The Group of Transformation Which Preserving Distance on Cuboctahedron and Truncated Octahedron Space

The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3-dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, first, metric properties of truncated octahedron distance, d_TO, in R^2 has been examined by metric approach. Then, by using synthetic approach some distance formulae in R_TO^3, 3-dimensional analytical space furnished with the truncated octahedron metric has been found.

Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have done the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split quaternions and metric Lorentz. This interpolation curve is called spherical spline split quaternion interpolation in Minkowski space (MSquad).

In the paper a fluid motion in a rigid porous medium of anisotropic pore space structure is described. Considerations are based on the new macroscopic model of saturated porous medium (Cieszko [4], [5]) in which the fluid flow through porous skeleton of anisotropic pore structure is described as a motion of the material continuum in the plane anisotropic metric space - Minkowski space - immersed in Euclidean one that is the model of the physical space. This model takes into account the fundamental fact for kinematics of fluid-saturated porous solid that pore space of permeable skeleton forms the real space for a fluid motion and its structure imposes restriction on that motion. In such approach the metric tensor of the Minkowski space is used to characterise the anisotropic structure of the skeleton pore space. It enabled one to determine the measures of any line, surface and volume elements in Minkowski and Euclidean spaces and to define the geometrical parameters characterising pore structure of porous materials: tortuosity, surface and volume porosity.The mass and linear momentum balance equations for fluid are derived and the equation for wave propagation in barotropic inviscid fluid filling orthotropic space of pores is obtained. It is shown that the velocity of the plane wave in such a medium depends on the direction of wave propagation.It worth to underline that presented description of fluid motion in the Minkowski space is a good starting point for modelling of mechanics of deformable porous solid saturated with fluid where the concept of deforming anisotropic space (Finsler space) as a model of pore space would have to be used.KeywordsSaturated porous materialsanisotropic pore structureanisotropic spacewave propagation

The problem of a relativistic bound-state system consisting of two scalar bosons interacting through the exchange of another scalar boson, in 2+1 space-time dimensions, has been studied. The Bethe-Salpeter equation (BSE) was solved by adopting the Nakanishi integral representation (NIR) and the Light-Front projection. The NIR allows us to solve the BSE in Minkowski space, which is a big and important challenge, since most of non-perturbative calculations are done in Euclidean space, e.g. Lattice and Schwinger-Dyson calculations. We have in this work adopted an interaction kernel containing the ladder and cross-ladder exchanges. In order to check that the NIR is also a good representation in 2+1, the coupling constants and Wick-rotated amplitudes have been computed and compared with calculations performed in Euclidean space. Very good agreement between the calculations performed in the Minkowski and Euclidean spaces has been found. This is an important consistence test that allows Minkowski calculations with the Nakanishi representation in 2+1 dimensions. This relativistic approach will allow us to perform applications in condensed matter problems in a near future.

The subject of this note is the modelling of an anisotropic pore space structure in rigid porous materials capable of fluid flow through its pores. In this paper, the new macroscopic model of a saturated porous medium is proposed in which a fluid flow through porous skeleton of an anisotropic pore structure is considered as a motion of the material continuum in the plane anisotropic metric space ‐ Minkowski space ‐ immersed in a Euclidean one that is used as the model of the physical space. This model takes into consideration the fundamental fact, concerning kinematics of a fluid‐saturated porous solid, that the pore space of permeable skeleton forms the real space for a fluid motion and its structure imposes restrictions on that motion.The metrics of Minkowski and Euclidean spaces are applied in the paper to determine the respective measures of any line, surface, and volume elements. The new metric tensors of surface and volume elements in these spaces have been proposed that are directly related to the metric tensors of distance. This enables one to define the geometrical parameters characterising pore structure of such materials.: the tortuosity, the volume, and surface porosity. It has been shown that the structure of isotropic pore space of porous materials is described only by two independent scalar parameters.

Actual solutions of the Bethe-Salpeter equation for a two-fermion bound system are becoming available directly in Minkowski space, by virtue of a novel technique, based on the so-called Nakanishi integral representation of the Bethe-Salpeter amplitude and improved by expressing the relevant momenta through light-front components, i.e. $k^\pm=k^0 \pm k^3$. We solve a crucial problem that widens the applicability of the method to real situations by providing an analytically exact treatment of the singularities plaguing the two-fermion problem in Minkowski space, irrespective of the complexity of the irreducible Bethe-Salpeter kernel. This paves the way for feasible numerical investigations of relativistic composite systems, with any spin degrees of freedom. We present a thorough comparison with existing numerical results, evaluated in both Minkowski and Euclidean space, fully corroborating our analytical treatment, as well as fresh light-front amplitudes illustrating the potentiality of non perturbative calculations performed directly in Minkowski space.

Given a triangular array of points on [ − 1 , 1 ] [-1,1] satisfying certain minimal separation conditions, a classical theorem of Szabados asserts the existence of polynomial operators that provide interpolation at these points as well as a near-optimal degree of approximation for arbitrary continuous functions on the interval. This paper provides a simple, functional-analytic proof of this fact. This abstract technique also leads to similar results in general situations where an analogue of the classical Jackson-type theorem holds. In particular, it allows one to obtain simultaneous interpolation and a near-optimal degree of approximation by neural networks on a cube, radial-basis functions on a torus, and Gaussian networks on Euclidean space. These ideas are illustrated by a discussion of simultaneous approximation and interpolation by polynomials and also by zonal-function networks on the unit sphere in Euclidean space.

Nonrenormalizable field theories

As an intrinsically multidimensional function theory, Clifford analysis offers a framework which is particularly suited for the integrated treatment of higher-dimensional phenomena. In this paper a detailed account is given of results connected to the Hilbert transform on the unit sphere in Euclidean space and some of its related concepts, such as Hardy spaces and the Cauchy integral, in a Clifford analysis context.Mathematics Subject Classification (2000)Primary 30G35Secondary 44A15KeywordsHilbert transformHardy spaceCauchy integral

In this paper we consider questions of whether a compact space can be embedded in a Euclidean space. The problem of embedding an '-like' compact space in is solved affirmatively under certain restrictions on the dimension . We clarify the relations between the realization problem and areas of homotopy theory and differential topology.

<p style='text-indent:20px;'>We study emergent dynamics of the Lohe Hermitian sphere (LHS) model with the same free flows under the dynamic interplay between state evolution and adaptive couplings. The LHS model is a complex counterpart of the Lohe sphere (LS) model on the unit sphere in Euclidean space, and when particles lie in the Euclidean unit sphere embedded in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb C^{d+1} $\end{document}</tex-math></inline-formula>, it reduces to the Lohe sphere model. In the absence of interactions between states and coupling gains, emergent dynamics have been addressed in [<xref ref-type="bibr" rid="b23">23</xref>]. In this paper, we further extend earlier results in the aforementioned work to the setting in which the state and coupling gains are dynamically interrelated via two types of coupling laws, namely anti-Hebbian and Hebbian coupling laws. In each case, we present two sufficient frameworks leading to complete aggregation depending on the coupling laws, when the corresponding free flow is the same for all particles.</p>

Perturbed kernel approximation on homogeneous manifolds

<p style='text-indent:20px;'>We study emergent behaviors of the Lohe Hermitian sphere(LHS) model with a time-delay for a homogeneous and heterogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit Hermitian sphere in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb C^d $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ d \geq 2 $\end{document}</tex-math></inline-formula>. Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^{2d} $\end{document}</tex-math></inline-formula>. In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.</p>

Polyhedra are used in several fields by mathematicians and other scientists. It is easy to think of examples from architecture. Polyhedra have been used to scientifically explain the world around us. In the early days of study, polyhedra included only convex polyhedra. Since the ancient Greeks, many thinkers have worked on convex polyhedra. There are only five regular convex polyhedra known as Platonic solids, thirteen semi-regular convex polyhedra known as Archimedean solids, and thirteen irregular convex polyhedra which are duals of the Archimedean solids and known as Catalan solids. In this study, we show that the isometry group of the threedimensional analytic space formed by the metrics of the Tetrakis hexahedron and the Disdyakis dodecahedron is the semi-direct product of Oh and T(3), where the octahedral group Oh is the (Euclidean) symmetry group of the octahedron and T(3) is the group of all translations of the three-dimensional space.

dS/CFT correspondence from a holographic description of massless scalar fields in Minkowski space–time