Set Of Planes Research Articles

Orientation of Convex Sets

We introduce a novel definition of orientation on the triples of a family of pairwise intersecting planar convex sets and study its properties. In particular, we compare it to other systems of orientations on triples that satisfy a so-called interiority condition: $\circlearrowleft(ABD)=\circlearrowleft(BCD)=\circlearrowleft(CAD)=1$ imply $\circlearrowleft(ABC)=1$ for any $A, B, C, D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O but not every P3O is the order type of some planar point set; a P3O that is realizable by points is called a p-P3O. If the family is non-degenerate with respect to the orientation, i.e., always $\circlearrowleft(ABC)\ne 0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-P3O, is the order type of a point set in general position. Despite these similarities to order types, P3O's and T3O's that can arise from the orientation of pairwise intersecting convex sets, denoted by C-P3O and C-T3O, turn out to be quite different from order types: there is no containment relation among the family of all C-P3O's and the family of all p-P3O's, or among the families of C-T3O's and p-T3O's.Finally, we study properties of these orientations if we also require that the family of the underlying convex sets satisfies the (4,3) property, as a first step towards obtaining better $(p,q)$-theorems.

2-Amino-5-oxo-4-(thiophen-2-yl)-5,6,7,8-tetrahydro-4H-chromene-3-carbonitrile

The crystal structure of the title compound, C14H12N2O2S, reveals two symmetrically independent molecules within the asymmetric unit. Each molecule contains a chromenone core attached to a 2-thiophene ring, cyano, and amino groups. The 2-thiophene ring of one of the two molecules in the asymmetric unit was found to be disordered over two positions, with the major component having a site occupancy factor of 0.837 (2). The 2-thiophene ring is nearly orthogonal to the fused 4H-pyran ring, with dihedral angles between the two sets of planes being 89.5 (5) and 89.63 (8)°. Intermolecular hydrogen bonding, involving N—H...N and N—H...O interactions, creates two distinct motifs leading to the formation of a two-dimensional supramolecular network along the crystallographic ac plane.

Research on the Identification of Rock Mass Structural Planes and Extraction of Dominant Orientations Based on 3D Point Cloud

The different spatial distribution forms of rock mass structural planes create weak zones in the rock mass, which is also a key factor in controlling rock mass stability. Accurately and efficiently identifying rock mass structural planes and obtaining their dominant orientations is critical for rock mass engineering design and construction. Traditional surveying methods for high and steep rock mass structural planes pose high safety risks, offer limited data, and make comprehensive statistical analysis difficult. This paper utilizes complex rock mass surface 3D point cloud data obtained through 3D laser scanning technology and uses the Hough space transform method to calculate the normal vectors of the 3D point cloud. Based on the difference in normal vectors and surface variation, region growing segmentation is applied to identify and extract rock mass structural planes. Additionally, the fast search and density peak clustering method (CFSFDP) is used for clustering analysis of the rock mass structural planes to obtain dominant orientations. This method was applied to a highway’s high and steep rock slope, successfully identifying 281 structural planes and two sets of dominant structural planes. The orientation of the dominant structural planes identified through RocScience Dips 7.0 analysis showed a deviation of no more than ±3°, complying with engineering standards. The research results offer a feasible solution for the identification of high and steep rock mass structural planes and the extraction of the orientation of dominant structural planes.

Mетод визначення зміни просторового положення об'єкта орбітального сервісу з невідомою формою

This paper presents a method for determining the pose of an on-orbit service object (target) with an unknown surface shape based on processing target surface point clouds. It is assumed that a point cloud is obtained using a sensor in the form of a lidar-based system onboard the service spacecraft. The algorithm of the method operates with a simplified description of the cloud; it uses only the coordinates of the cloud points without identifying any surface features. The idea of the method is as follows. From all cloud points obtained at a certain time instant, select points that are close to a certain degree to a given set of planes arranged in space in a certain way. From the point cloud corresponding to the next time instant, select points that are close to the same set of planes, but moved in space in a known way. By convention, let the arrangement of the projections of the selected cloud points onto the given planes be called a pattern of the intersection of the point clouds with the planes. By varying the displacement of the set of planes, maximize the coincidence of the intersection patterns of the first and second clouds of points. The degree of coincidence of the intersection patterns is characterized by a specified objective function whose arguments are target pose parameters. The target displacement parameters are found as the arguments of the objective function that maximize it. A variant of the objective function is proposed. A test example of the application of the proposed method is considered. A global optimization method known as the direct search method was used to find the arguments of the objective function. Test calculations were performed, and they confirmed the workability of the algorithm and determined its scope of applicability. The computational burden was not considered: the goal was to verify the workability of the method in principle. The advantage of the proposed method is its ability to determine a change in the pose of an object with an unknown surface shape. The proposed method may be considered as a source of observation data for a filtering procedure when estimating the parameters of the relative motion of a target with an unknown surface shape.

MANDI-Code: A Coding System for the human mandible

The mandible provides valuable insights into its biological identity. However, the existence of several terminologies for mandibular measurements and inconsistent language can lead to misinterpretation, confusion, and miscommunication. A standardised set of anatomical points, planes, and measurements would assist with these issues and ensure reproducibility and comparability. The proposed coding system offers a comprehensive approach for professionals and researchers in dentistry, archaeology, and anthropology.

Crystallographic Dependence of Field Evaporation Energy Barrier in Metals Using Field Evaporation Energy Loss Spectroscopy Mapping.

Atom probe tomography data are composed of a list of coordinates of the reconstructed atoms in the probed volume. The elemental identity of each atom is derived from time-of-flight mass spectrometry, with no local chemical information readily available. In this study, we use a data processing technique referred to as field evaporation energy loss spectroscopy (FEELS), which analyzes the tails of mass peaks. FEELS was used to extract critical energetic parameters that are related to the activation energy for atoms to escape from the surface under intense electrostatic field and dependent of the path followed by the departing atoms. We focused our study on pure face-centered cubic metals. We demonstrate that the energetic parameters can be mapped in two-dimensional with nanometric resolution. A dependence on the considered crystallographic planes is observed, with sets of planes of low Miller indices showing a lower sensitivity to the field. The temperature is also an important parameter in particular for aluminum, which we attribute to an energetic transition between two paths of field evaporation between 25 and 60 K close to (002) pole. This paper shows that the information that can be retrieved from the measured energy loss of surface atoms is important both experimentally and theoretically.

Cascades of heterodimensional cycles via period doubling

A heterodimensional cycle is formed by the intersection of stable and unstable manifolds of two saddle periodic orbits that have unstable manifolds of different dimensions: connecting orbits exist from one periodic orbit to the other, and vice versa. The difference in dimensions of the invariant manifolds can only be achieved in vector fields of dimension at least four. At least one of the connecting orbits of the heterodimensional cycle will necessarily be structurally unstable, meaning that is does not persist under small perturbations. Nevertheless, the theory states that the existence of a heterodimensional cycle is generally a robust phenomenon: any sufficiently close vector field (in the C1-topology) also has a heterodimensional cycle.We investigate a particular four-dimensional vector field that is known to have a heterodimensional cycle. We continue this cycle as a codimension-one invariant set in a two-parameter plane. Our investigations make extensive use of advanced numerical methods that prove to be an important tool for uncovering the dynamics and providing insight into the underlying geometric structure. We study changes in the family of connecting orbits as two parameters vary and Floquet multipliers of the periodic orbits in the heterodimensional cycle change. In particular the Floquet multipliers of one of the periodic orbits change from real positive to real negative prior to a period-doubling bifurcation. We then focus on the transitions that occur near this period-doubling bifurcation and find that it generates new families of heterodimensional cycles with different geometric properties. Our careful numerical study suggests that further two-parameter continuation of the ‘period-doubled heterodimensional cycles’ gives rise to an abundance of heterodimensional cycles of different types in the limit of a period-doubling cascade.Our results for this particular example vector field make a contribution to the emerging bifurcation theory of heterodimensional cycles. In particular, the bifurcation scenario we present can be viewed as a specific mechanism behind so-called stabilisation of a heterodimensional cycle via the embedding of one of its constituent periodic orbits into a more complex invariant set.

Biological synthesis of silver nanoparticles using Senna auriculata flower extract for antibacterial activities.

In this study, biological synthesis of silver nanoparticles (AgNPs) using Senna auriculata flower extract for antibacterial activities was reported. The silver spectra compared to the plant extract show a rightward shift in AgNP peaks, indicating successful nanoparticle formation. The absorption band at 302 nm and the disappearance or shift of other peaks further confirm the synthesis. X-ray diffraction (XRD) analysis reveals that the AgNPs synthesized with S. auriculata extract have an average crystallite size of 25 nm. Transmission electron microscopy (TEM) results exhibited a polydispersed, spherical shape with sizes ranging from 70 nm, in clear contrast to the electron microscope image that showed their spherical shape. When examining the selected area electron diffraction (SAED) image, a specific set of lattice planes was correlated with a specific spot. A histogram of AgNP particle size distribution can be seen. AgNPs were tested against four different strains of bacteria for their antibacterial effectiveness, including gram negative bacteria (Escherichia coli, Klebsiella pneumoniae, and Staphylococcus aureus), as well as gram positive bacteria (S. aureus, d. Bacillus subtilis), at various concentrations of AgNP. The results of in vitro experiments indicate that AgNPs containing S. auriculata flowers inhibit amylase well. At two concentrations, ~16.03% and ~70.99%, AgNPs inhibit the reaction at low and high concentrations, respectively.

Generating Stochastic Structural Planes Using Statistical Models and Generative Deep Learning Models: A Comparative Investigation

Structural planes are one of the key factors controlling the stability of rock masses. A comprehensive understanding of the spatial distribution characteristics of structural planes is essential for accurately identifying key blocks, analyzing rock mass stability, and addressing various rock engineering challenges. This study compares the effectiveness of four stochastic structural plane generation methods—the Monte Carlo method, the Copula-based method, generative adversarial networks (GAN), and denoised diffusion models (DDPM)—in generating stochastic structural planes and capturing potential correlations between structural plane parameters. The Monte Carlo method employs the mean and variance of three parameters of the measured factual structural planes to generate data that follow the same distributions. The other three methods take the entire set of measured factual structural planes as the overall input to generate structural planes that exhibit the same probability distributions. Five sets of structural planes on four rock slopes in Norway are examined as an example. The validation and analysis were performed using histogram comparison, data feature comparison, scatter plot comparison, and linear regression analysis. The results show that the Monte Carlo method fails to capture the potential correlation between the dip direction and dip angle despite the best fit to the measured factual structural planes. The Copula-based method performs better with smaller datasets, and GAN and DDPM are better at capturing the correlation of measured factual structural planes in the case of large datasets. Therefore, in the case of a limited number of measured structural planes, it is advisable to employ the Copula-based method. In scenarios where the dataset is extensive, the deep generative model is recommended due to its ability to capture complex data structures. The results of this study can be utilized as a valuable point of reference for the accurate generation of stochastic structural planes within rock masses.

The Role of Topology and Capacity in Some Bounds for Principal Frequencies

We prove a lower bound on the sharp Poincaré–Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet–Laplacian. We also consider some limit situations, like the sharp Moser–Trudinger constant and the Cheeger constant. As a byproduct of our discussion, we also obtain a Buser-type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.

An Improvement of the Upper Bound for the Number of Halving Lines of Planar Sets

In this paper, we provide improvements in the additive constant of the current best asymptotic upper bound for the maximum number of halving lines for planar sets of n points, where n is an even number. We also improve this current best upper bound for small values of n, namely, 106≤n≤336. To obtain this enhancements, we provide lower bounds for the sum of the squares of the degrees of the vertices of a graph related to the halving lines.

Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. We do not assume that the points are in general position. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.

L q -spectra of graph-directed planar non-conformal measures

We estimate the upper and lower Lq -spectra of measures generated by a class of graph-directed planar non-conformal iterated function systems. These systems are composed of C1+ϵ contractions with lower triangular pointwise Jacobian. The estimation provides a criterion for the existence of the Lq -spectra in certain case. As an application, we derive the box-counting dimension of planar self-affine sets generated by lower triangular matrices, whose geometric construction satisfies a finite overlapping type condition.

The Intrinsic Geometry of Simply and Rectifiably Connected Plane Sets

AbstractWe prove that the metric completion of the intrinsic length space associated with a simply and rectifiably connected plane set is a Hadamard space. We also characterize when such a space is Gromov hyperbolic.

Non-invertible planar self-affine sets

Abstract We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamic formalism where, for dominated and irreducible matrices, we completely characterise the behaviour of the pressures.

Nice operators and nice spaces

In this paper we obtain a necessary and sufficient condition for an operator on a uniform algebra to be nice. We characterize nice operators on an expansive class of Banach spaces. Then as examples, we give a complete description of nice operators on the space of differentiable functions on compact perfect plane sets, Bloch and Zygmund spaces and the space of Lipschitz functions on intervals. We also prove that every surjective nice operators are isometries on some of these Banach spaces.

Aliasing from Galactic Plane Setting in Widefield Radio Interferometry

Measurements with widefield radio interferometers often include the near-infinite gradient between the sky and the horizon. This causes aliasing inherent to the measurement itself and is purely a consequence of the Fourier basis. For this reason, the horizon is often attenuated by the instrumental beam down to levels deemed inconsequential. However, this effect is enhanced via our own Galactic plane as it sets over the course of a night. We show all-sky simulations of the Galactic plane setting in a low-frequency radio interferometer in detail for the first time. We then apply these simulations to the Murchison Widefield Array to show that a beam attenuation of 0.1% is not sufficient in some precision science cases. We determine that the noise statistics of the residual data image are drastically more Gaussian with aliasing removal, and explore consequences in simulation for cataloging of extragalactic sources and 21 cm Epoch of Reionization detection via the power spectrum.

Scale recurrence lemma and dimension formula for Cantor sets in the complex plane

Abstract We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2)154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z.303 (2023), 3], to prove that under the right hypothesis for the Cantor sets $K_1,\ldots ,K_n$ and the function $h:\mathbb {C}^{n}\to \mathbb {R}^{l}$ , the following formula holds: $$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

Hausdorff dimension of Besicovitch sets of Cantor graphs

AbstractWe consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets Γ, that is, sets that contain a rotated copy of Γ in each direction. We show that for a large class of Cantor sets C and Cantor‐graphs Γ built on C, the Hausdorff dimension of any Γ‐Besicovitch set must be at least , where .

View synthesis with multiplane images from computationally generated RGB-D light fields

Image based view synthesis using deep neural networks provide novel scene views from a set of captured single or multiple images. Multiplane images (MPI) represent scene content as set of RGBα planes within a reference view frustum and render novel views by projecting the content into the target viewpoints. Image based view synthesis with multiple images is very popularly deployed in various areas because it effectively represents geometric uncertainty in ambiguous regions and can convincingly simulate non-Lambertian effects. However, previous image based view synthesis approaches suffer from interpolating and extrapolating information in pixels or ray spaces to generate seamless novel views without occlusion. To effectively improve visual performance for view interpolation and extrapolation, this paper proposes a novel view synthesis with MPI images. From a monocular RGB image, light field images are computationally generated, the proposed depth map guided deep network produces robust MPI using the light field images and their corresponding depth images, and the MPI network embedded with depth attention blocks forces semantic and geometric information to be uniformly distributed and divided among layers. The proposed approach achieves 3.5% and 4.02% improvements in SSIM and PSNR values, comparing to the SOTA approaches. Qualitative analysis on benchmark dataset also verifies the robustness of the proposed approach.