Three-dimensional Euclidean Space Research Articles

JNR Skyrmions

The JNR ansatz provides a simple formula to obtain families of charge N self-dual SU(2) Yang-Mills instantons in four-dimensional Euclidean space, from the free data of N + 1 distinct points with associated positive weights. Here an analogous formula is presented for Skyrme fields in three-dimensional Euclidean space. These families of Skyrme fields include good approximations to a range of Skyrmions, with energies that are typically within a few percent of the numerically computed solutions.

Minimal translation factorable surfaces in hyperbolic space

The theory of Minimal Surfaces is one of the most interesting subject in mathematics. The study and computation of minimal surfaces has a long history. Lagrange made the first investigation of the minimal surfaces by asking a simple question named as Plateau’s problem which concerns with finding a surface of least area that spans a given fixed one-dimensional contour in three-dimensional Euclidean space. The study of this kind of surfaces has motivated the development of numerous concepts and techniques in the calculus of variations which later have been applied to solving a great quantity of problems in very diverse areas ranging from the study of partial differential equations to topology and even to settling conjectures in relativity. Mathematically, a minimal surface corresponds to the solution of a nonlinear partial differential equation. In this paper, we define two types of Translation-Factorable (T-F) surfaces in the 3-dimensional hyperbolic space. Then, we obtain the complete classification of these surfaces with vanishing mean curvature by solving some differential equations.

Three-dimensional continuum point cloud method for large deformation and its verification

This study presents a strong form based meshfree collocation method, which is named Continuum Point Cloud Method, to solve nonlinear field equations derived from classical mechanics for deformed bodies in three-dimensional Euclidean space. The method and its implementation are benchmarked against a nonlinear vector field using manufactured solutions. The analysis of mechanical fields firstly focuses on the study of St. Venant Kirchhoff and compressible neo-Hookean materials. Results for various initial boundary value problems are presented, including benchmark cases involving unidirectional tension and simple shear. Subsequently, the study concludes with an analysis of a displacement-controlled simulation of a compressible neo-Hookean material, specifically a bar that is pulled to 50% of its original length and rotated 90°. The pure tension case yields a 1.5% error in displacement between computed and expected values and a combined tension and torsion loading case provides further insight into material behavior under complex loading conditions. The resulting normal axial and transverse stress-strain curves are also presented. Finally, the consistency and robustness of the proposed nonlinear numerical schemes are successfully demonstrated through various numerical experiments.

On Flc-Focal Curves According Flc-Frame

In this study, we first characterize focal curves by examining the FLC-frame in three-dimensional Euclidean space. We then derive the relationship between the curvatures of a curve and the focal curvatures. Finally, we present some new conditions for curves with constant curvatures in \(\mathbb{E}^3\).

Structure of Primitive Pythagorean Triples in Generating Trees

A Pythagorean triple is a triple of positive integers $(a,b,c)$ such that $a^2+b^2=c^2$. If $a,b$ are coprime, then it is called a primitive Pythagorean triple. It is known that every primitive Pythagorean triple can be generated from the triple $(3,4,5)$ using multiplication by unique number and order of three specific $3\times3$ matrices, which yields a ternary tree of triplets. Two such trees were described by Berggren and Price, respectively. A different approach is to view the primitive Pythagorean triples as points in the three-dimensional Euclidean space. In this paper, we prove that the triple of descendants of any primitive Pythagorean triple in Berggren's or Price's tree forms a triangle (and therefore defines a plane), and we present our results related to these triangles (and these planes).

Superintegrable families of magnetic monopoles with non-radial potential in curved background

We review some known results on the superintegrability of monopole systems in the three-dimensional (3D) Euclidean space and in the 3D generalized Taub-NUT spaces. We show that these results can be extended to certain curved backgrounds that, for suitable choice of the domain of the coordinates, can be related via conformal transformations to systems in Taub-NUT spaces. These include the multifold Kepler systems as special cases. The curvature of the space is not constant and depends on a rational parameter that is also related to the order of the integrals. New results on minimal superintegrability when the electrostatic potential depends on both radial and angular variables are also presented.

Wormhole solutions in [formula omitted] gravity generated by new shape functions and their complexity factor

The objective of this paper is to investigate the existence and stability of wormholes within the framework of modified gravity, specifically F(R,Lm) gravity, where F(R) represents a function of the Ricci scalar and Lm accounts for matter Lagrangian. We consider the linear functional form of F(R,Lm) with two novel shape functions. To illustrate the wormhole configurations, the embedding diagram in two and three-dimensional Euclidean space is also examined. The Tolman–Oppenheimer–Volkoff (TOV) equation is also used to investigate the stability of the created framework of the traversable wormholes. We observed that near the wormhole throat, the complexity factor (ycf) is monotonically decreasing for wormhole Model-I and increasing for wormhole Model-II, and for higher values of the radial coordinate, ycf is getting closer to zero. Our analysis shows that the energy conditions have been violated and that this generic violation points to the presence of exotic matter.

The imaginary unit described as an algebraic ambiguity

A complex number is formed by a real part and an imaginary part, the latter being composed of a real number multiplied by the imaginary unit “i,” defined as the square root of . For this reason, imaginary numbers cannot be placed on the real number line. The geometric representation of complex numbers is done in the so-called complex plane, composed of a real axis and an imaginary axis, orthogonal to the real line. However, from a philosophical point of view, the fact that the imaginary axis is not identified with any axis in three-dimensional Euclidean space raises ontological questions that transcend the well-established mathematical formalism of the set of complex numbers: are imaginary numbers just a sophism, an arbitrary invention, or a true mathematical discovery? Are imaginary numbers a glimpse of a parallel reality that is beyond our perception? In this article, I seek to demonstrate that the imaginary unit can be interpreted as an algebraic ambiguity that arises when representing certain vectors in the real plane as one-dimensional variables. Consequently, we will see how it is possible to express complex numbers in terms of real numbers.

From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part I. Stationary Hamiltonians

It is known that the Frenet–Serret apparatus of a space curve in three-dimensional Euclidean space determines the local geometry of curves. In particular, the Frenet–Serret apparatus specifies important geometric invariants, including the curvature and the torsion of a curve. It is also acknowledged in quantum information science that low complexity and high efficiency are essential features to achieve when cleverly manipulating quantum states that encode quantum information about a physical system. In this paper, we propose a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by dynamically evolving state vectors. Specifically, we propose a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced by a parallel-transported pure quantum state evolving unitarily under a stationary Hamiltonian specifying the Schrödinger equation. Our proposed constant curvature coefficient is given by the magnitude squared of the covariant derivative of the tangent vector [Formula: see text] to the state vector [Formula: see text] and represents a useful measure of the bending of the quantum curve. Our proposed constant torsion coefficient, instead, is defined in terms of the magnitude squared of the projection of the covariant derivative of the tangent vector [Formula: see text], orthogonal to both [Formula: see text] and [Formula: see text]. The torsion coefficient provides a convenient measure of the twisting of the quantum curve. Remarkably, we show that our proposed curvature and torsion coefficients coincide with those existing in the literature, although introduced in a completely different manner. Interestingly, not only we establish that zero curvature corresponds to unit geodesic efficiency during the quantum transportation in projective Hilbert space, but we also find that the concepts of curvature and torsion help enlighten the statistical structure of quantum theory. Indeed, while the former concept can be essentially defined in terms of the concept of kurtosis, the positivity of the latter can be regarded as a restatement of the well-known Pearson inequality that involves both the concepts of kurtosis and skewness in mathematical statistics. Finally, not only do we present illustrative examples with nonzero curvature for single-qubit time-independent Hamiltonian evolutions for which it is impossible to generate torsion, but we also discuss physical applications extended to two-qubit stationary Hamiltonians that generate curves with both nonzero curvature and nonvanishing torsion traced by quantum states with different degrees of entanglement, ranging from separable states to maximally entangled Bell states. In the Appendix C, we examine the different curvature and torsion characteristics of the three qubit [Formula: see text] and [Formula: see text] states under evolution by a quantum Heisenberg Hamiltonian.

Heron’s Formula in Higher Dimensions

This paper shows how geometric algebra can be used to derive a novel generalization of Heron’s classical formula for the area of a triangle in the plane to higher dimensions. It begins by illustrating some of the many ways in which the conformal model of three-dimensional Euclidean space yields provocative insights into some of our most basic intuitive notions of solid geometry. It then uses this conceptual framework to elucidate the geometric meaning of Heron’s formula in the plane, and explains in detail how it extends naturally to the volumes of tetrahedra in space. The paper closes by outlining a proof of a previously conjectured extension of the formula to the hyper-volumes of simplices in all dimensions.

An approach to the global well-posedness of a coupled 3-dimensional Navier-Stokes-Darcy model with Beavers-Joseph-Saffman-Jones interface boundary condition

<abstract><p>This study focused on investigating the global well-posedness of a coupled Navier-Stokes-Darcy model with the Beavers-Joseph-Saffman-Jones interface boundary condition in the three-dimensional Euclidean space. By utilizing this approach, we successfully obtained the global strong solution of the system in the three-dimensional space. Furthermore, we demonstrated the exponential stability of this strong solution. The significance of such coupled systems lies in their pivotal role in the analysis of subsurface flow problems, particularly in the context of karst aquifers.</p></abstract>

A New Method to Compute Orthonormal Basis in Three-Dimensional Euclidean Spaces

A New Method to Compute Orthonormal Basis in Three-Dimensional Euclidean Spaces

Evolutionary Khovanov homology

<p>Knot theory, a subfield in geometric topology, is the study of the embedding of closed circles into three-dimensional Euclidean space, motivated by the ubiquity of knots in daily life and human civilization. However, focusing on topology, the current knot theory lacks metric analysis. As a result, the application of knot theory has remained largely primitive and qualitative. Motivated by the need of quantitative knot data analysis (KDA), this work implemented the evolutionary Khovanov homology (EKH) to facilitate a multiscale KDA of real-world data. EKH considers specific metrics to filter links, capturing multiscale topological features of knot configurations beyond traditional invariants. It is demonstrated that EKH can reveal non-trivial knot invariants at appropriate scales, even when the global topological structure of a knot is simple. The proposed EKH holds great potential for KDA and machine learning applications related to knot-type data, in contrast to other data forms, such as point cloud data and data on manifolds.</p>

Disk-band graphs in the theory of framed tangles

A disk-band graph is a smooth compact two-dimensional manifold with boundary, partitioned into handles; the partition contains only index zero and index one handles and imitates the structure of the graph. (Index zero handles are analogues of the vertices of the graph, index one handles are analogues of the graph edges.) A disk-band graph is called spatial if it is a smooth submanifold of three-dimensional Euclidean space. A tangle is usually understood as a smooth compact one-dimensional submanifold of the standard three-dimensional ball that intersects the boundary of the ball orthogonally, only along its boundary, the intersection is contained in the equator. We call a tangle framed if it is equipped with a smooth field of normal straight lines. It is well known that there is a reduction of the problem of isotopic classification of spatial disk-band graphs to the problem of isotopic classification of framed tangles. This work focuses on the application of (abstract) disk-band graphs to study the set of isotopic classes of framed tangles.

A new approach to special curved surface families according to modified orthogonal frame

The main purpose of this paper was to investigate the problem of finding the surface family with respect to two different types of modified orthogonal frames defined for curves with curvature and torsion different from zero, respectively. For this purpose, conditions were given for the parametric curve with the modified orthogonal frame in three-dimensional Euclidean space to be a geodesic, asymptotic or line of curvature on the surface, respectively. It has been shown that a member of the surface family with the same special curve such as geodesic, asymptotic, or line of curvature can be obtained by choosing different deviation functions in the parametric writing of the surface to satisfy the conditions. Finally, several examples were given to support the study.

Non-crystallographic helices in polymers and close-packed metallic crystals determined by the four-dimensional counterpart of the icosahedron

Non-crystallographic fractional screw axes are inherent to the constructions of n-dimensional crystallography, where 3 < n ≤ 8. This fact allows one to consider experimentally obtained helices as periodic approximants of helices from the four-dimensional {3, 3, 5} polytope and its derivative constructions. For the tetrahedral Coxeter–Boerdijk helix (tetrahelix) with a 30/11 axis from the {3, 3, 5} polytope, approximants with 11/4 and 8/3 axes in three-dimensional Euclidean space {\bb E}^{3} are considered. These determine the structure of rods composed of deformed tetrahedra in close-packed crystals of α-Mn and β-Mn. In the {3, 3, 5} polytope, highlighted here for the first time, is a 40-vertex helix with a 20/9 axis composed of seven-vertex quadruples of tetrahedra (tetrablocks), whose 7/3 approximants determine in a crystal of an α-Mn rod of deformed tetrablocks with the same period as the 11/4 approximant of the tetrahelix. In the spaces of the three-dimensional sphere and {\bb E}^{3}, the parameters of 20/9, 40/9 and 40/11 helices, as well as of their 20- and 40-vertex approximants, are calculated. The parameters of the approximant of the 40/11 helix in {\bb E}^{3} correspond to experimentally determined parameters of the α-helix, which allows us to explain the versatility of the α-helix in proteins by the symmetry of the polytope. The set of fractional axes of all periodic approximants of helices with 30/11, 20/9, 40/9, 40/11 axes, as well as the powers of these axes, are combined into a tetrahedral-polytope class of 50 basic axes. The basic axes as well as composite (defined as a combination of basic ones) fractional axes of this class cover all fractional axes known to us according to literature data for polymers, biopolymers and close-packed metals.

Spatially embedded recurrent neural networks reveal widespread links between structural and functional neuroscience findings

Brain networks exist within the confines of resource limitations. As a result, a brain network must overcome the metabolic costs of growing and sustaining the network within its physical space, while simultaneously implementing its required information processing. Here, to observe the effect of these processes, we introduce the spatially embedded recurrent neural network (seRNN). seRNNs learn basic task-related inferences while existing within a three-dimensional Euclidean space, where the communication of constituent neurons is constrained by a sparse connectome. We find that seRNNs converge on structural and functional features that are also commonly found in primate cerebral cortices. Specifically, they converge on solving inferences using modular small-world networks, in which functionally similar units spatially configure themselves to utilize an energetically efficient mixed-selective code. Because these features emerge in unison, seRNNs reveal how many common structural and functional brain motifs are strongly intertwined and can be attributed to basic biological optimization processes. seRNNs incorporate biophysical constraints within a fully artificial system and can serve as a bridge between structural and functional research communities to move neuroscientific understanding forwards.

A Scalar Product for Computing Fundamental Quantities in Matter

We introduce a systematic way to obtain expressions for computing the amount of fundamental quantities such as helicity and angular momentum contained in static matter, given its charge and magnetization densities. The method is based on a scalar product that we put forward which is invariant under the ten-parameter conformal group in three-dimensional Euclidean space. This group is obtained as the static restriction (frequency ω=0) of the symmetry group of Maxwell equations: The fifteen-parameter conformal group in 3+1 Minkowski spacetime. In an exemplary application, we compute the helicity and angular momentum squared stored in a magnetic Hopfion.

Riccati equations of opposite torsions from the Lie-Darboux method for spatial curves and possible applications

A novel formulation of the Lie-Darboux method of obtaining the Riccati equations for the spatial curves in Euclidean three-dimensional space is presented. It leads to two Riccati equations that differ by the sign of torsion. The case of cylindrical helices is used as an illustrative example. Possible applications in Physics are suggested.

A new approach to find the magnetic flux surfaces

Magnetic flux surfaces allow us to examine surfaces in a laboratory environment. In this paper, we obtain differential equations to find the magnetic flux surfaces. Differential equations are needed for defining, modeling and solving engineering problems. They are particularly widely used in systems and control theory. First, we thought of linear combinations of Killing vector fields in three-dimensional Euclidean space [Formula: see text]. We obtain partial differential equations by using these vectors. We find magnetic flux surface families with the solutions of these equations. Examples of the magnetic flux surface families members found by solving each equation are given.