU construction, the numerical quadrature scheme, and the multilevel solver, are designed for general point sets P and general covers C Ω we applied our meshfree discretization method to uniform node arrangements only; up to now. So far we did not have to deal with the challenging task of the cover construction for irregular point sets. In this chapter we now focus on this very critical issue of constructing a suitable cover C Ω for a general given point set efficiently.KeywordsStiffness MatrixCoarse LevelTree PartitionIntegration CellCover PatchThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this article, we prove that the set of points whose pointwise emergence have same degree in [1,∞] with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a C1+α diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Mañé and Shub examples of robustly transitive non-hyperbolic diffeomorphisms. This way, our unifying approach recovers a collection of known deep theorems on the genericity of the irregular set, for both additive and sub-additive potentials, and also provides a number of new applications.

This paper presents a new method based on the application of the Hough Transform for solving the robot motion problem and their intermediate steps: geometric modelling and collision detection. In terms of modelling, given a set of points, which is the representation of a given object, the maximum inner and the minimum outer spherically-extended polytopes (s-tope) are automatically generated. Collision detection is based on the computation of the translational penetration distance. Such a distance also allows to compute an intermediate point when collision avoidance is required. The set of intermediate points describes a solution to the path-planning problem. This new approach introduces an important decrease in the computational time required for robot motion planning.

Let Ω be a bounded and subanalytic domain in R n , n > 2. We show that the set of boundary points of Ω which are regular with respect to the Dirichlet problem is again subanalytic. Moreover, we give sharp upper bounds for the dimension of the set of irregular boundary points. This enables us to decide whether the domain has a classical Green function. In dimensions 2 and 3, this is the case, given some mild and necessary conditions on the topology of the domain.

Sequential function approximation on arbitrarily distributed point sets

Nowadays, various types of geodetic measuring instruments are available in surveying. When conducting a survey of a terrain, surveyor must choose a survey method that is capable of providing high precision measurements. In the surveying process, it is possible to use the following measurement methods - tachymetric survey method, positioning method and remote sensing method. Each of the methods produces data formats that can be used further for volume calculations. The paper analyses the volume differences that are obtained when using various survey methods and volume calculation programs to obtain the surface model. The use of such a tool is important for useful mineral miners, so that surveyors make qualitative surveys and obtain accurate and reliable volumes. Useful minerals miners are required to report annually on their mining volume for a specified period. The useful minerals miners shall at all times keep records of the quantities of minerals actually extracted. Regardless of the type of accounting, topographic surveying and the calculation of volumes, if mineral is extracted in mineral deposits of national importance, more than 50 000 m3, are required (Augstas detalizācijas topogrāfiskās…, 2012). High-quality and accurate topographic survey is needed to create the most realistic surface model appropriate to the terrain. It is defined that the difference between the volumes obtained from the recording of vehicle loads should not be more than 5% when comparing the result of the mineral extraction from topographic surveys (Derīgo izrakteņu ieguves…, 2012; Noteikumi par valsts…, 2018). Nowadays, various measurement methods can be used to obtain data from which a surface model can be generated, and this model can then be used for volume calculations. The surface model is a set of points where, in addition to plane coordinates (x and y), there is also altitude (z) known (Luukkonen, 2011). Each measurement method has its own different measuring instruments that can be used, each method has its own operating principles, on which the quality and accuracy of the obtained data depends.

An elliptic curve can be defined as a smooth projective curve of degree 3 in the projective plane with a distinguished point chosen on it. The set of points on the curve can thus be endowed with a natural additive group structure. The most concrete description of an elliptic curve comes from its affine equation, written as $$y^2=x^3+ax+b,\quad \text{with}\ 4a^3+27b^2\not=0.$$ The theory of elliptic curves is a marvelous mixture of elementary mathematics and profound, advanced mathematics, a mixture which moreover lies on the crossroads of multiple theories: arithmetic, algebraic geometry, group representations, complex analysis, etc. Here, we will provide an introduction to the subject and prove the main Diophantine theorems: the group of rational points is finitely generated (the Mordell-Weil theorem) and the set of integral points is finite (Siegel’s theorem). Finally, we will evoke the famous theorem of Wiles—whose proof resulted in the proof of Fermat’s last theorem—and the Birch & Swinnerton-Dyer conjecture.

Traditional Convolutional Neural Networks (CNN) are limited to extract informative local features of point clouds due to the fixed geometric structures in convolution kernel against irregular and unstructured point clouds. It usually requires data transformation such as voxelization or projection, inducing a possible loss of information. Instead of fitting the input points to the kernel by regularization, we choose to fit the kernel to input points to conduct convolution. In this paper, we present a new method to define and compute convolution directly on 3D point clouds by Adaptive Surface Fitting Convolution (ASFConv). The key idea is to utilize a set of kernel points distributed on the tangent plane and project them back to point cloud surface. After adapting to the distribution of input points, ASFConv kernel can better capture local neighborhood geometry and benefit the feature extraction. In the experiments, we evaluate our network on two public datasets: ModelNet40 and ShapeNet for classification and segmentation. The experimental results show that our method obtain competitive performances compared to the state-of-the-art.

Textured surface models of three-dimensional objects are gaining importance in computer graphics applications. These models often have to be merged from several overlapping partial models which have to be registered (i.e. the relative transformation between the partial models has to be determined) prior to the merging process. In this paper a method is presented that makes use of both camera-based depth information (e.g. from stereo) and the luminance image. The luminance information is exploited to determine corresponding point sets on the partial surfaces using an optical flow approach. Quaternions are then employed to determine the transformation between the partial models which minimizes the sum of the 3-D Euclidian distances between the corresponding point sets. In order to find corresponding points on the partial surfaces luminance information is linearized. The procedure is iterated until convergence is reached. In contrast to only using depth information, employing luminance speeds up convergence and reduces remaining degrees of freedom (e.g. when registering sphere-like shapes).

ABSTRACTThis paper proposes a novel method for register partially overlapped three-dimensional (3D) point sets when a complete 3D scanned model is constructed from partially scanned 3D point sets. 3D point sets are the most primitive and fundamental surface representations. 3D acquisition devices such as laser scanners and range image finders are popular sources for acquiring point clouds. With the availability of low cost and highly efficient 3D laser scanners, the popularity of such devices is increasing rapidly in various fields. Generally, a standard scanner can obtain only a partial data to build a complete 3D model of the object. These partially scanned 3D patches need to be aligned together in the same coordinate system to construct a complete 3D model, and this process is widely known as registration. A coarse-to-fine alignment technique is proposed to register two partially overlapped point sets. First, the overlapping regions using a novel sampling strategy are identified and the overlapping regions of two point sets are aligned together. Second, fine registration is carried out, in which an auxiliary pair and surface normal constraints are used to find reliable corresponding points in order to achieve higher accuracy.

In this paper, we propose a method of polypointed mappings of line segments in geometric modeling of complex objects, which differs from ordinary polypointed mappings in that you can set straight lines as an inverse image, not in an implicit form, but as a set of points and obtain a transformed object as a set of points. In addition, it is proposed to use weights for each base point, which allows you to control the shape of the transformed curve in such a way that the curve is somehow attracted to a point with a higher weight compared to other base points. Polypoint transformations are used in the design of complex geometric shapes in mechanical engineering. Enhancing the capabilities of polypoint transformations is an important issue at the present time. The aim of the article is the use of line segments in the method of polypoint mappings for geometric modeling of complex objects. The proposed process of weighted political dot mapping can be applied in the implementation of geometric modeling of complex shape objects with the ability to control the shape of a simulated object by changing the weights of the points. As shown in test examples, the shape of the transformed figure changes in such a way that it seems to be attracted to points with increased weight and, conversely, is repelled from points with reduced weight. As the disadvantages of the proposed method, one can indicate an insufficient level of predictability of the result. Further research is proposed in identifying the properties of using weighted polypoint mappings in the direction of increasing the level of predictability of the results for effective use in geometric modeling of complex objects. Key words: discrete-given curve, line segment, polypoint mapping of lines, geometric modeling.