Extra Coordinate Research Articles

Fast simulation of 3D elastic response for wheel–rail contact loading using Proper Generalized Decomposition

To increase computational efficiency, we adopt Proper Generalized Decomposition (PGD) to solve a reduced-order problem of the displacement field for a three-dimensional rail head exposed to different contact scenarios. The three-dimensional solid rail head is modeled as a two-dimensional cross-section, with the coordinate along the rail being treated as a parameter in the PGD approximation. A novel feature is that this allows us to solve the full three-dimensional model with a nearly two-dimensional computational effort. Additionally, we incorporate the distributed contact load predicted from dynamic vehicle–track simulations as extra coordinates in the PGD formulation, using a semi-Hertzian contact model. The problem is formulated in two ways; one general ansatz which considers the treatment of numerous parameters, some of which exhibit a linear influence, and a linear ansatz where multiple PGD solutions are solved for. In particular, situations where certain parameters become invariant are handled. We assess the accuracy and efficiency of the proposed strategy through a series of verification examples. It is shown that the PGD solution converges toward the FE solution with reduced computational cost. Furthermore, solving for the PGD approximation based on load parameterization in an offline stage allows expedient handling of the wheel–rail contact problem online.

Dynamically restoring conformal invariance in (integrable) $\sigma$-models

Integrable λ-deformed σ-models are characterized by an underlying current algebra/coset model CFT deformed, at the infinitesimal level, by current/parafermion bilinears. We promote the deformation parameters to dynamical functions of time introduced as an extra coordinate. It is conceivable that by appropriately constraining them, the beta-functions vanish and consequently the σ-model stays conformal. Remarkably, we explicitly materialize this scenario in several cases having a single and even multiple deformation parameters. These generically obey a system of non-linear second-order ordinary differential equations. They are solved by the fixed points of the RG flow of the original σ-model. Moreover, by appropriately choosing initial conditions we may even interpolate between the RG fixed points as the time varies from the far past to the far future. Finally, we present an extension of our analysis to the Yang–Baxter deformed PCMs.

A muconate bridged bipyridyl appended binuclear Cu(II) complex reveals dissimilar affinities to DNA and BSA protein

A new muconate based Cu(II) metal complex, [Cu2(muco)(5,5′-Me2bpy)4]·2ClO4¯ (1; H2muco = trans, trans-1,3-butadiene-1,4-dicarboxylic acid or trans, trans-muconic acid and 5,5′-Me2bpy = 5,5′-dimethyl-2,2′-bipyridine), has been synthesized and structurally characterized. X-ray structure analysis reveals that 1 forms a binuclear complex with a five coordinated square pyramidal geometry of the central Cu(II) ion attached to a muco and two bipyridine ligands. Docking studies have been carried out which reveal the successful binding of the metal complex to the major groove of DNA and the planar aromatic bipyridyl ligand partially intercalates into the base pairs. The synthesized complex fits into the pocket of bovine serum albumin (BSA), while the maximum fraction submerges into the BSA binding pocket. An extra coordinate bond has been found, forming through the oxygen atom of 158 asparagines amino acid, which subsequently forms a new six coordinated octahedral complex. Experimental studies have been performed to validate all the data obtained from the docking studies. This study reveals highly diverse affinities and binding modes of the synthesized complex towards DNA and protein. Therefore, the synthesized complex may pave the way to be used as a specific protein binder (low affinity to DNA or RNA) bioactive compound.

(2 + 1)-Maxwell Equations in Split Quaternions

The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two SO(2,1) subgroups of the full group of symmetry of the norms of split quaternions, SO(2,2). One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second SO(2,1) subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.

A Hybrid Arbitrary Lagrangian Eulerian Formulation for the Investigation of the Stability of Pipes Conveying Fluid and Axially Moving Beams

Abstract This work addresses pipes conveying fluid and axially moving beams undergoing large deformations. A novel two-dimensional beam finite element is presented, based on the absolute nodal coordinate formulation (ANCF) with an extra Eulerian coordinate to describe axial motion. The resulting formulation is well known as the arbitrary Lagrangian Eulerian (ALE) method, which is often used to model axially moving beams and pipes conveying fluid. The proposed approach, which is derived from an extended version of Lagrange's equations of motion, allows for the investigation of the stability of pipes conveying fluid and axially moving beams for a certain axial velocity and stationary state of large deformation. Additionally, a multibody modeling approach allows us to extend the beam formulation for comoving discrete masses, which represent concentrated masses attached to the beam, e.g., gondolas in ropeway systems, or transported masses in conveyor belts. Within numerical investigations, we show that axially moving beams and a larger number of discrete masses behave similarly as in the case of beams with evenly distributed mass.

Extending $$\Lambda (t)-$$CDM to the inflationary epoch using dynamical foliations and a pre-inflationary vacuum energy from 5D geometrical vacuum as a unifying mechanism

In this paper assuming a 5D quantum pre-inflationary vacuum energy, we propose a manner to extend some \(\Lambda (t)\)-CDM models to the inflationary period by using dynamical foliations of the five-dimensional (5D) Ricci-flat space-time manifold, regarding a non-compact extra space-like coordinate. In this formalism, we achieve also a geometrical unification of inflation and the present accelerating epoch. In this approach, inflation is generated by a pre-inflationary quantum vacuum energy that maintains the 5D classical vacuum on cosmological scales. We obtain from geometrical conditions that we can model the presence of the pre-inflationary vacuum energy in 4D as a dynamical cosmological constant. In this model, the 4D inflationary period is governed by a power law expansion and for certain values of some parameters of the model, we obtain an spectral index satisfying \(0.9607\le n_s\le 0.9691\) and a scalar-to-tensor ratio \(r=0.098\), values that fit well according to Planck 2018 results. The 4D inflationary potential is induced for the 5D geometry and the 4D pre-inflationary potential is determined by the model and its contribution is necessary so that \(n_s\) and r can fit the observational data. We also show that in this theoretical framework, the present acceleration in the expansion of the universe can be explained due to a remanent of this pre-inflationary vacuum energy scaled to the present epoch and that its description can be done with the same \(\Lambda (t)\). In this period, we obtain a deceleration parameter in agreement with Planck 2018 data under certain restrictions of the parameters of the model. From the geometrical point of view, \(\Lambda (t)\) is depending on the dynamical foliation of the 5D space-time manifold.

Searching for a modification of a conformally flat metric in the five-dimensional braneworld model that localizes scalar and vector fields

A modification of the Randall-Sundrum metric model in the form of conformally flat metric was proposed some years ago. The proposed model shows better localization properties than the original Randall-Sundrum model. However, this new model still does not meet the expectation that all fundamental fields can be localized for one type of warp factor, e.g. for a decreasing warp factor. In this research we propose to add a factor f(r) in the extra coordinate part of the above modified Randall-Sundrum metric and derive conditions for localization for scalar and vector fields. r is the extra coordinate. We then look for some functions f(r) that fulfill the localization conditions for both scalar and vector fields. Examples of the function f(r) under investigation are e Ar, e −Ar−Br 2 , e −A/r , Ar, Ar + Br 2, and sinh(Ar). We found the expected function, that is e Ar , which might be used as a further discussion in fields localization.

Constraints on Space-Time-Matter Theory in the Framework of the Standard-Model Extension

We use experimental limits on Lorentz violation within the framework of the Standard-Model Extension to derive quantitative constraints on Space-Time-Matter theory, a version of Kaluza–Klein theory in which the cylinder condition is relaxed so that four-dimensional physics can in principle depend on the extra coordinates. The extra dimensions are not necessarily compact or length-like. We find that the associated variation in fundamental quantities such as rest mass must occur slowly, on cosmological scales.

On Functional Hamilton–Jacobi and Schrödinger Equations and Functional Renormalization Group

We consider the functional Hamilton–Jacobi (HJ) equation, which is the central equation of the holographic renormalization group (HRG), functional Schrödinger equation, and generalized Wilson–Polchinski (WP) equation, which is the central equation of the functional renormalization group (FRG). These equations are formulated in D-dimensional coordinate and abstract (formal) spaces. Instead of extra coordinates or an FRG scale, a “holographic” scalar field Λ is introduced. The extra coordinate (or scale) is obtained as the amplitude of delta-field or constant-field configurations of Λ. For all the functional equations above a rigorous derivation of corresponding integro-differential equation hierarchies for Green functions (GFs) as well as the integration formula for functionals are given. An advantage of the HJ hierarchy compared to Schrödinger or WP hierarchies is that the HJ hierarchy splits into independent equations. Using the integration formula, the functional (arbitrary configuration of Λ) solution for the translation-invariant two-particle GF is obtained. For the delta-field and the constant-field configurations of Λ, this solution is studied in detail. A separable solution for a two-particle GF is briefly discussed. Then, rigorous derivation of the quantum HJ and the continuity functional equations from the functional Schrödinger equation as well as the semiclassical approximation are given. An iterative procedure for solving the functional Schrödinger equation is suggested. Translation-invariant solutions for various GFs (both hierarchies) on delta-field configuration of Λ are obtained. In context of the continuity equation and open quantum field systems, an optical potential is briefly discussed. The mode coarse-graining growth functional for the WP action (WP functional) is analyzed. Based on this analysis, an approximation scheme is proposed for the generalized WP equation. With an optimized (Litim) regulator translation-invariant solutions for two-particle and four-particle amputated GFs from approximated WP hierarchy are found analytically. For Λ=0 these solutions are monotonic in each of the momentum variables.

Inhomogeneous deformation in hyperspace

Geometrical approach to continuum mechanics allowing an atomistic treatment is developed. It suggests a body as a surface immersed into ambient space. Four kinds of surfaces inherent to basic deformations determine classical dynamics of the atoms alongside their spatial coordinates. For uniaxial stretch, this surface is isomorphic to the cylinder. For a simple shear, it is the twisted cylinder, for a bending — a cone, and for a torsion — a helicoid. Scalar parameters of the metric of these surfaces that are stretch ratios, shear, cone, and torsion angles used instead of the strain tensor constitute extra coordinates of the particles in hypersurface. Correspondingly, vector-valued tensions are used instead of stress. In the case of pure deformation, the normal and extra coordinates of atoms obey the classical equation of motion admitting periodic boundary conditions. It is shown how the employing of artificial orders of freedom leads to a minimum of elastic energy. In boundary problems, the system of governing equations unambiguously reproduce the stress–strain–displacement relations with the help of using the tension boundary theorem instead of the Cauchy traction principle. The derived direct method of elastostatic makes the system of governing equations closed and the compatibility condition unnecessary. Commonest examples of bending problem normally following from the Airy stress function illustrate the new concept.

On the Galilean Duffin–Kemmer–Petiau equation in arbitrary dimensions

We obtain the representations of the Galilean covariant Duffin–Kemmer–Petiau equation in an arbitrary number of dimensions. Their purpose is to facilitate the study of nonrelativistic many-body systems with spinless and spin-one fields. A Galilean covariant formalism exploits the tensor structure of relativistic Lorentz-invariant theories by adding one extra spacelike coordinate and working with light-cone coordinates.

Five-dimensional supergravity in N = 1/2 superspace

We construct 5D, N = 1 supergravity in a 4D, N = 1 superspace with an extra bosonic coordinate. This represents four of the supersymmetries and the associated Poincaré symmetries manifestly. The remaining four supersymmetries and the rest of the Poincaré symmetries are represented linearly but not manifestly. In the linearized approximation, the action reduces to the known superspace result. As an application of the formalism, we discuss the construction of the 5D gravitational Chern-Simons invariant ∫ A ∧ R ∧ R in this superspace.

Vacuum polarization in high-dimensional AdS space-time in the presence of a cosmic string and a compactified extra dimension

In this paper, we analyze the vacuum expectation values (VEVs) of the field squared and the energy-momentum tensor associated with a charged and massive scalar quantum field in a generalized $(D+1)$-dimensional anti-de Sitter space in the presence of a cosmic string, admitting a magnetic flux running along the string's core. In addition we admit that an extra coordinate is compactified to a circle and presents a magnetic flux running along its center. This compactification is implemented by imposing a quasiperiodic condition on the field with an arbitrary phase. The calculation of the VEVs of the field squared and the energy-moment tensor, are developed by using positive-energy Wightman function. The latter is construct by the mode sum over the complete set of normalized bosonic wavefunctions. Due to the compactification, the Wightman function is presented as the sum of two distinct contributions. The first one corresponding to the idealized cosmic string, and the second is induced by the compactification. The latter goes to zero for an infinite length of the compactification. As a consequence of the general structure of the Wightman function, both VEVs present also this decomposition. Moreover, due to the Aharanov-Bohm type of interaction between the field with the magnetic fluxes, the VEVs depend on the fractional part of the ration between the total flux and the quantum one.

Kaluza-Klein cosmology: The bulk metric

The Cosmological Principle is applied to a five-dimensional vacuum manifold. The general (non-trivial) solution is explicitly given. The result is a unique metric, parametrized with the sign of the space curvature ($k=0,\pm 1$) and the signature of the fifth coordinate. Friedmann-Robertson-Walker (FRW) metrics can be obtained from this single 'mother' metric (M-metric), by projecting onto different space-homogeneous four-dimensional hypersurfaces. The expansion factor $R$ is used as time coordinate in order to get full control of the equation of state of the resulting projection. The embedding of a generic (equilibrium) mixture of matter, radiation and cosmological constant is given, modulo a quadrature, although some signature-dependent restrictions must be accounted for. In the 4+1 case, where the extra coordinate is spacelike, the condition ensuring that the projected hypersurface is of Lorentzian type is explicitly given. An example showing a smooth transition from an Euclidian to a Lorentzian 4D metric is provided. This dynamical signature change can be considered a classical counterpart of the Hartle-Hawking 'no-boundary' proposal. The resulting FRW model shows an initial singularity at a finite value of the expansion factor $R$. It can be termed as a 'Big unfreeze', as it is produced just by the beginning of time, without affecting space geometry. The model can be extended in order to fit the present value of the density parameters.

Open exceptional strings and D-branes

We study D-branes in the extended geometry appearing in exceptional field theory (or exceptional generalised geometry). Starting from the exceptional sigma model (an Ed(d) covariant worldsheet action with extra target space coordinates), we define open string boundary conditions. We write down Neumann and Dirichlet projectors compatible with the preservation of half-maximal supersymmetry by the brane (building on previous work on the definition of generalised orientifold quotients in exceptional field theory). This leads to a definition of D-branes, plus their S-duals, as particular subspaces of the exceptional geometry, and provides an opportunity to study D-branes in U-fold backgrounds.

Geometry of the non-compact G(2)

Geometrical applications of the non-compact form of Cartan’s exceptional Lie group G(2) are considered. This group comprises specific rotations of 7-dimensional Minkowski-like space with three extra time-like coordinates and in some limiting cases imitates standard Poincare transformations. In this model space–time translations are non-commutative and are represented by the rotations towards the extra time-like coordinates. The second order Casimir element of non-compact G(2) and its expression by the Casimir operators of the Lorentz and Poincare groups are found.

Exchange of macromolecules and colloids in a dense medium: A molecular simulation method

This study presents a molecular simulation method to insert and delete large objects, such as macromolecules or colloids, in a dense medium. This method is based on an extended Hamiltonian formulation that allows extra spatial dimensions for the atoms of the inserted object. By actively controlling the extra coordinates of the atoms, this method can prevent fatal instability in principle and is as efficient as FADE. Its implementation is a straightforward extension of a standard molecular simulation algorithm. This method is first applied successfully to three molecular systems: Ar liquid, linear polymer melts, and a carbon nanotube in an Ar solvent. Unlike FADE, this method does not require extensive pre-testing of each material system to obtain control parameters that depend less on the material properties. Insertion of a nanosized solid sphere, most disruptive to the potential field of the base medium among the objects, is also conducted. This method enables a robust mass exchange in molecular simulation, especially for macromolecules or colloids in complex molecular systems.

The Field Localization of Yukawa Interaction in a Modified Randall-Sundrum Model

We study the localization properties of spinor field coupled to scalar field through Yukawa coupling in a Modified Randall-Sundrum (MRS) model. We derive general localization conditions and solve the solution of the field equation corresponding to the extra coordinate and match the solution to the localization conditions. We obtain that the spinor and scalar fields are localizable on the MRS brane for specific conditions with a decreasing warp factor.

The Importance of Scalar Fields as Extradimensional Metric Components in Kaluza-Klein Models

Extradimensional models are achieving their highest popularity nowadays, among other reasons, because they can plausible explain some standard cosmology issues, such as the cosmological constant and hierarchy problems. In extradimensional models, we can infer that the four-dimensional matter rises as a geometric manifestation of the extra coordinate. In this way, although we still cannot see the extra dimension, we can relate it to physical quantities that are able to exert such a mechanism of matter induction in the observable universe. In this work we propose that scalar fields are those physical quantities. The models here presented are purely geometrical no matter the fact that Lagrangian is assumed and even the scalar fields are contained in the extradimensional metric. The results are capable of describing different observable cosmic features and yield an alternative to ultimately understand the extra dimension and the mechanism in which it is responsible for the creation of matter in the observable universe.

Induced current in high-dimensional AdS spacetime in the presence of a cosmic string and a compactified extra dimension

In this paper, we analyse the bosonic current densities induced by a magnetic flux running along the core of an idealized cosmic string in a high-dimensional AdS spacetime, admitting that an extra dimension coordinate is compactified. Additionally we admit the presence of a magnetic flux enclosed by the compactified axis. In order to develop this analysis we calculate the complete set of normalized bosonic wave-functions obeying a quasiperiodicity condition, with arbitrary phase $\beta$, along the compactified extra dimension. In this context, only azimuthal and axial currents densities take place. As to the azimuthal current, two contributions appear. The first one corresponds to the standard azimuthal current in high-dimensional AdS spacetime with a cosmic string without compactification while the second contribution is a new one, induced by the compactification itself. The latter is an even function of the magnetic flux enclosed by the compactified axis and is an odd function of the magnetic flux along its core with period equal to quantum flux, $\Phi_0=2\pi/e$. On the other hand, the nonzero axial current density is an even function of the magnetic flux along the core of the string and an odd function of the magnetic flux enclosed by the compactified axis. We also find that the axial current density vanishes for untwisted and twisted bosonic fields in the absence of the magnetic flux enclosed by the compactified axis. Some asymptotic expressions for the current density are provided for specific limiting cases of the physical parameter of the model.