Invariant Equations Research Articles

The Weyl double copy in vacuum spacetimes with a cosmological constant

We examine the Weyl double copy relation for vacuum solutions of the Einstein equations with a cosmological constant using the approach we previously described, in which the spin-1/2 massless free-field spinors (Dirac-Weyl fields) are regarded as basic units. Based on the exact non-twisting vacuum type N and vacuum type D solutions, the finding explicitly shows that the single and zeroth copies fulfill conformally invariant field equations in conformally flat spacetime. In addition, irrespective of the presence of a cosmological constant, we demonstrate that the zeroth copy connects Dirac-Weyl fields with the degenerate electromagnetic fields in the curved spacetime in addition to connecting gravity fields with the single copy in conformally flat spacetime. Moreover, the study also demonstrates the critical significance the zeroth copy plays in time-dependent radiation solutions. In particular, for Robinson-Trautman (Λ) gravitational waves, unlike the single copy, we find that the zeroth copy carries additional information to specify whether the sources of associated gravitational waves are time-like, null, or space-like, at least in the weak field limit.

Is there a relativistic Gorini-Kossakowski-Lindblad-Sudarshan master equation?

The celebrated GKLS master equation, widely called just Lindblad equation, is the universal dynamical equation of non-relativistic open quantum systems in their Markovian approximation. It is not necessary and perhaps impossible that GKLS equations possess sensible relativistic forms. In a lucid talk on black hole information loss paradox, David Poulin conjectured a Lorentz invariant GKLS master equation. It remained unpublished. Poulin passed away at heights of his activity. But the equation is really puzzling. A closer look uncovers a smartly hidden defect which leaves us without Lorentz invariant Markovian master equations. They, in view of the present author, should not exist.

Matter effects of sterile neutrino in light of renormalization-group equations

The renormalization-group equation (RGE) approach to neutrino matter effects is further developed in this work. We derive a complete set of differential equations for effective mixing elements, masses and Jarlskog-like invariants in presence of a light sterile neutrino. The evolutions of mixing elements as well as Jarlskog-like invariants are obtained by numerically solving these differential equations. We calculate terrestrial matter effects in long-baseline (LBL) experiments, taking NOvA, T2K and DUNE as examples. In both three-flavor and four-flavor frameworks, electron-neutrino survival probabilities as well as the day-night asymmetry of solar neutrino are also evaluated as a further examination of the RGE approach.

Predicting Perceived Sport Performance via Self-Determination Theory

Sport performance depends in part on psychological factors. Self-determination theory emphasizes that the satisfisfaction of basic psychological needs and autonomous motivation may be linked to positive behavioral results and positive perceptions of performance. Our main objective in this study was to analyze the predictive power of such variables as perceived autonomy support, the satisfaction of psychological needs, and autonomous motivation on perceived sport performance. We recruited 447 young Portuguese athletes/participants (340 males, 107 females; Mage = 15.72 years, SD = 1.8). We had participants complete Portuguese sport versions of the Perceived Autonomy Support Exercise Climate Questionnaire, the Basic Psychological Needs in Exercise Scale, the Behavior Regulation Sport Questionnaire and the Perceived Performance in Sport Questionnaire. A gender invariant structural equation model showed that the coach's autonomy support significantly predicted the satisfaction of psychological needs for autonomy, competence, and relatedness. The satisfaction of these three psychological needs predicted autonomous motivation. Satisfying the need for competence positively predicted perceived sport performance, but autonomous motivation was not significantly related to perceived performance.

Stability analysis of reaction–diffusion PDEs coupled at the boundaries with an ODE

This paper addresses the derivation of generic and tractable sufficient conditions ensuring the stability of a coupled system composed of a reaction–diffusion partial differential equation (PDE) and a finite-dimensional linear time invariant ordinary differential equation (ODE). The coupling of the PDE with the ODE is located either at the boundaries or in the domain of the reaction–diffusion equation and takes the form of the input and output of the ODE. We investigate boundary Dirichlet/Neumann/Robin couplings, as well as in-domain Dirichlet/Neumann couplings. The adopted approach relies on the spectral reduction of the problem by projecting the trajectory of the PDE into a Hilbert basis composed of the eigenvectors of the underlying Sturm–Liouville operator and yields a set of sufficient stability conditions taking the form of LMIs. We propose numerical examples, consisting of an unstable reaction–diffusion equation and an unstable ODE, such that the application of the derived stability conditions ensure the stability of the resulting coupled PDE–ODE system.

A typology of Marxian transformation procedures with endogenous exploitation rate

AbstractThis paper constructs a unified framework to evaluate the Marxian transformation models, where the exploitation rate is an endogenous variable determined within value and price of production systems. All different procedures are based on two different value‐price invariance equations. These could be chosen among either two aggregate product equations (gross or net output) or three capital invariant equations (total, variable or constant). Different combinations of invariance equations result in most Marxian solutions developed in recent decades. The solution does not imply that one model (prices) is logically prior to the other (values), but that in fact that the joint solution is needed.

On the σ2-Nirenberg problem on [formula omitted

We establish theorems on the existence and compactness of solutions to the σ2-Nirenberg problem on the standard sphere S2. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ2-Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ2-Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove L∞ a priori estimates for solutions of the σ2-Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the σ2-Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators.

Dynamical Properties, Deformations, and Chaos in a Class of Inversion Invariant Jerk Equations

In this paper, we consider a class of jerk equations which are invariant to inversion. We discuss the stability and some bifurcations of the considered equation. In addition, we construct integrable deformations in order to stabilize some equilibrium points. Finally, we introduce a piecewise chaotic system which belongs to the considered class of jerk equations.

Spin-valley Silin modes in graphene with substrate-induced spin-orbit coupling

In the presence of external magnetic field the Fermi-liquid state supports oscillatory spin modes known as Silin modes. We predict the existence of the generalized Silin modes in a multivalley system, monolayer graphene. A gauge- and Berry-gauge- invariant kinetic equation for a multivalley Fermi liquid is developed and applied to the case of graphene with extrinsic spin-orbit coupling (SOC). The interplay of SOC and Berry curvature allows for the excitation of generalized Silin modes in the spin and valley-staggered-spin channels via an AC electric field. The resonant contributions from these modes to the optical conductivity are calculated.

Spatial regularity for a class of degenerate Kolmogorov equations

We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion’s directions, i.e. $$x_{1}, \ldots , x_{m_{0}}$$ . The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide a new proof for the optimal spatial regularity.

Topological field theory with Haagerup symmetry

We construct a (1 + 1)d topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup H3 fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup H1 and H2 fusion categories can be obtained by gauging algebra objects. This article makes a modest offering in our pursuit of exotica and the quest for their eventual conformity.

Lacunary invariant statistical equivalence for double set sequences

In this paper, we introduce the notions of asymptotical strong σ 2 σ2 -equivalence, asymptotical σ 2 σ2 -statistical equivalence, asymptotical lacunary strong σ 2 σ2 -equivalence and asymptotical lacunary σ 2 σ2 -statistical equivalence in the Wijsman sense for double set sequences. Also, we investigate some relations between these new asymptotical equivalence notions.

Liouville theorems for parabolic systems with homogeneous nonlinearities and gradient structure

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. Assume that \(p>1\) is subcritical in the Sobolev sense. In the case of nonnegative solutions and the system $$\begin{aligned} U_t-\varDelta U=F(U)\quad \hbox {in}\quad \mathbb {R}^n\times \mathbb {R}\end{aligned}$$where \(U=(u_1,\dots ,u_N)\), \(F=\nabla G\) is p-homogeneous and satisfies the positivity assumptions \(G(U)>0\) for \(U\ne 0\) and \(\xi \cdot F(U)>0\) for some \(\xi \in \mathbb {R}^N\) and all \(U\ge 0\), \(U\ne 0\), it has recently been shown in [P. Quittner, Duke Math. J. 170 (2021), 1113–1136] that the parabolic Liouville theorem is true whenever the corresponding elliptic Liouville theorem for the system \(-\varDelta U=F(U)\) is true. By modifying the arguments in that proof we show that the same result remains true without the positivity assumptions on G and F, and that the class of solutions can also be enlarged to contain (some or all) sign-changing solutions. In particular, in the scalar case \(N=1\) and \(F(u)=|u|^{p-1}u\), our results cover the main result in [T. Bartsch, P. Poláčik and P. Quittner, J. European Math. Soc. 13 (2011), 219–247]. We also prove a parabolic Liouville theorem for solutions in \(\mathbb {R}^n_+\times \mathbb {R}\) satisfying homogeneous Dirichlet boundary conditions on \(\partial \mathbb {R}^n_+\times \mathbb {R}\) since such theorem is also needed if one wants to prove universal estimates of solutions of related systems in \(\varOmega \times (0,T)\), where \(\varOmega \subset \mathbb {R}^n\) is a smooth domain. Finally, we use our Liouville theorems to prove universal estimates for particular parabolic systems.

Asymptotic freedom and safety in quantum gravity

We compute non-perturbative flow equations for the couplings of quantum gravity in fourth order of a derivative expansion. The gauge invariant functional flow equation for arbitrary metrics allows us to extract β-functions for all couplings. In our truncation we find two fixed points. One corresponds to asymptotically free higher derivative gravity, the other is an extension of the asymptotically safe fixed point in the Einstein-Hilbert truncation or extensions thereof. The infrared limit of the flow equations entails only unobservably small modifications of Einstein gravity coupled to a scalar field. Quantum gravity can be asymptotically free, based on a flow trajectory from the corresponding ultraviolet fixed point to the infrared region. This flow can also be realized by a scaling solution for varying values of a scalar field. As an alternative possibility, quantum gravity can be realized by asymptotic safety at the other fixed point. There may exist a critical trajectory between the two fixed points, starting in the extreme ultraviolet from asymptotic freedom. We compute critical exponents and determine the number of relevant parameters for the two fixed points. Evaluating the flow equation for constant scalar fields yields the universal gravitational contribution to the effective potential for the scalars.

Einstein’s dream to unify all forces finally materializes: a revived de Broglie’s pilot-wave theory with novel solutions

Starting from Louis de Broglie’s pilot wave-theory, this paper unifies gravity and quantum mechanics under a single mathematical field theory for all forces in Nature. Two families of potentials coexist as mathematical solutions for the homogeneous Klein-Gordon equation which is the same homogeneous classical wave equation: (a) Neo-Laplacian local time-independent background potentials, and (b) Novel time-distance entangled Q(q) potentials which are isomorph to distance-time-velocity transformations based on any of the competing relativistic theories (Lorentz, Poincaré or Einstein), or on the pre-relativistic Galilean invariant Doppler equations. This remarkable property makes present theory compatible with all previous empirical evidence, including experiments conventionally interpreted as supporting Einstein’s special relativity. We report explicit closed solutions for potentials solving the one-dimensional and three-dimensional classical wave equations, and describe in detail how to calculate time-independent neo-Laplacian background forces and relativistically isomorph time-dependent entangled forces. The scale of the problem appears as a required parameter, thus making our theory applicable to all scales of Nature from quarks to cosmos. A usually overlooked neo-Laplacian logarithmic potential predicts the observed high values of non-Keplerian tangential speeds at the galactic scale. At the human scale, calculations relative to hurricanes and tornadoes may be facilitated by the closed form of our unified forces. A novel torsion component of gravity automatically appears from our new solutions.

Fractality of Aging of Living Systems

On the basis of the basic model of the kinetic theory of aging of living systems, mathematical modeling of various characteristics of aging of mankind, state, generation, human body, organs and cells has been carried out. These results are compared with experimental and calculated data of other authors. The analysis of the works presented here and those carried out earlier gave reason to believe that the basic mathematical model of the evolution of aging of living dynamic systems of various hierarchical levels and nature is the invariant differential equation of the kinetic theory of aging, as a manifestation of the fractality property of living systems.

The Blow-Up Rate for a Non-Scaling Invariant Semilinear Heat Equation

We consider the semilinear heat equation $$\partial_t u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log^a (2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.

Invariant manifold approach for quantifying the dynamics of highly inertial particles in steady and time-periodic incompressible flows.

The dynamics of finite-sized particles with large inertia are investigated in steady and time-dependent flows through the numerical solution of the invariance equation, describing the spatiotemporal evolution of the slow/inertial manifold representing the effective particle velocity field. This approach allows for an accurate reconstruction of the effective particle divergence field, controlling clustering/dispersion features of particles with large inertia for which a perturbative approach is either inaccurate or not even convergent. The effect of inertia on heavy and light particles is quantified in terms of the rate of contraction/expansion of volume elements along a particle trajectory and of the maximum Lyapunov exponent for systems exhibiting chaotic orbits, such as the time-periodic sine-flow on the 2D torus and the time-dependent 2D cavity flow.

On the solution of hyperbolic equations using the peridynamic differential operator

Numerical solution of hyperbolic differential equations, such as the advection equation, poses challenges. Classically, this issue has been addressed by using a scheme known as the upwind scheme. It simply invokes more points from the upwind side of the flow stream when calculating derivatives. This study presents a generalized upwind scheme, referred to as directional nonlocality, for the numerical solution of linear and nonlinear hyperbolic Partial Differential Equations (PDEs) using the peridynamic differential operator (PDDO). The PDDO provides the nonlocal form of the differential equations by introducing an internal length parameter (horizon) and a weight function. The weight function controls the degree of interaction among the points within the horizon. A modification to the weight function, i.e., upwinded-weight function, accounts for directional nonlocality along which information travels. This modification results in a stable PDDO discretization of hyperbolic PDEs. Solutions are constructed in a consistent manner without special treatments through simple discretization. The capability of this approach is demonstrated by considering time dependent linear and nonlinear hyperbolic equations as well as the time invariant Eikonal equation. Numerical stability is ensured for the linear advection equation and the PD solutions compare well with the analytical/reference solutions.

On the variational principle in the unfolded dynamics

The interplay between off-shell and on-shell unfolded systems is analyzed. The formulation of invariant constraints that put an off-shell system on shell is developed by adding new variables and derivation in the target space, that extends the original Q-derivation of the unfolded system to a bicomplex. The analogue of the Euler-Lagrange equations in the unfolded dynamics is suggested. The general class of invariant on-shell equation constraints is defined in cohomological terms. The necessary and sufficient condition for the on-shell equation constraints being Euler-Lagrange for some Lagrangian system is proven. The proposed construction is illustrated by the scalar field example.