Three-dimensional Euclidean Space Research Articles

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.

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

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.

Polytope 240-Based Rod Structures Made of Tetrahedral Atoms: Comparison with Structures Obtained Using a Modular Design Method

Some rod structures consisting of tetrahedral atoms and differing in symmetry are considered. The structures are obtained as a stereographic projection of polytope 240 vertices into three-dimensional Euclidean space when the polytope rolls along a straight line belonging to this space. The axes of the rod structures are the projections of great circles of a three-dimensional sphere containing the polytope 240 vertices. It has been shown that the rod structures obtained earlier by modular design are topologically equivalent to those considered in this paper, but they have a changed symmetry. Three possible conformations following from the results of this work are presented for a CnH2n hydrocarbon chain with close packing of atoms, conservation of the tetrahedral coordination of carbon atoms, and noncrystallographic symmetry.

Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps.

Evolving embedded traversable wormholes in [formula omitted] gravity: A comparative study

The objective of this paper is to unveil the evolving traversable wormhole solutions within the perspective of modified f(R,G) gravity, where R is Ricci scalar and G is Gauss Bonnet term. In order to accomplish this, we develop a wormhole shape function using the Karmarkar condition for traversable static wormhole geometry, that links two asymptotically flat spacetime regions and meets the necessary requirements. The embedding diagram in two and three-dimensional Euclidean space is also discussed in order to demonstrate the wormhole configurations. For our current study, we choose two realistic and viable f(R,G) gravity models to discuss the wormhole geometry. For the traversable wormhole configurations the energy conditions must be satisfied at the throat. For this purpose, we check the energy conditions and observed that these conditions are satisfied near the throat of the wormhole space in modified f(R,G) gravity. Further, the intriguing part of this research is to perform a comparative analysis of the evolving wormhole geometries of our considered models with the help of three-dimensional graphical representation along with their regions. Moreover, we also describe the stability of obtained wormholes solutions by employing the equilibrium condition. It can be observed that our shape function acquired through the Karmarkar technique yields validated wormhole configurations with even less exotic matter correlating to the proper choice of f(R,G) gravity models. As a nutshell, we can infer that our findings fulfill all of the criterion for the presence of traversable wormhole, ensuring that our study is viable and consistent.

Well-posed problems for the Laplace-Beltrami operator on a punctured two-dimensional sphere

An arbitrary point is removed from a three-dimensional Euclidean space on a two-dimensional sphere. The new well-posed solvable boundary value problems for the corresponding Laplace-Beltrami operator on the resulting punctured sphere are presented. To formulate the well-posed problems some properties of Green's function of the Laplace-Beltrami operator on a two-dimensional sphere are previously studied in detail.

Developable Surfaces Foliated by General Ellipses in Euclidean Space R3

In this article, we classify the developable surfaces in three-dimensional Euclidean space R3 that are foliated by general ellipses. We show that the surface has constant Gaussian curvature (CGC) and is foliated by general ellipses if and only if the surface is developable, i.e., the Gaussian curvature G vanishes everywhere. We characterize all developable surfaces foliated by general ellipses. Some of these surfaces are conical surfaces, and the others are surfaces generated by some special base curves.

Twisted Surfaces Family in Euclidean 3-Space

In this study, we examine a distinct set of twisted surfaces in the three-dimensional Euclidean space E3. Our focus lies in the investigation of the differential geometry of this surface family, including the determination of their curvatures. Furthermore, we establish the essential conditions for minimal surfaces within this framework. Additionally, we calculate the Laplace−Beltrami operator for this particular surface family and provide an illustrative example.

Internal and external harmonics in bi-cyclide coordinates

The Laplace equation in three-dimensional Euclidean space is -separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé–Wangerin functions called internal and external bi-cyclide harmonics. An expansion for a fundamental solution of Laplace’s equation in products of internal and external bi-cyclide harmonics is derived. In limiting cases this expansion reduces to known expansions in bi-spherical and prolate spheroidal coordinates.

Conoid Surfaces Family in Three-Dimensional Euclidean Space

In this research, we investigate a specific family of conoid surfaces within the three-dimensional Euclidean space E3. We consider the differential geometry of the family. We determine the curvatures of these particular surfaces. Moreover, we provide the necessary conditions for minimality within this framework. Additionally, we compute the Laplace−Beltrami operator for this family and present an example.

Wigner rotation and Euler angle parametrization

Analogous to the famous Euler angle parametrization in three-dimensional Euclidean space, a reflection-free Lorentz transformation in (2 + 1)-dimensional Minkowski space can be decomposed into three simple parts. Applying this decomposition to the Wigner rotation problem, we are able to show that the related mathematics becomes much simpler and the physical meanings more comprehensible and enlightening.

Investigation of traversable wormhole solutions in modified f(R) gravity with scalar potential

The objective of this manuscript is to investigate the traversable wormhole solutions in the background of the f(R, phi ) theory of gravity, where R is the Ricci scalar and phi is the scalar potential respectively. For this reason, we use the Karmarkar criterion for traversable static wormhole geometry to create a wormhole shape function. The suggested shape function creates wormhole geometry that links two asymptotically flat spacetime regions and meets the necessary requirements. The embedding diagram in three-dimensional Euclidean space is also discussed in order to demonstrate the wormhole configurations. For our current analysis, we choose the suitable values of free parameters for f(R, phi ) gravity models to discuss the wormhole geometry. It can be observed that our proposed shape function provides the wormhole solutions with less amount of exotic matter. It can be noticed that energy conditions especially null energy conditions are violated for all considered models. The violation of energy conditions indicates the existence of exotic matter and wormhole geometry. It is concluded that the shape function acquired through the Karmarkar technique yields validated wormhole configurations with even less exotic matter correlating to the chosen f(R, phi ) gravity models.