Regular Solution Research Articles

Quasinormal modes and phase structure of regular AdS Einstein–Gauss–Bonnet black holes

In this paper, we present an exact regular black hole solution in Einstein–Gauss–Bonnet coupled with nonlinear matter fields. It is a generalization of a regular Einstein–Gauss–Bonnet black hole in [Formula: see text] [Formula: see text] spacetime. The causal structure of the obtained solution identifies with Boulware–Deser black hole solution, except for the curvature singularity at the center. It incorporates the Boulware–Deser black holes in the absence of deviation parameters. We also study the thermodynamic properties of the solution that satisfies a modified first law of thermodynamics. Furthermore, we discuss the stability of the obtained black hole solution and, in this regard, a double phase transition occurs. Within this context, we find that phase transition exists at the point where the heat capacity diverges and, incidentally, the temperature attains the maximum value. We discuss the fluid nature of the black hole also exhibiting critical points. The quasinormal modes of the black hole solution and their dependencies on Gauss–Bonnet coupling and deviation parameters are also analyzed in terms of null geodesics.

On the stability of ring relative equilibria in the N-body problem on with Hodge potential

AbstractIn this paper, we study the stability of the ring solution of the N-body problem in the entire sphere $\mathbb {S}^2$ by using the logarithmic potential proposed in Boatto et al. (2016, Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 472, 20160020) and Dritschel (2019, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 377, 20180349), derived through a definition of central force and Hodge decomposition theorem for 1-forms in manifolds. First, we characterize the ring solution and study its spectral stability, obtaining regions (spherical caps) where the ring solution is spectrally stable for $2\leq N\leq 6$ , while, for $N\geq 7$ , the ring is spectrally unstable. The nonlinear stability is studied by reducing the system to the homographic regular polygonal solutions, obtaining a 2-d.o.f. Hamiltonian system, and therefore some classic results on stability for 2-d.o.f. Hamiltonian systems are applied to prove that the ring solution is unstable at any parallel where it is placed. Additionally, this system can be reduced to 1-d.o.f. by using the angular momentum integral, which enables us to describe the phase portraits and use them to find periodic ring solutions to the full system. Some of those solutions are numerically approximated.

Imposing a constraint on the discrete Reynolds–Orr equation demonstrated in shear flows

The Reynolds–Orr equation predicts the unconditional stability limit of a flow. Although this seems to be a desirable aim in engineering applications, the predicted critical Reynolds numbers are one magnitude below the experimental observations. In this paper, an attempt is made to reduce this gap for incompressible shear flows. It is known that the Navier–Stokes equation has no regular solution at the initial time if the initial velocity field does not fulfill the compatibility condition. However, the original solution of the Reynolds–Orr equation, the critical perturbation, does not necessarily fulfill this condition. Therefore, the condition is added to the original problem as a non-linear constraint. This requires the use of a discrete functional, introduced in the paper. Two different formulations are implemented and discussed. The solution is assumed in a waveform. The augmented problem is solved in the cases of planar Poiseuille and the Couette flow. The result shows that adding the constraint increases the critical Reynolds number significantly in the case of a streamwise perturbation but only slightly in the case of a spanwise one. It was demonstrated using numerical simulations that the single waveform assumption was unreasonably strict. The usage of the compatibility condition without assuming the single waveform has a negligible effect on the critical Reynolds number. However, the presented methods can be used for adding other reasonable and complicated constraints to the variational problem.

The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method

Abstract In this paper, the analytic solution of the delay fractional model is derived by the method of steps. The theoretical result implies that the regularity of the solution at s + {s^{+}} is better than that at 0 + {0^{+}} , where s is a constant time delay. The behavior of derivative discontinuity is also discussed. Then, improved regularity solution is obtained by the decomposition technique and a fitted L ⁢ 1 {L1} numerical scheme is designed for it. For the case of initial singularity, the optimal convergence order is reached on uniform meshes when α ∈ [ 2 3 , 1 ) {\alpha\in[\frac{2}{3},1)} , α is the order of fractional derivative. Furthermore, an improved fitted L ⁢ 1 {L1} method is proposed and the region of optimal convergence order is larger. For the case t > s {t>s} , stability and min ⁡ { 2 ⁢ α , 1 } {\min\{2\alpha,1\}} order convergence of the fitted L ⁢ 1 {L1} scheme are deduced. At last, the numerical tests are carried out and confirm the theoretical result.

Chemical aging of sealing gasket of proton exchange membrane fuel cell

AbstractAs the key sealing part of proton exchange membrane fuel cell (PEMFC), the gasket is easily damaged in acidic solution and wet air for a long time in PEMFC, resulting in unsafe and unstable operation of PEMFC. In this paper, the regular solution (RS) and accelerated durability test (ADT) solution in PEMFC were prepared to study the aging damage process of silicone rubber gasket, and the attenuated total reflection Fourier transform infrared spectrometer (ATR‐FTIR) was used to analyze the surface changes of gaskets soaked in solutions. The results showed that the weight loss of samples increased with the soaking time in both ADT solution and RS solution, and the damage of samples in ADT solution was faster and more serious than that in RS solution. Meanwhile, the microstructure diagram also indicated that the damage of the samples in ADT solution was more serious than that in the RS solution. The ATR‐FTIR analysis results showed that the intensity of characteristic wave peaks of samples at 706, 787, 867, 1006, 1071, 1260, and 2961cm−1 in ADT solution decreased faster than that in RS solution, which was consistent with the results of weight analysis and microstructure analysis.

Critical micelle concentration and partition coefficient of mixed micelles: Analysis of ternary systems based on Markov chain model and simple mixture model

When two or more surfactants are mixed, the mixed system exhibits improved performance compared with a single surfactant system in solution. The Markov chain model can analyze the critical micelle concentration (CMC) of mixed micelles, yielding results similar to those of a simple mixture model, which is typically referred to as the “regular solution theory”. In this study, two hypotheses were tested: (1) the Markov chain model for ternary systems can be simplified by approximating the association constant of surfactants i and j as Kij = Kji and (2) the quasi-simple ternary mixture model, that is, a simple mixture model analogous to the Markov chain model, helps interpret the interaction parameter of the simple mixture model that can describe the partition coefficient of the binary mixed micelle. Equations were derived for (1) the Markov chain model for ternary systems by assuming Kij = Kji and (2) the interaction parameter of the simple mixture model that can describe the partition coefficient of the binary mixed micelle. The models were compared with the experiment data, and the derived equations described the data of the CMC and partition coefficient well.

Determination of Three-Dimensional Brinkman—Forchheimer-Extended Darcy Flow

The aim of this study is to determine a 3D incompressible Brinkman–Forchheimer-extended Darcy fluid flow. Based on global well-posedness and regularity of solutions with a periodic boundary condition, the determining modes for weak and regular solutions is achieved via the generalized Grashof number for a 3D non-autonomous Brinkman–Forchheimer-Darcy fluid flow in porous medium. Furthermore, the asymptotic determination of the complete trajectories inside an attractor via Fourier functionals is shown for a 3D autonomous Brinkman–Forchheimer-extended Darcy model.

Existence, nonexistence and uniqueness for Lane–Emden type fully nonlinear systems

We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane–Emden coupling configuration. Our results are based on the analysis of the associated quadratic dynamical system and energy methods. For both regular and exterior domain radial solutions we obtain new regions of existence and nonexistence. Besides, we show an exclusion principle for regular solutions, either in or in a ball, by exploiting the uniqueness of trajectories produced by the flow. In particular, for the standard Lane–Emden system involving the Laplacian operator, we prove that the critical hyperbola of regular radial positive solutions is also the threshold for existence and nonexistence of radial exterior domain solutions with Neumann boundary condition. As a byproduct, singular solutions with fast decay at infinity are also found.

An Evolutionary Guardrail Layout Design Framework for Crowd Control in Subway Stations

Deploying guardrails near elevator entrances is an effective way to alleviate congestion and improve the flow rate in subway stations. How to properly design the guardrail layout is a complex black-box optimization problem. Existing methods are mainly based on manual design, which are highly dependent on the empirical experience of the designers and may not get satisfactory results in complicated scenarios. To address the above issues, this article proposes an evolutionary framework to automatically optimize guardrail layouts in subway stations. In the proposed framework, a novel guardrail layout encoding method is proposed, which can facilitate the algorithm to generate regular guardrail layout design solutions. Furthermore, a new fitness evaluation function is proposed to effectively measure the quality of a given guardrail layout design strategy. To validate its effectiveness, the proposed framework is applied to two scenarios with different characteristics. Simulation results have demonstrated that the proposed framework can provide promising guardrail layout designs, which can alleviate the congestion of subway stations effectively.

Substitution limits and thermodynamic stability of solid solutions in HTSC based on iron and yttrium oxyarsenides

The mixing energies (interaction parameters) and the critical decomposition (stability) temperatures of Y1–xLnxFeAsO0.6 solid solutions, where Ln = La–Er, 0 < x < 1.0, were determined within the framework of the V. S. Urusov crystal-energetic approach in the approximation of regular solid solutions. Diagram of thermodynamic stability and domes of the solid solution decomposition have been plotted, which make it possible to calculate the equilibrium isomorphous substitution limits of yttrium for rare earth elements x depending on the decomposition temperature Td, or the decomposition temperature depending on the substitution limit. The results of calculation do not contradict to the experimental data found earlier in the literature for La1–xYxFeAsO0.6 and related systems La1–yYvFeAsO, La0.8Y0.2FeAsO0.7, and Y0.95La0.05FeAsO1–vH0.15. The results of this work can be useful in choosing the components’ ratio in “mixed” matrices and the dopant content in high-temperature superconductors, effective magnetic and other materials.

Measurements of cholesterol chemical potential constrain models of cholesterol-phosphatidylcholine interactions.

Bilayer membranes composed of cholesterol and phospholipids exhibit diverse forms of non-ideal mixing. In particular, many previous studies document macroscopic liquid-liquid phase separation as well as nanometer-scale heterogeneity in membranes of phosphatidylcholine (PC) and cholesterol. Here, we present experimental measurements of cholesterol chemical potential (μch) in binary membranes containing dioleoyl PC (DOPC), 1-palmitoyl-2-oleoyl PC (POPC), or dipalmitoyl PC (DPPC), and in ternary membranes of DOPC and DPPC, adapting a calibrated experimental protocol developed to measure μch in cells (Ayuyan and Cohen, Biophys. J. 114:904-918). μch is the thermodynamic quantity that dictates the availability of cholesterol to bind other factors, and notably must be equal between coexisting phases of a phase-separated mixture. It is simply related to concentration under conditions of ideal mixing, but is found to be far from ideal for the majority of lipid mixtures investigated. Experimental measurements of μch are used to constrain thermodynamic models of membrane interactions. Measurements are consistent with models involving cholesterol-phospholipid complexes only if complexes are more weakly bound than is assumed in previous reports. Experimental measurements are also well described by regular solution theory and lattice models of pairwise interactions between components. These findings reinforce that μch depends on membrane composition overall, providing avenues for cells to alter the availability of cholesterol without varying cholesterol concentration.

Regular solutions for black strings and torus-like black holes

In this letter, we extend the recent black hole solution with spherical symmetry [Phys. Lett. B \textbf{835} (2022), 137546], to a new class of black hole solutions with different topology, namely, a torus-like black hole and a black string in the presence of the cosmological constant. We have used the regularized expressions for the gravitational and the electromagnetic potentials, then we derived the corrections to the energy density and, finally, the black hole solutions are obtained which are regular in the limit $r \to 0$. Interestingly, it is found that the total mass of the black hole is corrected due to the regularized self-energy of the electrostatic field. In the special limit, we recover the well-known solutions in the literature.

SOLITON OF SELF-CONSISTENT SYSTEM OF NONLINEAR SPINOR AND GRAVITATIONAL FIELDS WITH THE ARBI- TRARY FUNCTION DEPENDING ON THE BILINEAR PAULI- FIERZ INVARIANT IS = S2 IN GENERAL RELATIVITY

A self-consistent system of nonlinear spinor and gravitational fields, modeled by static spherical symmetric metric, is considered and studied. Exact spherical symmetric solutions of nonlinear spinor field equations in the Gravitational Theory are obtained. The nonlinearity in the spinor lagrangian is given by an arbitrary function which depends on the invariant generated from the Fierz-Pauli bilinear spinor form IS = S2. It is shown that a soliton-like configuration has a localized energy density and a finite total energy. In addition, The total charge and total spin are also finite. Let us emphasize that the effect of gravitational field on the properties of regular localized solutions significantly depends on the symmetry of the system. The nonlin- ear terms, the gravitational field of elementary particles and the geometrical properties of the metric of the space-time play an important role in the obtaining of analytical solutions having the soliton-like configuration. Let us emphasize that the numerical solutions of the solutions obtained here are presented in graphical form.

Construction of $L^2$ log-log blowup solutions for the mass critical nonlinear Schrödinger equation

In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schrödinger equation on \mathbb{R}^2 under rough but structured random perturbations at L^{2}(\mathbb{R}^2) regularity. In particular, by employing probabilistic methods, we provide a construction of a family of L^{2}(\mathbb{R}^2) regularity solutions which do not lie in H^{s}(\mathbb{R}^2) for any s>0 , and which blowup according to the log-log dynamics.

Solving Nonlinear Conservation Laws of Partial Differential Equations Using Graph Neural Networks

Nonlinear Conservation Laws of Partial Differential Equations (PDEs) are widely used in different domains. Solving these types of equations is a significant and challenging task. Graph Neural Networks (GNNs) have recently been established as fast and accurate alternatives for principled solvers when applied to standard equations with regular solutions. There have been few investigations on GNNs implemented for complex PDEs with nonlinear conservation laws. Herein, we explore GNNs to solve the following problem ut + f(u, β)x = 0 where f(u, β) is the nonlinear flux function of the scalar conservation law, u is the main variable, and β is the physical parameter. The main challenge of nonlinear conservation laws is that solutions typically create shocks. That is, one or several jumps in the form (uL, uR) with uL ≠ uR moving in space and probably changing over time such that information about f(u) in the interval associated with this jump is not present in the observation data. We demonstrate that GNNs could achieve accurate estimates of PDEs solutions based on new initial conditions and physical parameters within a specific parameter range.

The geometry of adversarial training in binary classification

AbstractWe establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type $L^1+\text{(nonlocal)}\operatorname{TV}$, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original problem, including the existence of minimal and maximal solutions (interpreted in a suitable sense) and the existence of regular solutions (also interpreted in a suitable sense). In addition, we highlight how the connection between adversarial training and perimeter minimization problems provides a novel, directly interpretable, statistical motivation for a family of regularized risk minimization problems involving perimeter/total variation. The majority of our theoretical results are independent of the distance used to define adversarial attacks.

A new class of anisotropic rotating fluids and some throat-like sources for Kerr metric as examples

Motivated by the increasing interest in finding physically viable rotating sources, we present a new class of anisotropic rotating solutions. The energy–momentum tensor compatible with the metric is composed of anisotropic matter with a nonvanishing energy flow around the symmetry axis and vanishing viscosity. The new class of solutions can be used to find new possible sources for the Kerr metric, to obtain new regular black hole solutions and to study galaxies with a central rotating black hole and an halo of dark matter. As an example, we obtain a five-parameter class of solutions representing a two-way traversable wormhole smoothly matched to the Kerr one and satisfying all energy conditions outside the wormhole for a wide range of parameters, in particular for compact objects. Finally, with a simple modification of the aforementioned solution, we obtain a source for Kerr metric with a throat geometry, nonrepresenting a two-way traversable wormhole and satisfying all energy conditions.

Generic Behavior of Electromagnetic Fields of Regular Rotating Electrically Charged Compact Objects in Nonlinear Electrodynamics Minimally Coupled to Gravity

Regular rotating electrically charged compact objects are described by nonlinear electrodynamics minimally coupled to gravity in a self-consistent way and without additional assumptions on the relation between the electromagnetic field and gravity. The electromagnetic fields obey the system of four source-free nonlinear equations for the electromagnetic tensor Fμν, with only two independent components due to spacetime symmetry determined by the algebraic structure of electromagnetic stress–energy tensors (pr=−ρ). In this paper, we present, for an arbitrary gauge-invariant Lagrangian, the general regular solution and generic behavior of electromagnetic fields, including the generic features of the Lagrange dynamics, for regular rotating electrically charged black holes and electromagnetic spinning solitons.

Synergy Effect between Sodium Oleate and Alcohol Ethoxylates on the Reverse Flotation of Quartz

In this study, an optimized method was presented for the reverse quartz flotation in iron ore purification where sodium oleate (NaOL) was always selected as the collector due to its accessibility and economic benefits. Three alcohol ethoxylates, octylphenol ethoxylate (OP-10), nonylphenol ethoxylate (NP-10) and fatty alcohol ethoxylates (AEO-9) were introduced to improve the collecting performance of NaOL in the reverse flotation of quartz. It turned out that the addition of alcohol ethoxylates was helpful to increase the recovery of quartz in the flotation with the order of OP-10 > NP-10 > AEO-9. To characterize the adsorption of NaOL on activated quartz in different surfactant systems, Fourier-Transform Infrared Spectroscopy (FTIR) and Quartz Crystal Microbalance with Dissipation (QCM-D) were employed. Furthermore, the contact angles of the quartz surface that interacted with different surfactant systems were measured to investigate the synergy effect of NaOL and alcohol ethoxylates on the hydrophobicity of quartz. It turned out that alcohol ethoxylates promoted the adsorption of NaOL on the activated quartz, resulting in an increase in the hydrophobicity of quartz also in the order of OP-10 > NP-10 > AEO-9, which well explained the flotation results. Regular Solution Theory (RST) and the dilution effect based on the surface tension analysis were utilized to elucidate the synergy mechanism between NaOL and the alcohol ethoxylate.

Black String Bounce to Traversable Wormhole

In this work, a regular black string solution is presented from the method used by Simpson–Visser to regularize the Schwarzschild solution. As in the Simpson–Visser work, in this new black string solution, it is possible to represent both a regular black hole and a wormhole simply by changing the value of a parameter “a” used in its metric. Tensors and curvature invariants are analyzed to verify the regularity of the solution as well as the energy conditions of the system. It is found that the null energy condition is always violated for the entire space. An additional analysis of the thermodynamic properties of the regular black string is carried out, in which the modifications generated about the original solution of the black string are evaluated, specifically, the Hawking temperature, entropy, its thermal capacity, and the Helmholtz free energy. Finally, we investigate the possible stable or unstable circular orbits for photons and massive particles. The results are compared with those for the non-regular black string, seeking to make a parallel with the Simpson–Visser work.