No-go theorem for inflation in an extended Ricci-inverse gravity model

In this paper, we propose an extension of the Ricci-inverse gravity, which has been proposed recently as a very novel type of fourth-order gravity, by introducing a second order term of the so-called anticurvature scalar as a correction. The main purpose of this paper is that we would like to see whether the extended Ricci-inverse gravity model admits the homogeneous and isotropic Friedmann–Lemaitre–Robertson–Walker metric as its stable inflationary solution. However, a no-go theorem for inflation in this extended Ricci-inverse gravity is shown to appear through a stability analysis based on the dynamical system method. As a result, this no-go theorem implies that it is impossible to have such stable inflation in this extended Ricci-inverse gravity model.

Adjacency Labelling for Planar Graphs (and Beyond)

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n -vertex planar graph G is assigned a (1 + o(1)) log 2 n -bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G . This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n , there exists a graph U n with n 1+o(1) vertices such that every n -vertex planar graph is an induced subgraph of U n . These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k -planar graphs.

New Gravity Field Equation is derived by Einstein Field Equation in General Relativity Theory

We found the 4-order curvature term satisfied the co-variant derivative. Einstein gravity fieldequation is consist of 2-order curvature terms. Hence, the 4-order curvature term and 2-order curvature termsmake new gravity field equation. In this point, Einstein’s gravity field equation can be modified by new 4-order curvature term because gravity field equation’s term doesn’t have to be 2-order term. Indeed, Einsteinhimself was like that, 0-order term, the cosmological term. Therefore, our theory is based on legitimate facts.

On the justification of topological derivative for wave-based qualitative imaging of finite-sized defects in bounded media

PurposeThis work contributes to the general problem of justifying the validity of the heuristic that underpins medium imaging using topological derivatives (TDs), which involves the sign and the spatial decay away from the true anomaly of the TD functional. The author considers here the identification of finite-sized (i.e. not necessarily small) anomalies embedded in bounded media and affecting the leading-order term of the acoustic field equation.Design/methodology/approachTD-based imaging functionals are reformulated for analysis using a suitable factorization of the acoustic fields, which is facilitated by a volume integral formulation. The three kinds of TDs (single-measurement, full-measurement and eigenfunction-based) studied in this work are given expressions whose structure allows to establish results on their sign and decay properties. The latter are obtained using analytical methods involving classical identities on Bessel functions and Legendre polynomials, as well as asymptotic approximations predicated on spatial scaling assumptions.FindingsThe sign component of the TD imaging heuristic is found to be valid for multistatic experiments and if the sought anomaly satisfies a bound (on a certain operator norm) involving its geometry, its contrast and the operating frequency. Moreover, upon processing the excitation and data by applying suitably-defined bounded linear operatirs to them, the magnitude component of the TD imaging heuristic is proved under scaling assumptions where the anomaly is small relative to the probing region, the latter being itself small relative to the propagation domain. The author additionally validates both components of the TD imaging heuristic when the probing excitation is taken as an eigenfunction of the source-to-measurement operator, with a focusing effect analogous to that achieved in time-reversal based methods taking place. These findings extend those of earlier studies to the case of finite-sized anomalies embedded in bounded media.Originality/valueThe originality of the paper lies in the theoretical justifications of the TD-based imaging heuristic for finite-sized anomalies embedded in bounded media.

An Improved Backscattering Theoretical Model for Snow Area

Remote sensing has been studied for a long time to monitor the earth terrain. Remote sensing technology has been used globally in many different fields and one of the most popular area of study that uses remote sensing technology is snow monitoring. In previous researches, remote sensing has been modelled on snow area to study the scattering mechanisms of various scattering processes. In this paper, surface volume second order term that was dropped in previous study is derived, included and studied to observe the improvement in the surface volume backscattering coefficient. This new model is applied on snow layer above ground and the snow layer is modelled as a volume of ice particles as the Mie scatterers that are closely packed and bounded by irregular boundaries. Various parameters are used to investigate the improvement of adding the new term. Results show improvement in cross-polarized return, for all the range of parameters studied. Comparison is made with the field measurement result from U.S. Army Cold Regions Research and Engineering Laboratory (CRREL) in 1990. Close agreement is shown between developed model and data field backscattering coefficient result.

Phase field model for electrically induced damage using microforce theory

In this paper, we present a consistent derivation of the phase field model for electrically induced damage. The derivation is based on Gurtin’s microstress and microforce theory and the Coleman–Noll procedure. The resulting model accounts for Ohmic currents, includes charge conservation law and allows for finite electric permittivity and conductivity distribution in the medium. Special attention is devoted to the case when the damaged region is a codimension-two object, i.e., a curve in three dimensions. It is shown that in this case the free energy of the model necessarily includes a high-order term, which ensures the well-posedness of the problem. A special problem setting is proposed to account for the prescribed charge distribution. Local features of the phase field distribution are illustrated with one-dimensional axisymmetric numerical experiments.

Efficient Full Higher-Order Unification

We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski. Our procedure removes many redundant unifiers by carefully restricting the search space and tightly integrating decision procedures for fragments that admit a finite complete set of unifiers. We identify a new such fragment and describe a procedure for computing its unifiers. Our unification procedure, together with new higher-order term indexing data structures, is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure.

A nonlinear magneto-elastoplastic coupling model based on Jiles-Atherton theory of ferromagnetic materials

As seen in the Jiles-Atherton (J-A) model and its modified form, the linear relationship between the magnetization coefficient and the stress deviates significantly from the experimental results. It is required to introduce many parameters that are difficult to obtain or unknown to describe the effect of elastoplastic deformation on magnetization or hysteresis, such as shape coefficient, pinning coefficient, and molecular field coefficient. In this paper, a new nonlinear magneto-elastoplastic model for ferromagnetic materials is established based on the magneto-mechanical coupling effect, and both the sixth-order term of magnetization and the nonlinear equation of the magnetization coefficient are introduced into the magnetostriction equation. In the models established in this paper, the elastoplastic deformation equivalent magnetic field is introduced into the effective magnetic field, and the Frohlich-Kennelly equation is used to describe the anhysteretic magnetization. After comparing the prediction results of different models with the available experimental results, it is observed that the proposed model in this paper exhibits superior prediction ability for magnetostrictive strain, magnetization, and hysteresis phenomena under different stresses. This paper has also analyzed the mechanism of the effect of elasto-plastic loading and residual plastic deformation on the hysteresis in different models as well as the differences between them. The determination coefficient of the proposed model in this paper is closer to 1 that is better than the existing models, indicating that it has a better fitting effect and is of great significance to the development of quantitative nondestructive testing technology.

A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes

We prove Price’s law with an explicit leading order term for solutions $$\phi (t,x)$$ ϕ ( t , x ) of the scalar wave equation on a class of stationary asymptotically flat $$(3+1)$$ ( 3 + 1 ) -dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that $$\phi (t,x)=c t^{-3}+{\mathcal {O}}(t^{-4+})$$ ϕ ( t , x ) = c t - 3 + O ( t - 4 + ) for bounded |x|, where $$c\in {\mathbb {C}}$$ c ∈ C is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise $$t^{-2 l-3}$$ t - 2 l - 3 decay for waves with angular frequency at least l, and $$t^{-2 l-4}$$ t - 2 l - 4 decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

Nodal Multi-peak Standing Waves of Fourth-Order Schrödinger Equations with Mixed Dispersion

We consider the existence and concentration properties of standing waves for a fourth-order Schrödinger equation with mixed dispersion, which was introduced to regularize and stabilize solutions to the classical time-dependent Schrödinger equation. This leads to study multi-peak solutions to the following singularly perturbed fourth-order nonlinear Schrödinger equation $$\begin{aligned} {\varepsilon ^{\text {4}}}{\Delta ^{\text {2}}}u - \beta {\varepsilon ^2}\Delta u + V(x)u = |u{|^{p - 2}}u{\text { in }}{\mathbb {R}^N},{\text { }}u \in {H^2}({\mathbb {R}^N}). \end{aligned}$$ ε 4 Δ 2 u - β ε 2 Δ u + V ( x ) u = | u | p - 2 u in R N , u ∈ H 2 ( R N ) . We first establish a local $${W^{4,p}}$$ W 4 , p -estimate for a class of fourth-order semilinear elliptic equations, which is a key to get the uniform and global $${L^\infty }$$ L ∞ -estimate of solutions to the considered singularly perturbed equation above. Next, under certain assumptions on $$\beta $$ β and the potential V(x), we construct a family of sign-changing multi-peak solutions with a unique maximum (or minimum) point on each component. We prove that these solutions concentrate around any prescribed finite set of local minima (possibly degenerate) of the potential V(x). Compared with the classical singularly perturbed Schrödinger equation, the presence of a fourth-order term in the problem above forces the development of new techniques to obtain qualitative properties of multi-peak solutions.