Stationary Solutions Research Articles

Electrochemical polishing with glycol-based electrolyte of the curved internal channels fabricated via laser powder bed fusion

Electrochemical polishing (ECP) is used to process metal additive manufacturing parts with poor surface qualities arising from powder adhesion, spheroidal effect, and step effect during forming. Conventional ECP uses stationary acid-based solutions and very low current densities, which makes it challenging to efficiently polish complex-shaped internal channels. Typically, an oxide layer forms in the conventional aqueous-based electrolyte, which prevents uniform dissolution and consequently deteriorates surface quality. In this study, we proposed a modified ECP method for polishing curved internal channels in laser powder bed fusion (LPBF) 316 L stainless steel (SS) specimens using a high-speed flow of a NaCl–ethylene glycol (EG) solution at low current densities. During electrochemical tests, LPBF 316LSS specimens did not exhibit passivation in the NaCl–EG solution, and the current efficiency was high at low current densities (i < 4 A·cm−2). The reaction mechanism of ECP LPBF 316LSS in the NaCl–EG solution was clarified by observing the experimental phenomena and testing the composition of the reaction products. The ECP experiments showed that the decrease of roughness value is directly related to the charge. A small roughness value can also be obtained at low current densities (i < 2 A·cm−2). Finally, S-shaped curved holes with a diameter of Φ3 mm were efficiently polished using a flexible cathode in the NaCl–EG solution. In the case of an aperture increase rate of only about 7 %, it would only take 400 s to drastically reduce the surface roughness value Sa from 12.73 to 12.94 μm to 2.16– 2.43 μm.

Thermocapillary thin films: periodic steady states and film rupture

We study stationary, periodic solutions to the thermocapillary thin-film model 0,\\ x\\in \\mathbb{R},$?> ∂th+∂xh3∂x3h−g∂xh+Mh21+h2∂xh=0,t>0, x∈R, which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value M∗ , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

Stationary dimpled drops under linear flow

The axially symmetric deformation of a drop in a viscous fluid, under the influence of an externally imposed flow having simultaneous rotating and compressional or extensional components, is addressed. In the previous studies, two families of stationary drop shapes were constructed by simulating the dynamics of drop deformation: stable singly connected shapes with respect to axisymmetric disturbances, and unstable toroidal shapes. These two branches coexist at the same flow conditions, but were not connected. In this study, we obtain a new family of branches of unstable highly deformed stationary drops connecting with the stable flattened shapes and the toroidal ones. We use a method based on classical control theory. The controller is designed for a two-state dynamic model of the system and is employed on a high-order nonlinear dynamic model of the drop deformation. Through this feedback-control-centred approach, an extended collection of unstable stationary solutions is constructed, which spans the range from the loss of stability to the dimpled shapes almost collapsed at the centre. In the latter region, which was never obtained in previous studies, a multiplicity of solutions is identified.

ЭФФЕКТ САНЬЯКА В ПРОСТРАНСТВЕ-ВРЕМЕНИ ВРАЩАЮЩЕЙСЯ ЧЕРНОЙ ДЫРЫ В ЧЕТЫРЕХМЕРНОЙ ТЕОРИИ ЭЙНШТЕЙНА–ГАУССА–БОННЕ

Recently, Glavan and Lin proposed a new four-dimensional theory of Einstein-Gauss-Bonnet (EGB) gravity by rescaling the Gauss-Bonnet (GB) coupling constant ߙ→ ܦ(/ߙ− 4) and adopting the ܦ→ 4 constraint at the level of the equations of motion. The GB coupling constant contributes to the field equations and thus bypasses Lovelock's theorem. This theory preserves the number of degrees of freedom and avoids Ostrogradsky instability. Glavan and Lin obtained an exact solution for nonsingular static and spherically symmetric black holes in the four-dimensional EGB theory of gravity. Later, Kumar and Ghosh applied the Newman-Janis algorithm to the static solution and constructed a solution for a rotating black hole in the 4-dimensional EGB theory of gravity. Since the theory solves many observational problems in astrophysics and is promising for research, the influence of the GB coupling parameter on astrophysical effects is actively studied in the literature. However, this theory has not been tested previously using time effects. The work examines one of the most interesting time effects – the Sagnac effect. Thus, the goal of the work is to study the Sagnac effect in the space-time of a rotating black hole in the four-dimensional EGB theory and obtain a constraint on the GB coupling parameter using observational data of the Sagnac effect. In the most general case, the Sagnac effect is understood as the difference in the time it takes light beams moving in opposite directions to pass through a closed circle. The effect has many applications, is observable on Earth and is taken into account in satellite navigation systems (GPS, GLONASS) when synchronizing time signals.

Effect of unfavorable regions on the spreading solution in a diffusion equation

We consider a diffusion equation on the real line with growth rates (reaction terms) being negative in a bounded unfavorable region and bistable on two sides. The equation can be used to model a species living in a habitat with a polluted or hunting zone but still tries to survive or even spread to the whole space. Using the zero number argument, we first show the general convergence to stationary solution for any nonnegative global solutions, and then we prove a spreading–transition–vanishing trichotomy result for the asymptotic behavior of global solutions. The key point is to find a quite special stationary solution to distinguish the spreading and transition solutions. Finally, we construct precise upper and lower solutions to estimate the asymptotic speed for the spreading solution.

An optimal Bayesian intervention policy in response to unknown dynamic cell stimuli

Interventions in gene regulatory networks (GRNs) aim to restore normal functions of cells experiencing abnormal behavior, such as uncontrolled cell proliferation. The dynamic, uncertain, and complex nature of cellular processes poses significant challenges in determining the best interventions. Most existing intervention methods assume that cells are unresponsive to therapies, resulting in stationary and deterministic intervention solutions. However, cells in unhealthy conditions can dynamically respond to therapies through internal stimuli, leading to the recurrence of undesirable conditions. This paper proposes a Bayesian intervention policy that adaptively responds to cell dynamic responses according to the latest available information. The GRNs are modeled using a Boolean network with perturbation (BNp), and the fight between the cell and intervention is modeled as a two-player zero-sum game. Assuming an incomplete knowledge of cell stimuli, a recursive approach is developed to keep track of the posterior distribution of cell responses. The proposed Bayesian intervention policy takes action according to the posterior distribution and a set of Nash equilibrium policies associated with all possible cell responses. Analytical results demonstrate the superiority of the proposed intervention policy against several existing intervention techniques. Meanwhile, the performance of the proposed policy is investigated through comprehensive numerical experiments using the p53-MDM2 negative feedback loop regulatory network and melanoma network. The results demonstrate the empirical convergence of the proposed policy to the optimal Nash equilibrium policy.

Classical-quantum correspondence for particles in the Penning trap

We derive new solutions of the Schrödinger equation which describe the motion of particles in the Penning trap. These solutions are direct counterparts of classical orbits. They are obtained by injection of classical trajectories into the wave functions of stationary solutions.

Static axion stars revisited

We consider static solutions to the spherically symmetric Einstein-scalar field systems with an axion potential known as axion stars, originally described by Guerra et al., JCAP (2019, 09 (09)). We construct numerically families of axion stars in the ground state, for different values of the decay constant fa. It is shown that the existence diagram becomes richer than the mini-boson star case, and several regions of stability appear as the value of fa decreases, yielding to more massive configurations with larger compactness. Some intrinsic properties, such as isotropy and compactness of such stars, are also discussed. Finally, we describe the motion of test particles around these objects.

Exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise

We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter H > 1 2 , which arise e.g. from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on E. Buckwar et al. [The numerical stability of stochastic ordinary differential equations with additive noise, Stoch. Dyn. 11 (2011), pp. 265–281], we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.

Generalized Boussinesq System with Energy Dissipation: Existence of Stationary Solutions

In this paper, we investigate the solvability of a boundary value problem for a heat and mass transfer model with the spatially averaged Rayleigh function. The considered model describes the 3D steady-state non-isothermal flow of a generalized Newtonian fluid (with shear-dependent viscosity) in a bounded domain with Lipschitz boundary. The main novelty of our work is that we do not neglect the viscous dissipation effect in contrast to the classical Boussinesq approximation, and hence, deal with a system of strongly nonlinear partial differential equations. Using the properties of the averaging operation and d-monotone operators as well as the Leray–Schauder alternative for completely continuous mappings, we prove the existence of weak solutions without any smallness assumptions for model data. Moreover, it is shown that the set of all weak solutions is compact, and each solution from this set satisfies some energy equalities.

Critical degeneracy in a nonlinear hyperbolic equation can produce atypical instability

This paper deals with the initial value problem for a semilinear wave equation on a bounded domain and solutions are required to vanish on the boundary of this domain. The essential feature of the situation considered here is that the ellipticity of the spatial part of the differential operator degenerates like the square of the distance from given point in the domain and so hyperbolicity is lost at this point. The assumptions ensure that [Formula: see text] is a stationary solution of the problem and the object is to study the stability of this solution with respect to perturbations of the initial data. Stability, instability and asymptotic stability are all considered. The assumptions about the nonlinear terms ensure that the problem has a well-defined linearization at [Formula: see text]. There are simple cases where this linearization is asymptotically stable but [Formula: see text] is an unstable solution of the nonlinear problem. We also establish conditions under which the stability of a stationary solution [Formula: see text] can be determined using our results. The quadratic degeneracy at a point treated here is typical of what is required in models for acoustic (or sonic) black holes. It also occurs in a simplified Wheeler–DeWitt model which we discuss in some detail.

The Riemann problem for the Chaplygin gas dynamics with a single-point heating source

This paper is devoted to the Riemann solutions to the Chaplygin gas dynamics with a single-point heating source. To deal with Dirac measure source term, we transform this model into a system of conservation laws and then present a definition of solution to this system of conservation laws. Based on this definition, we establish, respectively, the condition for the occurrence of stationary contact discontinuity solution and delta standing wave solution. With the help of these results, by analyzing the possible combination of various waves, ten kinds of exact solutions and the criteria for the emergence of each solution are established. It is observed that a delta standing wave solution arises for certain initial values, where two components of this solution contain the Dirac measure. It is also noticed that a stationary contact discontinuity followed by a delta shock wave emerges in some solutions. Moreover, these solutions can be used to test the validation of numerical algorithm for a system of conservation laws with a singular source term.

Phase-space evolution of quasiparticle excitations in electron gas

In this research, we use the dual lengthscale quasiparticle model for collective quantum excitations in electron gas to study the time evolution of the Wigner function. The linearized time-dependent Schrödinger–Poisson system for quasiparticles is used to study the dynamics of initial known stationary and damped solutions in an electron gas with arbitrary degree of degeneracy. The self-consistent potential in the Schrödinger–Poisson model is treated in a quite different manner in this analysis due to the effective coupling of the electrostatic field to the electron density, which leads to a modified Wigner function. It is shown that the modified Wigner function in the absence of external potential evolves similar to the system of free particles, a feature of collective quantum excitations which is quite analogous to freely evolving classical system of particles in the center of mass frame in the absence of external forces. The time evolution of the modified Wigner function reveals a grinding effect on large-amplitude density structures present at initial states, which is a characteristic feature of the Landau damping in plasmas. It is further shown that linear phase-space dynamics of spill-out electrons (damped quasiparticles) can be described similar to free quasiparticles with imaginary momentum. The later predicts the surface electron tunneling via the collective excitations of spill-out electrons at the half-space boundary, which is closely related to the Heisenberg's uncertainty principle. Current research can have applications in plasmonics and related fields.

Harmonic wave scattered by an inclusion in an elastic plane: The complete Gurtin-Murdoch model

We consider the propagation of a harmonic elastic wave in a composite inclusion–matrix structure subjected to plane deformation. The interface between the inclusion and matrix is described by the complete Gurtin-Murdoch model with non-vanishing interface tension and interface stretching rigidity. We consider an inclusion of general shape and formulate the corresponding boundary value problem for the wave functions in the inclusion and matrix when a harmonic compressional or shear wave is incident on the edge of the matrix. The problem is then solved by series expansion methods for the case of a circular inclusion embedded in an infinite matrix. The series solution obtained is validated by checking its convergence and via comparisons with existing static and dynamic solutions for certain reduced cases. Numerical examples are presented for the case of a small-sized circular hole embedded in a soft matrix demonstrating the influence of surface tension on the incident wave-induced dynamic stress concentration in the matrix. We find that the presence of surface tension relieves the peak stress around the circular hole when the frequency of the incident wave is below a certain critical value, while it tends to intensify the peak stress for a high-frequency incident wave.

Theories derived from Haissinski equation and their applications to electron storage rings

As a stationary solution of the Vlasov–Fokker–Planck equation, the Haissinski equation predicts the equilibrium line density of a bunch that circulates in a storage ring for a given wake function. This paper shows that some equations regarding the centroid shift of the bunch, the peak position of the bunch profile, bunch length, and the impedance from the bunch profile can be derived from the Haissinski equation in a self-consistent manner. Specifically, a generalized quadratic equation for potential-well bunch lengthening is derived to accommodate any arbitrary impedance model (that is, the impedance spectrum under consideration can take any form). This expands upon Zotter’s cubic equation, which is primarily applicable to inductive impedance. The equations derived in this paper are tested using computed impedance models for some electron storage rings, showing machine-dependent properties of impedance effects. We conclude that these equations can be employed in electron storage rings to effectively bridge the gap between impedance computations and beam-based measurements.

Synchronization of differential equations driven by linear multiplicative fractional Brownian motion

This paper is devoted to the synchronization of stochastic differential equations driven by the linear multiplicative fractional Brownian motion with Hurst parameter H∈(12,1). We use equivalent transformations to prove that the differential equation has a unique stationary solution, which generates a random dynamical system. Moreover, the system has the pathwise singleton set random attractor. We then establish the synchronization of the coupled differential equations and provide numerical simulation results. At the end, we discuss two specific noise forms and present the corresponding synchronization results.

The cyclic load effect on the elasticity and plasticity deformation of high-strength reinforced concrete elements

This article investigates the degradation of stiffness in high and standard-strength concrete due to the influence of repeated loading and considers the non-linear behavior arising from localized or natural defects of concrete. The article also refers to the models from existing literature that evaluate the characteristics of concrete by enhancing its physicochemical properties through the application of natural and synthetic fibers. The impact of the service load, which influences the change in concrete stiffness, is illustrated using the elastic–plastic concrete model. The elastic–plastic model of concrete, combined with experimentally determined data such as elastic and residual deformations, simplifies the analysis of concrete beams or surface elements. This model enables static solutions of inelastic materials based on the classical theory of elasticity. The degradation of element stiffness in this model is described by the local increment of the rotation angle. Based on the model we can assume a constant stiffness along the length of the beam before as well as after exceeding the value of the cracking moment. In the case of cyclic loads, it can be assumed that residual deformations have the greatest influence on the displacement values, which in the calculation model are described by the component of the angle of rotation in the crack, describing permanent displacements.

Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3

Abstract The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution ( ρ s ( x ) , 0 , 0 ) T {\left({\rho }_{s}\left(x),0,0)}^{T} is in a small neighborhood of the constant state ( ρ ¯ , 0 , 0 ) T {\left(\bar{\rho },0,0)}^{T} at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear L p {L}^{p} - L q {L}^{q} decay estimates.

Is Birkhoff's theorem valid in Einstein-Aether theory?

We attempt to answer whether Birkhoff's theorem (BT) is valid in the Einstein-Aether (EA) theory. The BT states that any spherically symmetric solution of the vacuum field equations must be static, unique, and asymptotically flat. For a general spherically symmetric metric with metric functions A(r,t) &B(r,t), and aether components a(r,t) &b(r,t), we prove the conditions for the staticity of spacetime using two different methods. We point out that BT is valid in EA theory only for special values of c1+c3, c1+c4, and c2, where we can show that all these special cases are asymptotically flat. In particular, when the aether has only a temporal component i.e., b(r,t)=0, the c14≠0 case gives us spherically symmetric static solutions with singularities without Killing nor universal horizons, at least for special values of c14. However, when we have an aether vector with temporal and radial components, we prove that the staticity and the flatness at infinity hold for only a special metric and a particular combination of the aether parameters. These solutions have universal horizons.

Stationary Points of a Shallow Neural Network with Quadratic Activations and the Global Optimality of the Gradient Descent Algorithm

We consider the problem of training a shallow neural network with quadratic activation functions and the generalization power of such trained networks. Assuming that the samples are generated by a full rank matrix [Formula: see text] of the hidden network node weights, we obtain the following results. We establish that all full-rank approximately stationary solutions of the risk minimization problem are also approximate global optimums of the risk (in-sample and population). As a consequence, we establish that, when trained on polynomially many samples, the gradient descent algorithm converges to the global optimum of the risk minimization problem regardless of the width of the network when it is initialized at some value [Formula: see text], which we compute. Furthermore, the network produced by the gradient descent has a near zero generalization error. Next, we establish that initializing the gradient descent algorithm below [Formula: see text] is easily achieved when the weights of the ground truth matrix [Formula: see text] are randomly generated and the matrix is sufficiently overparameterized. Finally, we identify a simple necessary and sufficient geometric condition on the size of the training set under which any global minimizer of the empirical risk has necessarily zero generalization error. Funding: The research of E. C. Kizildag is supported by Columbia University, with the Distinguished Postdoctoral Fellowship in Statistics. Support from the National Science Foundation [Grant DMS-2015517] is gratefully acknowledged.