Three-dimensional Euclidean Space Research Articles

Hopf-type theorem for self-shrinkers

Abstract In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three-dimensional Euclidean space ℝ 3 {{\mathbb{R}}^{3}} is a round sphere, provided its mean curvature and the norm of the its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in ℝ n {{\mathbb{R}}^{n}} with radial weight. These results are applications of a new generalization of Cauchy’s Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.

On some properties of normally-inflective complexes with simple inflective center in $E_3$

In the paper, we consider the special degenerate class of mormally-inflective complexes with simple inflective center in three-dimensional Euclidean space $E_3$. We prove that to construct this class of complexes one should take an arbitrary curve and draw sheaf of straight lines through each point of this curve. For arbitrary normally-inflective complex with simple inflective center we establish that such complex is fibered into two one-parametric families of congruences.

Geometric kinematics of persistent rigid motions in three-dimensional Minkowski space

This paper investigates whether Lorentz geometry can be systematically implemented to study the geometric kinematics of persistent rigid motions in three-dimensional Minkowski space. The notion of persistence of a one-parameter rigid motion is identified by the property that the instantaneous twist of the motion has a constant pitch. The main difference between three-dimensional Euclidean and Minkowski spaces is that the pitch of twists will take three different values in Minkowski space depending on the causal character of the curve on which the motions are based. Furthermore, based on the fundamentals of screw theory, the paper establishes necessary and sufficient criteria for modeling the persistence of some significant frame motions, including Frenet–Serret, adapted frame, and Bishop motions. Then, the axode surfaces of these special motions and their geometric concepts are defined. Finally, for a thorough treatment of special frame motions in three-dimensional Minkowski space, this paper reveals some illustrative examples of persistent rigid motions and axode surfaces.

Knot probabilities in equilateral random polygons

We consider the probability of knotting in equilateral random polygons in Euclidean three-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal scaling formula for the knotting probability with the number of edges. This scaling formula involves an exponential function, independent of knot type, with a power law factor that depends on the number of prime components of the knot. The unknot, appearing as a composite knot with zero components, scales with a small negative power law, contrasting with previous studies that indicated a purely exponential scaling. The methodology incorporates several improvements over previous investigations: our random polygon data set is generated using a fast, unbiased algorithm, and knotting is detected using an optimised set of knot invariants based on the Alexander polynomial.

Bour\u2019s theorem and helicoidal surfaces with constant mean curvature in the Bianchi\u2013Cartan\u2013Vranceanu spaces

In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space {mathbb {R}}^3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

Cohomogeneity one conformal actions on the $3$-sphere and the three-dimensional Euclidean space

Cohomogeneity one conformal actions on the $3$-sphere and the three-dimensional Euclidean space

Reorganizing local image features with chaotic maps: an application to texture recognition

Texture images are those where the focus of the analysis is on the spatial arrangement of pixels (primitives or textons) rather than on particular objects in the scene. The recognition of such images is naturally challenging as those primitives can be perceived in different manners depending on the image context. Despite the recent success of convolutional neural networks in texture recognition, model-based descriptors are still competitive, especially when we do not have access to large amounts of annotated data for training and the interpretation of the model is an important issue. Among the model-based approaches, fractal geometry has been one of the most popular, especially in biological applications. Nevertheless, fractals are part of a much broader family of models, which are the non-linear operators, studied in chaos theory. This scenario raises the question whether techniques from the broader area of chaos theory could be useful in texture modeling. Those techniques have been used for a long time in image cryptography, but studies on modeling in a broad sense have been scarce in the literature. In this context, we propose here a chaos-based local descriptor for texture recognition. More specifically, we map the image into the three-dimensional Euclidean space, iterate a chaotic map over this three-dimensional structure and convert it back to the original image. From such chaos-transformed image at each iteration we collect local descriptors (here we use local binary patters) and those descriptors compose the feature representation of the texture. The performance of our method was verified on the classification of benchmark databases and in the identification of Brazilian plant species based on the texture of the leaf surface. The achieved results confirmed our expectation of a competitive performance, even when compared with some learning-based modern approaches in the literature.

Intersections of Boerdijk–Coxeter Helices in Three-Dimensional Euclidian Space

Intersections of Boerdijk–Coxeter Helices in Three-Dimensional Euclidian Space

Iterative algorithms for minimizing the Hausdorff distance between convex polyhedrons

The problem of finding the optimal location of moving bodies in three-dimensional Euclidean space is considered. We study the problem of finding such a position for two given polytopes A and B at which the Hausdorff distance between them would be minimal. To solve it, the apparatus of convex and nonsmooth analysis is used, as well as methods of computational geometry. Iterative algorithms have been developed and justification has been made for the correctness of their work. A software package has been created, its work is illustrated with specific examples.

Картина мира XVIII–XIX вв. Александр фон Гумбольдт и Карл Фридрих Гаусс (по роману Д. Кельмана «Измеряя мир»)

The article examines the worldview of the 18th–19th centuries developed by three contemporaries. The philosophy of Immanuel Kant is placed at the center of the worldview, in relation to which the views of Alexander von Humboldt represent a return to Cartesianism, and the views of Carl Friedrich Gauss – an appeal to the future, to the theory of relativity of space and time. I. Kant taught that Euclidean space is, as the critique of pure reason asserts, the form of our contemplation, prescribed to every possible experience. For Humboldt, the world is fully measurable within three-dimensional Euclidean space. Gauss, on the other hand, thinks of Kant further, theoretically admitting and mathematically calculating the various possibilities of conceivability of space, thus the world for him is more relative, fiction, or a beautiful dream. Space is folded, curved and very strange. Cartesianism and the relativism of both scientists also extend to their ethical concepts: for Humboldt, man is a machine of “highest skill” designed to serve humanistic goals; for Gauss, human behavior admits of accidents and exceptions, although it did not reach the proper rationality and does not deserve attention, but yet, genius is possible. The personal characteristics of both scientists, as well as the connection with their epoch are revealed. Humboldt appears as a humanist, and Gauss as a misanthrope. It is concluded that the worldview is not one-dimensional, but includes the past, present, future of science, and also allows for mystery, chance, mystical elements present in culture.

On the Regularity Radius of Delone Sets in $${\mathbb {R}}^3$$

We complete the proof of the upper bound $\hat\rho_3\leq 10R$ for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all $10R$-clusters of a Delone set $X$ with parameters $(r,R)$ are equivalent, then $X$ is a regular system.

Matrix Fisher–Gaussian Distribution on $\mathrm{SO}(3)\times \mathbb {R}^{ n }$ and Bayesian Attitude Estimation

In this article, a new probability distribution, referred to as the matrix Fisher–Gaussian distribution, is proposed on the product manifold of three-dimensional special orthogonal group and Euclidean space. It is constructed by conditioning a multivariate Gaussian distribution from the ambient Euclidean space into the manifold, while imposing a certain geometric constraint on the correlation term to avoid over parameterization. The unique feature is that it may represent large uncertainties in attitudes, linear variables of an arbitrary dimension, and angular–linear correlations between them in a global fashion without singularities. Various stochastic properties and an approximate maximum likelihood estimator are developed. Furthermore, two methods are developed to propagate uncertainties through a stochastic differential equation representing attitude kinematics. Based on these, a Bayesian estimator is proposed to estimate the attitude and time-varying gyro bias concurrently. Numerical studies indicate that the proposed estimator provides more accurate estimates against the multiplicative extended Kalman filter and unscented Kalman filter for challenging cases.

The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Densely packed tetrahedra clusters displaying diamond-like superstructures

Two special tetrahedra packing clusters were analytically calculated and discussed. Diamond and double-diamond, periodic superstructures emerge. Both structures exemplify an efficient, congruent tetrahedra space tiling strategy. The diamond superstructure exhibits a remarkable packing density of φ∼0.76 exceeding maximal density for the packing of equal spheres. The presented architectures display long range order with specific, repetitive structural motifs. Due to their simple tetrahedral symmetry, the investigated clusters have the potential to become interesting, unique sub-units both in construction technology and in self-assembled systems design. The modified structure of presented cluster can entirely tile three-dimensional Euclidean space ℝ3 through periodic tessellations. Moreover, detailed analysis of the cluster enables definition of a new plesiohedron.

An evolutionary design of weighted minimum networks for four points in the three-dimensional Euclidean space

Abstract We find the equations of the two interior nodes (weighted Fermat–Torricelli points) with respect to the weighted Steiner problem for four points determining a tetrahedron in R 3 \mathbb{R}^{3} . Furthermore, by applying the solution with respect to the weighted Steiner problem for a boundary tetrahedron, we calculate the twist angle between the two weighted Steiner planes formed by one edge and the line defined by the two weighted Fermat–Torricelli points and a non-neighboring edge and the line defined by the two weighted Fermat–Torricelli points. By applying the plasticity principle of quadrilaterals starting from a weighted Fermat–Torricelli tree for a boundary triangle (monad) in the sense of Leibniz, established in [A. N. Zachos, A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature, Acta Appl. Math. 129 (2014), 81–134], we develop an evolutionary scheme of a weighted network for a boundary tetrahedron in R 3 \mathbb{R}^{3} . By introducing the inverse weighted Steiner network with two interior nodes built by two different quantities of the subconscious (remaining weights) for boundary tetrahedra, we describe the evolution of a weighted network with two nodes that have inherited a subconscious. The cancellation of the dynamic plasticity of these weighted networks may be applied to the creation of evolutionary scenarios, in order to prevent the development of a quadrilateral or tetrahedral virus (a monad that has got a subconscious) and the cancerogenesis of quadrilateral cells. We continue by giving the plasticity equations for a generalized weighted minimum network with two nodes that have got a subconscious whose vertices are replaced by Euclidean spheres. This evolutionary approach may be applied to the determination of the bond strengths of molecular structures between atoms in the sense of Pauling. We obtain the analytical solutions of the weighted Fermat–Torricelli problem for the case of pairs of equal weights or one pair of equal weights. We consider as a DNA-like chain a sequence of tetrahedra whose vertices possess some symmetrical weights. By calculating the twist angles of each sequence and by applying the weighted Fermat–Torricelli tree structures with symmetrical weights or weighted Steiner tree structures, we may approximate the curve axis of a DNA-like tree chain. Finally, we construct a minimum tree, which is not a minimal Steiner tree for some boundary symmetric tetrahedra in R 3 \mathbb{R}^{3} , which has two interior nodes with equal weights having the property that the common perpendicular of some two opposite edges passes through their midpoints. We prove that the length of this minimum tree may have length less than the length of the full Steiner tree for the same boundary symmetric tetrahedra, under some angular conditions.

Comments on “Stability Regions of Nonlinear Autonomous Dynamical Systems”

The proofs of the groundbreaking theorems of [Chiang, Hirsch, and Wu. Stability Regions of Nonlinear Autonomous Dynamical Systems, IEEE Transactions on Automatic Control, 1988] rely on a lemma which states that if the stable manifold of a first hyperbolic closed orbit intersects transversely the unstable manifold of a second (possibly the same) hyperbolic closed orbit, then the dimension of the unstable manifold of the first is strictly less than the dimension of the unstable manifold of the second. However, we provide an example meeting the conditions of the lemma where the dimensions of the unstable manifolds are equal, thereby disproving the lemma. In particular, we present a hyperbolic closed orbit of a smooth vector field over three-dimensional Euclidean space whose stable and unstable manifolds have nonempty, transverse intersection.

A Simple and Fast Hole Detection Algorithm for Triangulated Surfaces

Abstract We present a simple and fast algorithm for computing the exact holes in discrete two-dimensional manifolds embedded in a three-dimensional Euclidean space. We deal with the intentionally created “through holes” or “tunnel holes” in the geometry as opposed to missing triangles. The algorithm detects the holes in the geometry directly without any simplified geometry approximation. Discrete Gaussian curvature is used for approximating the local curvature flow in the geometry and for removing outliers from the collection of feature edges. We present an algorithm with varying degrees of flexibility. The algorithm is demonstrated separately for sheets and solid geometries. This article demonstrates the algorithm on triangulated surfaces. However, the algorithm and the underlying data structure are also applicable for surfaces with mixed polygons.

ON ESTIMATES FOR THE TRANSFORMATION FOURIER WITH DAMPED FACTOR

In this paper we consider estimates for the Fourier transform measures, concentrated on analytic hypersurfaces containing the of damping factor. The paper presents the solution of the problem S.D.Soggi and I.M. Stein about the optimal decay of the transformation Fourier measures with a damping factor for any analytic surfaces in three-dimensional Euclidean space.

VARIOUS CHARACTERIZATIONS FOR QUATERNIONIC MANNHEIM CURVES IN THREE-DIMENSIONAL EUCLIDEAN SPACE

In this article, quaternionic curves in 3-dimensional Euclidean space have examined. Firstly, algebraic properties of quaternions and their basic definitions and theorems are given. Later, some characterizations of the quaternionic Mannheim curves in the 3-dimensional Euclidean space have obtained.

Картина мира XVIII–XIX вв. Александр фон Гумбольдт и Карл Фридрих Гаусс (по роману Д. Кельмана «Измеряя мир»)

The article examines the worldview of the 18th–19th centuries developed by three contemporaries. The philosophy of Immanuel Kant is placed at the center of the worldview, in relation to which the views of Alexander von Humboldt represent a return to Cartesianism, and the views of Carl Friedrich Gauss – an appeal to the future, to the theory of relativity of space and time. I. Kant taught that Euclidean space is, as the critique of pure reason asserts, the form of our contemplation, prescribed to every possible experience. For Humboldt, the world is fully measurable within three-dimensional Euclidean space. Gauss, on the other hand, thinks of Kant further, theoretically admitting and mathematically calculating the various possibilities of conceivability of space, thus the world for him is more relative, fiction, or a beautiful dream. Space is folded, curved and very strange. Cartesianism and the relativism of both scientists also extend to their ethical concepts: for Humboldt, man is a machine of “highest skill” designed to serve humanistic goals; for Gauss, human behavior admits of accidents and exceptions, although it did not reach the proper rationality and does not deserve attention, but yet, genius is possible. The personal characteristics of both scientists, as well as the connection with their epoch are revealed. Humboldt appears as a humanist, and Gauss as a misanthrope. It is concluded that the worldview is not one-dimensional, but includes the past, present, future of science, and also allows for mystery, chance, mystical elements present in culture.