scholarly journals DIRAC EQUATION FOR EMBEDDED FOUR-GEOMETRIES

2013 ◽  
Vol 22 (06) ◽  
pp. 1350024
Author(s):  
MACIEJ TRZETRZELEWSKI ◽  
NORDITA
Keyword(s):  
Dirac Equation ◽  
Three Dimensional ◽  
Square Root ◽  
Quantum Level ◽  
Singularity Formation ◽  
Frw Metric ◽  
Friedmann Robertson Walker

We apply the Dirac's square root idea to constraints for embedded four-geometries swept by a three-dimensional membrane. The resulting Dirac-like equation is then analyzed for general coordinates as well as for the case of a Friedmann–Robertson–Walker (FRW) metric for spatially closed geometries. The problem of the singularity formation at the quantum level is addressed.

Signature change in p-adic and noncommutative FRW cosmology

2014 ◽  
Vol 29 (27) ◽  
pp. 1450155 ◽  
Author(s):  
Goran S. Djordjevic ◽  
Ljubisa Nesic ◽  
Darko Radovancevic
Keyword(s):  
Loop Quantum Gravity ◽  
Initial Conditions ◽  
Planck Scale ◽  
The Other ◽  
Frw Cosmology ◽  
Signature Change ◽  
Discrete Nature ◽  
Frw Metric ◽  
The Universe ◽  
Friedmann Robertson Walker

The significant matter for the construction of the so-called no-boundary proposal is the assumption of signature transition, which has been a way to deal with the problem of initial conditions of the universe. On the other hand, results of Loop Quantum Gravity indicate that the signature change is related to the discrete nature of space at the Planck scale. Motivated by possibility of non-Archimedean and/or noncommutative structure of space–time at the Planck scale, in this work we consider the classical, p-adic and (spatial) noncommutative form of a cosmological model with Friedmann–Robertson–Walker (FRW) metric coupled with a self-interacting scalar field.

Square-root-like higher-order topological states in three-dimensional sonic crystals

2021 ◽  
Author(s):  
Zhiguo Geng ◽  
Huanzhao Lv ◽  
Zhan Xiong ◽  
Yu-Gui Peng ◽  
Zhaojiang Chen ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Surface States ◽  
Higher Order ◽  
Square Root ◽  
Root Lattice ◽  
Third Order ◽  
Topological Materials ◽  
Topological States ◽  
Sonic Crystal ◽  
Sonic Crystals

Abstract The square-root descendants of higher-order topological insulators were proposed recently, whose topological property is inherited from the squared Hamiltonian. Here we present a three-dimensional (3D) square-root-like sonic crystal by stacking the 2D square-root lattice in the normal (z) direction. With the nontrivial intralayer couplings, the opened degeneracy at the K-H direction induces the emergence of multiple acoustic localized modes, i.e., the extended 2D surface states and 1D hinge states, which originate from the square-root nature of the system. The square-root-like higher order topological states can be tunable and designed by optionally removing the cavities at the boundaries. We further propose a third-order topological corner state in the 3D sonic crystal by introducing the staggered interlayer couplings on each square-root layer, which leads to a nontrivial bulk polarization in the z direction. Our work sheds light on the high-dimensional square-root topological materials, and have the potentials in designing advanced functional devices with sound trapping and acoustic sensing.

scholarly journals Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1867
Author(s):  
Alexander Breev ◽  
Alexander Shapovalov
Keyword(s):  
Dirac Equation ◽  
Exact Solutions ◽  
Homogeneous Space ◽  
Integration Method ◽  
Homogeneous Spaces ◽  
Separation Of Variables ◽  
General Structure ◽  
Three Dimensional ◽  
De Sitter ◽  
Elementary Functions

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

Fast algorithm for the three-dimensional Poisson equation in infinite domains

2020 ◽  
Author(s):  
Chunxiong Zheng ◽  
Xiang Ma
Keyword(s):  
Poisson Equation ◽  
Fast Algorithm ◽  
Root Function ◽  
Computational Cost ◽  
Three Dimensional ◽  
Chebyshev Approximation ◽  
Laplacian Operator ◽  
Square Root ◽  
Infinite Domain ◽  
Infinite Domains

Abstract This paper is concerned with a fast finite element method for the three-dimensional Poisson equation in infinite domains. Both the exterior problem and the strip-tail problem are considered. Exact Dirichlet-to-Neumann (DtN)-type artificial boundary conditions (ABCs) are derived to reduce the original infinite-domain problems to suitable truncated-domain problems. Based on the best relative Chebyshev approximation for the square-root function, a fast algorithm is developed to approximate exact ABCs. One remarkable advantage is that one need not compute the full eigensystem associated with the surface Laplacian operator on artificial boundaries. In addition, compared with the modal expansion method and the method based on Pad$\acute{\textrm{e}}$ approximation for the square-root function, the computational cost of the DtN mapping is further reduced. An error analysis is performed and numerical examples are presented to demonstrate the efficiency of the proposed method.

On the Singularities in Fracture and Contact Mechanics

2008 ◽  
Vol 75 (5) ◽  
Author(s):  
Fazil Erdogan ◽  
Murat Ozturk
Keyword(s):  
Contact Mechanics ◽  
Integral Equations ◽  
Mixed Boundary ◽  
Complex Function ◽  
Three Dimensional ◽  
Contact Problems ◽  
Cauchy Kernel ◽  
Square Root ◽  
Complex Function Theory ◽  
Graded Materials

Generally, the mixed boundary value problems in fracture and contact mechanics may be formulated in terms of integral equations. Through a careful asymptotic analysis of the kernels and by separating nonintegrable singular parts, the unique features of the unknown functions can then be recovered. In mechanics and potential theory, a characteristic feature of these singular kernels is the Cauchy singularity. In the absence of other nonintegrable kernels, Cauchy kernel would give a square-root or conventional singularity. On the other hand, if the kernels contain, in addition to a Cauchy singularity, other nonintegrable singular terms, the application of the complex function theory would show that the solution has a non-square-root or unconventional singularity. In this article, some typical examples from crack and contact mechanics demonstrating unique applications of such integral equations will be described. After some remarks on three-dimensional singularities, the key examples considered will include the generalized Cauchy kernels, membrane and sliding contact mechanics, coupled crack-contact problems, and crack and contact problems in graded materials.

Seismic depth migration with the Dirac equation

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 925-933 ◽  
Author(s):  
Ketil Hokstad ◽  
Rune Mittet
Keyword(s):  
Dirac Equation ◽  
Taylor Series ◽  
Rapid Expansion ◽  
Difference Operators ◽  
Square Root ◽  
Exact Linearization ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Spatial Derivatives ◽  
Velocity Variations

We demonstrate the applicability of the Dirac equation in seismic wavefield extrapolation by presenting a new explicit one‐way prestack depth migration scheme. The method is in principle accurate up to 90° from the vertical, and it tolerates lateral velocity variations. This is achieved by performing the extrapolation step of migration with the Dirac equation, implemented in the space‐frequency domain. The Dirac equation is an exact linearization of the square‐root wave equation and is equivalent to keeping infinitely many terms in a Taylor series or continued‐fraction expansion of the square‐root operator. An important property of the new method is that the local velocity and the spatial derivatives decouple in separate terms within the extrapolation operator. Therefore, we do not need to precompute and store large tables of convolutional extrapolator coefficients depending on velocity. The main drawback of the explicit scheme is that evanescent energy must be removed at each depth step to obtain numerical stability. We have tested two numerical implementations of the migration scheme. In the first implementation, we perform depth stepping using the Taylor series approximation and compute spatial derivatives with high‐order finite difference operators. In the second implementation, we perform depth stepping with the Rapid expansion method and numerical differentiation with the pseudospectral method. The imaging condition is a generalization of Claerbout’s U / D principle. For both implementations, the impulse response is accurate up to 80° from the vertical. Using synthetic data from a simple fault model, we test the depth migration scheme in the presence of lateral velocity variations. The results show that the proposed migration scheme images dipping reflectors and the fault plane in the correct positions.

scholarly journals Singularity formation in three-dimensional vortex sheets

2003 ◽  
Vol 15 (1) ◽  
pp. 147-172 ◽  
Author(s):  
Thomas Y. Hou ◽  
Gang Hu ◽  
Pingwen Zhang
Keyword(s):  
Three Dimensional ◽  
Singularity Formation ◽  
Vortex Sheets

scholarly journals Calculated Study of the Influence of Overheating Aluminum Melt on the Dynamics the Granulation Process

2020 ◽  
pp. 84-93
Author(s):  
Alexander P. Skuratov ◽  
Alexander V. Ivlev ◽  
Artem A. Pianykh
Keyword(s):  
Heat Transfer ◽  
Phase Transition ◽  
Solid Phase ◽  
Numerical Study ◽  
Three Dimensional ◽  
Solidification Process ◽  
Square Root ◽  
Effective Heat Capacity ◽  
Granulation Process ◽  
Aluminum Melt

A three-dimensional mathematical model of the solidification process of a liquid metal is considered, taking into account the mobility of the boundaries at which the phase transition is carried out (Stefan boundary value problem). The algorithm of calculation is improved, allowing due to the use of the Dirac δ-function in determining the effective heat capacity to take into account the nonlinearity of the equation of unsteady thermal conductivity and the heat of the phase transition. A numerical study of heat transfer during solidification of lead-containing aluminum melt droplets in air and water is carried out. The influence of droplet size and melt overheating on the solidification dynamics of granules has been studied. An approximate ratio based on the square root law is proposed, taking into account the amount of overheating of the liquid phase and linking the thickness of the formed solid phase with the duration of the granulation process

Sign in / Sign up

Export Citation Format

Share Document