First-order Form Research Articles

Vortices in Vacuumless Systems

We investigate the presence of vortex solutions in potentials without vacuum state. The study is conducted considering Maxwell and Chern-Simons dynamics. Also, we use a first-order formalism that helps us to find the solutions and their respective electromagnetic fields and energy densities. As a bonus, we get to calculate the energy without knowing the explicit solutions. Even though the solutions present a large “tail” which goes far away from the origin, the magnetic flux remains a well defined topological invariant.

Generation of N-waves in laboratory

In this research work, a novel theoretical first-order formulation for the generation of N-waves in laboratory, by means of a piston-type wave-maker, is presented. The plate's trajectory, velocity and acceleration equations for the generation of tsunami N-waves in a wave flume are deduced. A set of laboratory experiments performed for the generation of leading depression N-waves in a wave basin is described.A Tadepalli–Synolakis N-wave profile is assumed and the equations for the paddle's trajectory, velocity, and acceleration and for the piston stroke and stroke period are derived and discussed in detail. Limits for the initial paddle velocity and acceleration are established.A set of experiments, devised and performed in the 28 m long wave basin of the Laboratory of Hydraulics, at the Faculty of Engineering of the University of Porto, is described. Six leading depression N-waves were selected and classified into three categories, according to their Stokes parameter. The results show that the shorter waves undergo a strong transformation, tending to a solitary wave-like pulse, followed by a trough. The longer waves show a distinct behaviour, tending to a bore-like wave exhibiting a leading trough. The generated longer waves presented the best results when compared with the expected Tadepalli–Synolakis N-wave profiles. Further investigation on the formulation of the wave velocity of the N-waves and on the behaviour of waves with higher order Stokes parameter is necessary.

Burali-Forti as a Purely Logical Paradox

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like Russell’s paradox, is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language $\mathcal {L}$ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in $\mathcal {L}$ and Burali-Forti’s involves an inconsistent form expressible in $\mathcal {L} + \mathsf {W}$ .

Vehicle–space traffic-state estimation of a motorway corridor with slip roads

Traffic-state estimation is a key component of an advanced traffic management system. The Eulerian traffic-state estimation based on an X-T traffic-flow model has proved successful in processing traditional X-T data sources such as inductive loop data. However, over the past decade, the emergence of probe vehicle data from paired sensors has created challenges for X-T traffic-state estimation. The vehicle–space (N-X) model has shown potential benefits in modelling paired-probe data. In this paper, an original N-X traffic-state estimation framework is proposed and validated by numerical experiments. In addition, an original first-order conservation formulation is derived to address the entry and exit slip road issue for motorway corridors. An evaluation is conducted by comparing the proposed N-X estimation with the X-T estimation. The results show that the proposed model is superior to the X-T estimation in terms of accuracy and efficiency. It performs well, particularly during congestion, and can track traffic breakdowns and recoveries effectively.

Nonlinear Diffusion Acceleration of the Least-Squares Transport Equation in Geometries with Voids

In this paper we show the extension of nonlinear diffusion acceleration (NDA) to geometries containing small voids using a weighted-least-squares (WLS) high-order equation. Even though the WLS equation is well defined in voids, the low-order drift-diffusion equation was not defined in materials with a zero cross section. This paper derives the necessary modifications to the NDA algorithm. We show that a small change to the NDA closure term and a nonlocal definition of the diffusion coefficient solve the problems for void regions. These changes do not affect the algorithm for optically thick material regions while making the algorithm well defined in optically thin ones. We use a Fourier analysis to perform an iterative analysis to confirm that the modifications result in a stable and efficient algorithm. Later in the paper, numerical results of our method are presented. We test this formulation with a small, one-dimensional test problem. Additionally, we present results for a modified version of the C5G7 benchmark containing voids as a more complex, reactor-like problem. We compared our results to Texas A&M’s transport code PDT, utilizing a first-order discontinuous formulation as reference and the self-adjoint angular flux equation with void treatment (SAAF), a different second-order form. The results indicate that the NDA WLS performed comparably or slightly worse then the asymmetric SAAF while maintaining a symmetric discretization matrix.

Symmetry algebra in gauge theories of gravity

Diffeomorphisms and an internal symmetry (e.g. local Lorentz invariance) are typically regarded as the symmetries of any geometrical gravity theory, including general relativity. In the first-order formalism, diffeomorphisms can be thought of as a derived symmetry from the so-called local translations, which have improved properties. In this work, the algebra of an arbitrary internal symmetry and the local translations is obtained for a generic gauge theory of gravity, in any spacetime dimensions, and coupled to matter fields. It is shown that this algebra closes off shell suggesting that these symmetries form a larger gauge group. In addition, a mechanism to find the symmetries of theories that have nondynamical fields is proposed. It turns out that the explicit form of the local translations depend on the internal symmetry and that the algebra of local translations and the internal group still closes off shell. As an example, the unimodular Einstein–Cartan theory in four spacetime dimensions, which is only invariant under volume preserving diffeomorphisms, is studied.

Heterogeneous Multiscale Method for Maxwell's Equations

We present a finite element heterogeneous multiscale method for time-dependent Maxwell's equations in first-order formulation in highly oscillatory materials using Nédélec's edge elements. Based on a uniform approach for the error analysis of nonconforming space discretizations [D. Hipp, M. Hochbruck, and C. Stohrer, IMA J. Numer. Anal., 39 (2019), pp. 1206--1245], we prove an error bound for the semidiscrete scheme. We further present error bounds for the fully discrete scheme, where we consider time discretization using algebraically stable Runge--Kutta methods, the Crank--Nicolson method, and the leapfrog method. These error bounds are confirmed by numerical experiments.

An Alternative PML Implementation for Arbitrary Media

An efficient implementation of the complex frequency-shifted perfectly matched layer (PML) is proposed to terminate the finite-difference time domain (FDTD) lattices. The proposal is based on the matrix exponential algorithm originally used for modeling frequency dispersive media. As the equations of describing $E (D)$ fields, $H (B)$ fields, and auxiliary variables associated with PML are formulated as a compact first-order differential matrix form, no convolution operations, variable substitutions, or equations rearrangements are needed in its mathematical derivations. Via the concept of generalized material-independent PML, this new implementation can also be used to truncate arbitrary media. Furthermore, since its iteration forms are similar to those of the convolutional PML (CPML) that has been widely used in FDTD simulations, the proposal can be directly applied to existing codes with only minor changes. Meanwhile, it can achieve better absorption effect than CPML. Through a numerical example in terms of the printed F antenna and corresponding analyses of performance and efficiency, we find that the proposal works well as absorbing boundary condition.

Geometrically Nonlinear Analysis for Elastic Beam Using Point Interpolation Meshless Method

The intrinsic beam theory, as one of the exact beam formulas, is quite suitable to describe large deformation of the flexible curved beam and has been widely used in many engineering applications. Owing to the advantages of the intrinsic beam theory, the resulted equations are expressed in first‐order partial differential form with second‐order nonlinear terms. In order to solve the intrinsic beam equations in a relative simple way, in this paper, the point interpolation meshless method was employed to obtain the discretization equations of motion. Different from those equations by using the finite element method, only the differential of the shape functions are needed to form the final discrete equations. Thus, the present method does not need integration process for all elements during each time step. The proposed method has been demonstrated by a numerical example, and results show that this method is highly efficient in treating this type of problem with good accuracy.

Nonlinear Differential Equation Solvers via Adaptive Picard–Chebyshev Iteration: Applications in Astrodynamics

An adaptive self-tuning Picard–Chebyshev numerical integration method is presented for solving initial and boundary value problems by considering high-fidelity perturbed two-body dynamics. The current adaptation technique is self-tuning and adjusts the size of the time interval segments and the number of nodes per segment automatically to achieve near-maximum efficiency. The technique also uses recent insights on local force approximations and adaptive force models that take advantage of the fixed-point nature of the Picard iteration. In addition to developing the adaptive method, an integral quasi-linearization “error feedback” term is introduced that accelerates convergence to a machine precision solution by about a of two. The integral quasi linearization can be implemented for both first- and second-order systems of ordinary differential equations. A discussion is presented regarding the subtle but significant distinction between integral quasi linearization for first-order systems, second-order systems that can be rearranged and integrated in first-order form, and second-order systems that are integrated using a kinematically consistent Picard–Chebyshev iteration in cascade form. The enhanced performance of the current algorithm is demonstrated by solving an important problem in astrodynamics: the perturbed two-body problem for near-Earth orbits. The adaptive algorithm has proven to be more efficient than an eighth-order Gauss–Jackson and a 12th/10th-order Runge–Kutta while maintaining machine precision over several weeks of propagation.

4D spin-2 fields from 5D Chern-Simons theory

We consider a 5-dimensional Chern-Simons gauge theory for the isometry group of Anti-de-Sitter spacetime, AdS4+1 ≃ SO(4, 2), and invoke different dimensional reduction schemes in order to relate it to 4-dimensional spin-2 theories. The AdS gauge algebra is isomorphic to a parametrized 4-dimensional conformal algebra, and the gauge fields corresponding to the generators of non-Abelian translations and special conformal transformations reduce to two vierbein fields in D = 4. Besides these two vierbeine, our reduction schemes leave only the Lorentz spin connection as an additional dynamical field in the 4-dimensional theories. We identify the corresponding actions as particular generalizations of Einstein-Cartan theory, conformal gravity and ghost-free bimetric gravity in first-order form.

Quantum Gravity at the Corner

We investigate the quantum geometry of a 2d surface S bounding the Cauchy slices of a 4d gravitational system. We investigate in detail for the first time the boundary symplectic current that naturally arises in the first-order formulation of general relativity in terms of the Ashtekar–Barbero connection. This current is proportional to the simplest quadratic form constructed out of the pull back to S of the triad field. We show that the would-be-gauge degrees of freedo arising from S U ( 2 ) gauge transformations plus diffeomorphisms tangent to the boundary are entirely described by the boundary 2-dimensional symplectic form, and give rise to a representation at each point of S of S L ( 2 , R ) × S U ( 2 ) . Independently of the connection with gravity, this system is very simple and rich at the quantum level, with possible connections with conformal field theory in 2d. A direct application of the quantum theory is modelling of the black horizons in quantum gravity.

Supertwistor formulation for massless superparticle in AdS5 × S5 superspace

Starting with the first-order formulation of the massless superparticle model on the AdS5×S5 superbackground and presenting the momentum components tangent to AdS5 and S5 subspaces as bilinear combinations of the constrained SU(2)-Majorana spinors allows to bring the superparticle's Lagrangian to the form quadratic in supertwistors. The SU(2,2|4) supertwistors are assembled into a pair of SU(2) doublets, one of which has even SU(2,2) and odd SU(4) components, while the other has odd SU(2,2) and even SU(4) components. They are subject to the first-class constraints that generate the psu(2|2)⊕u(1) gauge algebra. This justifies previously proposed group-theoretic definition of the AdS5×S5 supertwistors and allows to derive the incidence relations with the (10|32) supercoordinates of the AdS5×S5 superspace. Whenever superparticle moves within the AdS5 subspace of the AdS5×S5 space-time, twistor formulation of its Lagrangian involves just one SU(2) doublet of SU(2,2|4) supertwistors with even SU(2,2) and odd SU(4) components. If in addition particle's 5-momentum is null, four first-class constraints which are the su(2)⊕u(1) generators single out upon quantization the states of D=5N=8 gauged supergravity multiplet in the superambitwistor formulation.

Is There Any Symmetry Left in Gravity Theories with Explicit Lorentz Violation?

It is well known that a theory with explicit Lorentz violation is not invariant under diffeomorphisms. On the other hand, for geometrical theories of gravity, there are alternative transformations, which can be best defined within the first-order formalism and that can be regarded as a set of improved diffeomorphisms. These symmetries are known as local translations, and among other features, they are Lorentz covariant off shell. It is thus interesting to study if theories with explicit Lorentz violation are invariant under local translations. In this work, an example of such a theory, known as the minimal gravity sector of the Standard Model Extension, is analyzed. Using a robust algorithm, it is shown that local translations are not a symmetry of the theory. It remains to be seen if local translations are spontaneously broken under spontaneous Lorentz violation, which are regarded as a more natural alternative when spacetime is dynamic.

Numerical noise suppression for wave propagation with finite elements in first-order form by an extended source term

Finite elements can, in some cases, outperform finite-difference methods for modelling wave propagation in complex geological models with topography. In the weak form of the finiteelement method, the delta function is a natural way to represent a point source. If, instead of the usual second-order form, the first-order form of the wave equation is considered, this is no longer true. Fourier analysis for a simple case shows that the spatial operator corresponding to the first-order form has short-wavelength null-vectors. Once excited, these modes are not seen by the spatial operator but only by the time- stepping scheme and show up as noise. A sourcewith a larger spatial extent, for instance a Gaussian or a tapered sinc, can avoid the excitation of problematic short wavelengths. A series of numerical experiments on a 2-D problem with an exact solution provides a suggestion for the best choice of parameters for these source term distributions. The tapered sinc provided the best results and the resulting accuracy can be better than that of the second-order form. The higher operation count of the former, however, does not make it more efficient in terms of accuracy for a given computational effort, at least not for the 2-D examples considered here

Study of Homogeneous and Isotropic Universe in f(R,Tφ) Gravity

This paper is devoted to study the cosmological behavior of homogeneous and isotropic universe model in the context of f(R,Tφ) gravity, where φ is the scalar field. For this purpose, we follow the first-order formalism defined by H=W(φ). We evaluate Hubble parameter, effective equation of state parameter (ωeff), deceleration parameter, and potential of scalar field for three different values of W(φ). We obtain phantom era in some cases for the early times. It is found that exponential expression of W(φ) yields ωeff independent of time for flat universe and independent of model parameter otherwise. It is concluded that our model corresponds to ΛCDM for both initial and late times.

Domain wall brane in a reduced Born–Infeld-f(T) theory

The Born–Infeld f(T) theory is reduced from the Born–Infeld determinantal gravity in Weitzenböck spacetime. We investigate a braneworld scenario in this theory and obtain an analytic domain wall solution by utilizing the first-order formalism. The model is stable against the linear tensor perturbation. It is shown that the massless graviton is localized on the brane, but the continuous massive gravitons are non-localized and will generate a tiny correction with the behavior of 1/(kr)3 to the Newtonian potential. The four-dimensional teleparallel gravity is recovered as an effective infrared theory on the brane. As a physical application, we consider the (quasi-)localization property of spin-1/2 Dirac fermion in this model.

Global stabilization analysis of inertial memristive recurrent neural networks with discrete and distributed delays

This paper deals with the stabilization problem of memristive recurrent neural networks with inertial items, discrete delays, bounded and unbounded distributed delays. First, for inertial memristive recurrent neural networks (IMRNNs) with second-order derivatives of states, an appropriate variable substitution method is invoked to transfer IMRNNs into a first-order differential form. Then, based on nonsmooth analysis theory, several algebraic criteria are established for the global stabilizability of IMRNNs under proposed feedback control, where the cases with both bounded and unbounded distributed delays are successfully addressed. Finally, the theoretical results are illustrated via the numerical simulations.

Dynamical Symmetry Breaking in Geometrodynamics

Using a first-order perturbative formulation, we analyze the local loss of symmetry when a source of electromagnetic and gravitational fields interacts with an agent that perturbs the original geometry associated with the source. We had proved that the local gauge groups are isomorphic to local groups of transformations of special tetrads. These tetrads define two orthogonal planes at every point in space–time such that every vector in these local planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. Because the local gauge symmetry in Abelian or even non-Abelian field structures in four-dimensional Lorentzian space–times is manifested by the existence of local planes of symmetry, the loss of symmetry is manifested by a tilt of these planes under the influence of an external agent. In this strict sense, the original local symmetry is lost. We thus prove that the new planes at the same point after the tilting generated by the perturbation correspond to a new symmetry. Our goal here is to show that the geometric manifestation of local gauge symmetries is dynamical. Although the original local symmetries are lost, new symmetries arise. This is evidence for a dynamical evolution of local symmetries. We formulate a new theorem on dynamical symmetry evolution. The proposed new classical model can be useful for better understanding anomalies in quantum field theories.

Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes

We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential equations system is written in non-conservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations.