Small Parameter Research Articles

Gravitational collapse at low to moderate Mach numbers: The relationship between star formation efficiency and the fraction of mass in the massive object

The formation of massive objects via gravitational collapse is relevant both for explaining the origin of the first supermassive black holes and in the context of massive star formation. Here, we analyze simulations of the formation of massive objects pursued by different groups and in various environments, concerning the formation of supermassive black holes, primordial stars, as well as present-day massive stars. We focus here particularly on the regime of small virial parameters, that is, low ratios of the initial kinetic to gravitational energy, low to moderate Mach numbers, and the phase before feedback is very efficient. We compare the outcomes of collapse under different conditions using dimensionless parameters, particularly the star formation efficiency є*, the fraction ƒ* of mass in the most massive object relative to the total stellar mass, and the fraction ƒtot of mass of the most massive object as a function of the total mass. We find that in all simulations analyzed here, ƒtot increases as a function of є*, although the steepness of the increase depends on the environment. The relation between ƒ* and є* is found to be more complex and also strongly depends on the number of protostars present at the beginning of the simulations. We show that a collision parameter, estimated as the ratio of the system size divided by the typical collision length, allows us to approximately characterize whether collisions are important. A high collision parameter implies a steeper increase in the relation between ƒtot and є*. We analyze the statistical correlation between the dimensionless quantities using the Spearman coefficient and further confirm via a machine learning analysis that good predictions of ƒ* can be obtained from є* together with a rough estimate of the collision parameter. This suggests that a good estimate of the mass of the most massive object can be obtained once the maximum efficiency for a given environment is known and an estimate for the collision parameter has been determined.

The MUSE Ultra Deep Field (MUDF). VI. The Relationship between Galaxy Properties and Metals in the Circumgalactic Medium

We present initial results associating galaxies in the Multi-Unit Spectroscopic Explorer (MUSE) Ultra Deep Field (MUDF) with gas seen in absorption along the line of sight to two bright quasars in this field to explore the dependence of metals in the circumgalactic medium (CGM) on galaxy properties. The MUDF includes ∼140 hr of Very Large Telescope (VLT)/MUSE data and 90 orbits of Hubble Space Telescope (HST)/G141M grism observations alongside VLT/Ultraviolet and Visual Echelle Spectrograph spectroscopy of the two quasars and several bands of HST imaging. We compare the metal absorption around galaxies in this field as a function of impact parameter, azimuthal angle, and galaxy metallicity across redshifts 0.5 < z < 3.2. Due to the depth of our data and a large field of view, our analysis extends to low stellar masses (<107 M ⊙) and high impact parameters (>600 kpc). We find a correlation between the absorber equivalent width and the number of nearby galaxies, but do not detect a significant anticorrelation with the impact parameter. Our full sample does not show any significant change in absorber incidence as a function of azimuthal angle. However, we do find a bimodality in the azimuthal angle distribution of absorption at small impact parameters (<2 r vir) and around highly star-forming galaxies, possibly indicating disk-like accretion and biconical outflows. Finally, we do not detect any systematic deviation from the fundamental metallicity relation among galaxies with detected absorption. This work is limited by gaps in the wavelength coverage of our current data; broader-wavelength observations with the James Webb Space Telescope will allow us to unlock the full potential of the MUDF for studying the CGM.

On the Origin of the Ancient, Large-scale Cold Front in the Perseus Cluster of Galaxies

The intracluster medium of the Perseus Cluster exhibits spiral-shaped X-ray surface brightness discontinuities known as “cold fronts,” which simulations indicate are caused by the sloshing motion of the gas after the passage of a subcluster. Recent observations of Perseus have shown that these fronts extend to large radii. In this work, we present simulations of the formation of sloshing cold fronts in Perseus using the AREPO magnetohydrodynamics code, to produce a plausible scenario for the formation of the large front at a radius of 700 kpc. Our simulations explore a range of subcluster masses and impact parameters. We find that low-mass subclusters cannot generate a cold front that can propagate to such a large radius, and that small impact parameters create too much turbulence, which leads to the disruption of the cold front before it reaches such a large distance. Subclusters that make only one core passage produce a stable initial front that expands to large radii, but without a second core passage of the subcluster, other fronts are not created at a later time in the core region. We find a small range of simulations with subclusters with mass ratios of R ∼ 1:5 and an initial impact parameter of θ ∼ 20°–25° that not only produce the large cold front but a second set in the core region at later times. These simulations indicate that the “ancient” cold front is ∼6–8.5 Gyr old. For the simulations providing the closest match with observations, the subcluster has completely merged into the main cluster.

Value-Positivity for Matrix Games

Matrix games are the most basic model in game theory, and yet robustness with respect to small perturbations of the matrix entries is not fully understood. In this paper, we introduce value positivity and uniform value positivity, two properties that refine the notion of optimality in the context of polynomially perturbed matrix games. The first concept captures how the value depends on the perturbation parameter, and the second consists of the existence of a fixed strategy that guarantees the value of the unperturbed matrix game for every sufficiently small positive parameter. We provide polynomial-time algorithms to check whether a polynomially perturbed matrix game satisfies these properties. We further provide the functional form for a parameterized optimal strategy and the value function. Finally, we translate our results to linear programming and stochastic games, where value positivity is related to the existence of robust solutions. Funding: This research was supported by Fondation CFM pour la Recherche, the H2020 European Research Council [Grant ERC-CoG-863818 (ForM-SMArt)], the Austrian Science Fund [Grant 10.55776/COE12], ANID Chile [Grant ACT210005], and Agence Nationale de la Recherche [Grant ANR-21-CE40-0020].

Reduced-dimensional modelling for nonlinear convection-dominated flow in cylindric domains

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection–diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher Péclet numbers are also considered.

An extended two-parameter mixed-dimensional model of fractured porous media incorporating entrance flow and boundary-layer transition effects

We develop an enhanced reduced model for single-phase flow in fractured porous media capable of incorporating more realistic interface conditions at the fracture terminations. In addition to the traditional dimensional model reduction, where the elements of the discrete fracture network are treated as lower dimensional manifolds embedded in the porous matrix, we explore the microscale behavior of the boundary layer flow at the entrances of a fracture bounded by two parallel plates to construct a new set of interface conditions of Robin-type, giving rise to localized pressure jumps at the fracture edges. Within this enriched description, sharper reduced flow and tracer transport mixed-dimensional models are constructed in the asymptotic limit ruled by two small parameters related to the ratio between fracture aperture and entrance developing length and a macroscopic length scale. The discrete flow/transport mixed-dimensional model is discretized by a new discontinuous Galerkin(dG)-based formulation. An adequate version of the Galerkin-Newton method is developed for the numerical treatment of the non-linear Robin interface condition. Considering several fracture arrangements, numerical results illustrate the sharper description of the model proposed herein in predicting flow and tracer transport patterns in fractured media.

Semiclassical perturbations of single-degree-of-freedom Hamiltonian systems II: Nonintegrability

Continuing from Paper I [Ohsawa and Yagasaki, J. Math. Phys. 65, 102706 (2024)], we study semiclassical perturbations of single-degree-of-freedom analytic Hamiltonian systems and provide a sufficient condition for its meromorphic nonintegrability such that the first integrals depend on the small parameter meromorphically. Our approach is based on a generalization due to Ayoul and Zung of the Morales-Ramis theory, which enables us to show the meromorphic nonintegrability of dynamical systems by using the differential Galois theory. We remark that standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics are analytically integrable as well as the corresponding classical systems. We illustrate our theory for a bounded potential.

Synchronization analysis of a four-motor excitation with torsion spring coupling vibrating system

In oil drilling processes, vibrating screen is a crucial solid control equipment for separating harmful solid-phase particles. However, the conventional utilization of either dual-motor or triple-motor is inadequate demanding the excitation force of large vibrating equipment. Moreover, self-synchronization in zero phase is difficult to achieve in multiple motors system. To avoid cancellation of the excitation force caused by the phase inconsistency of multiple motors, the adoption of a four-motor excitation with torsion spring coupling system is proposed this work. Initially, the mathematical model of the vibrating system is formulated through the Lagrange's equation. Subsequently, the steady-state solution is determined using Laplace transform. Then the conditions and characteristics in stable equilibrium state are elucidated employing the small parameter averaging method. Ultimately, the reliability and applicability of the theoretical results are substantiated through numerical simulations. The findings reveal that with the increase of the torsion spring stiffness, the phase difference between the motors connected by torsion springs (MCTSs) is exponentially decreased, and desired zero-phase synchronization in engineering is eventually achieved. Furthermore, the present study uncovers two distinct synchronization mechanisms in the torsion spring coupling system: mechanical controlled synchronization between MCTSs is inevitably achieved, while the self-synchronization between motors unconnected by torsion springs (MUTSs) is limited by the synchronization conditions. The study provides valuable insights for designing mechanical controlled synchronization and self-synchronization vibrating machines.

PERTURBED MOTIONS OF A NEARLY DYNAMICALLY SPHERICAL RIGID BODY WITH A MOVABLE MASS SUBJECT TO CONSTANT BODY-FIXED TORQUE

The problem of motion of a rigid body about a fixed point is one of the classical problems of mechanics. The interest to the problems of the rigid body dynamics has increased in the second half of the XX century in connection with the development of rocket and space technologies. A spacecraft or satellite, while orbiting about its center of mass, experiences torques from forces of diverse physical nature. This includes torques generated by the motion of internal masses, which can arise from factors such as presence of rotating components (like rotors or gyroscopes), and the activities of crew members aboard the crew vehicle. The dynamics of rigid body incorporated moving masses is a significant focal point in classical mechanics. Extensive research is dedicated to investigating the rotation of a rigid body featuring motion of internal masses. It is assumed that the body contains a viscoelastic element that is modeled by a moving mass connected to the body by a strong damper. The moving mass model loosely attached elements in a space vehicles, which can significantly affect the vehicle’s motion about its center of mass during a long period of time. Some cases are considered of the motion of a rigid body containing internal masses connected to the body by means of elastic and dissipative elements. A number of works are devoted to the analysis of various problems of the dynamics of space vehicles containing internal movable masses. The paper develops an approximate solution by means of averaging method to the system of Euler’s equation terms for a nearly dynamically spherical rigid body containing a viscoelastic element under the action of constant body-fixed torque. We obtained the system of motion equations in the standard form which refined in square-approximation by small parameter. Asymptotic approach permits to obtain some qualitative results and to describe evolution of angular motion using simplified averaged equations and numerical solution. The main objective of this paper is to extend the previous results for the problem of motion about a center of mass of a rigid body under the influence of small internal torque (cavity filled with a fluid of high viscosity) or external torque (resistive medium). The importance of the results is in progress of moving mass control motion of spinning projectiles.

Theoretical and Numerical Analysis of Singular Perturbation Problems in Ordinary Differential Equations

Background: Singular perturbation troubles in everyday differential equations (ODEs) involve a small parameter ϵ that causes answers to show off behaviors on multiple scales, main to massive challenges in both theoretical and numerical evaluation. Objective: This paper at targets to comprehensively look at the theoretical and numerical techniques for reading and solving singular perturbation troubles in ODEs. Focus is located on know-how the behaviors precipitated through the small parameter and developing strong numerical techniques to accurately seize these behaviors. Methods: Theoretical approaches employed include matched asymptotic expansions and multiple scale analysis. These methods decompose the solution domain into regions with different scales, constructing and matching approximate solutions to ensure smooth transitions. Numerical techniques such as finite difference methods, finite element methods, and spectral methods are utilized, with particular emphasis on adaptive mesh refinement to handle boundary layers effectively Results: Theoretical evaluation demonstrates the effectiveness of matched asymptotic expansions and multiple scale analysis in presenting correct approximations and easy transitions among one of a kind solution regions. Numerical techniques showed high accuracy and performance, particularly whilst blended with adaptive techniques. Finite distinction strategies with non-uniform grids and adaptive mesh refinement, finite detail techniques with adaptive mesh strategies, and spectral strategies with special handling of boundary layers all proved successful. Stability and convergence analyses showed the reliability of those techniques. Conclusions: This comprehensive evaluation highlights the strengths and applicability of each theoretical and numerical strategies in tackling singular perturbation issues in ODEs. The mixed use of these techniques lets in for correct and green answers, presenting treasured equipment for applications in diverse clinical and engineering fields.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

Analyzing the morphology and avian β-defensins genes (AvβD) expression in the small intestine of Cobb500 broiler chicks fed with sodium butyrate

BackgroundSodium butyrate is a potential antibiotic growth promoter and has had advantageous effects on the poultry industry.MethodsEvaluating the effect of sodium butyrate on the intestinal villi and the humoral part of innate immunity of the male Cobb 500 broiler using scanning electron microscopy and quantitative real-time PCR analysis, the control group and treated group of Cobb 500 with SB supplemented received water containing 0.98 mg sodium butyrate.ResultsThe administration of sodium butyrate changed the villi characters, as the shape changed from tongue to long tongue. They were mainly parallel to each other and long finger-like at the duodenum. The tips of the villi in the control group appeared thin-slight curved with a prominent center in the duodenum, thin rectangular in the jejunum, and ileum in the control group. In contrast, in the treatment group, they changed to thick rectangular in the duodenum and ileum zigzag shape in the jejunum. The epithelium lining of the duodenal villi showed a dome shape, the jejunal villi showed a polygonal shape, and the ileal villi appeared scales-like. The epithelium lining showed irregular microfolds and many different-sized pores, and the treatment group showed islands of long microvilli in the duodenum and solitary long microvilli in the ileum. Real-time PCR of AvBD 1, 2, 10, and 12 significantly (P < 0.01). The better expression of AvBD 1, 2, and 12 was determined in the duodenum, while AvBD 10 was in the jejunum.ConclusionSodium butyrate enhanced the chicks’ growth and small intestine parameters, modified the morphology of the intestinal villi, and improved the humoral part of innate immunity.

Adaptive mesh extended cubic B-spline method for singularly perturbed delay Sobolev problems

The purpose of this paper is to develop a robust numerical scheme for a class of singularly perturbed delay Sobolev (pseudo-parabolic) problems that have wide application in various branches of mathematical physics and fluid mechanics.For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer(s), im-plicit Euler scheme for the time derivatives on uniform mesh and extended B-spline basis functions consisting of free parameter λ are presented for spatial variable on Bakhvalov type mesh. The stability and uniform convergence analyisis of the proposed method are established. The error estimation of the developed method is shown to be firts order accurate in time and second order accurate in space. Numerical exprementation is carried out to validate the applicability of the developednumerical method. The numerical results reveals that the computational result is in agreement with the theoretical estimations

FPSO ROLL MOTIONS DRIVEN BY SECOND-ORDER FORCES AND THE IMPACTS ON SLWR RISER DESIGN

Abstract The dynamic behavior of risers is highly influenced by the FPU motions, associated with environmental loads, caused by waves, current and wind. Particularly, as well known, under potential flow modelling, wave loads may be represented through power series expansions, with respect to a small parameter ε. First-order wave forces pulsate in the frequency range of the incident wave spectra, whereas second-order wave forces are related to sum or difference frequencies of the same spectra. The so-called second-order difference frequencies forces may then be responsible by resonating motions of the vessel in that range. Significant low-frequency motions due to second order forces are commonly observed in the horizontal plane, resonating low natural frequencies of moored floating systems characterized by low mooring stiffnesses. Recently, however, this phenomenon was also observed for roll motions in several new FPSOs that will operate in the Pre-Salt of Santos Basin. The phenomenon had already been detected in a deep draft semi-submersible in Campos Basin, but only more recently in FPSOs. The new floating units are characterized by high roll natural periods, above the typical wave period range in the pre-salt region, and the roll response to the second-order difference frequencies forces can be significant. This phenomenon was indeed detected in model tests and confirmed by numerical studies. In fact, this additional FPSO roll behavior occurs in typical periods slightly above 20 s, imposing displacements and accelerations that can directly affect the dynamics of the riser top region. The present paper addresses this phenomenon and its possible impact on steel lazy-wave riser (SLWR) designs.

Frustrated charge transfer in vibrationally inelastic Ar++N2 collisions via hard collision glory scattering

Vibrational energy transfer in collisions between ions and neutrals is a fundamental process in interstellar media, planetary atmospheres, and plasmas. The conventional wisdom is that glancing collisions with large impact parameters are forward-scattered with low vibrational excitation, while hard collisions with small impact parameters are sideway- or backward-scattered with relatively high vibrational excitation. Here, we report experimental observations with a three-dimensional velocity-map imaging crossed-beam apparatus in the inelastic scattering process Ar++N2(v′′ = 0, J′′)→Ar++N2(v′, J′), where all the vibrationally excited N2 products are dominated by forward scattering, contradicting the textbook model. Trajectory surface hopping calculations not only reproduced the experimental observation qualitatively, but also revealed that the vibrational excitation mainly occurs through a transient charge-transfer process. The hard collision glory mechanism, which has so far only been observed in inelastic rotational energy transfer between neutrals, is shown to play a major role for vibrational excitation in the inelastic Ar++N2 collision, via the frustrated charge transfer process.

Arithmetization-oriented APN permutations

AbstractRecently, many cryptographic primitives such as homomorphic encryption (HE), multi-party computation (MPC) and zero-knowledge (ZK) protocols have been proposed in the literature which operate on the prime field $${\mathbb {F}}_p$$ F p for some large prime p. Primitives that are designed using such operations are called arithmetization-oriented primitives. As the concept of arithmetization-oriented primitives is new, a rigorous cryptanalysis of such primitives is yet to be done. In this paper, we investigate arithmetization-oriented APN functions. More precisely, we investigate APN permutations in the CCZ-classes of known families of APN power functions over the prime field $${\mathbb {F}}_p$$ F p . Moreover, we present a class of binomial permutation having differential uniformity at most 5 defined via the quadratic character over finite fields of odd characteristic. Computationally it is confirmed that the latter family contains new APN permutations for some small parameters. We conjecture it to contain an infinite subfamily of APN permutations.

Ultraviolet finite resummation of perturbative quantum gravity

Abstract If the metric is chosen to depend exponentially on the conformal factor, and if one works in a gauge where the conformal factor has the wrong sign propagator, perturbative quantum gravity corrections can be partially resummed into a series of terms each of which is ultraviolet finite. These new terms however are not perturbative in some small parameter, and are not individually BRST invariant, or background diffeomorphism invariant. With appropriate parametrisation, the finiteness property holds true also for a full phenomenologically relevant theory of quantum gravity coupled to (beyond the standard model) matter fields, provided massive tadpole corrections are set to zero by a trivial renormalisation.

Carroll geodesics

Using effective field theory methods, we derive the Carrollian analog of the geodesic action. We find that it contains both “electric” and “magnetic” contributions that are in general coupled to each other. The equations of motion descending from this action are the Carrollian pendant of geodesics, allowing surprisingly rich dynamics. As an example, we derive Carrollian geodesics on a Carroll–Schwarzschild background and discover an effective potential similar to the one appearing in geodesics on Schwarzschild backgrounds. However, the Newton term in the potential turns out to depend on the Carroll particle’s energy. As a consequence, there is only one circular orbit localized at the Carroll extremal surface, and this orbit is unstable. For large impact parameters, the deflection angle is half the value of the general relativistic light-bending result. For impact parameters slightly bigger than the Schwarzschild radius, orbits wind around the Carroll extremal surface. For small impact parameters, geodesics get reflected by the Carroll black hole, which acts as a perfect mirror.

SS-DNN: A hybrid strang splitting deep neural network approach for solving the Allen–Cahn equation

The Allen–Cahn equation is a fundamental partial differential equation that describes phase separation and interface motion in materials science, physics, and various other scientific domains. The presence of interfacial width (ϵ) between two stable phases, associated with a nonlinear term, is a small positive parameter which makes the problem more challenging to solve as ϵ approaches zero. This paper proposes a novel hybrid deep splitting method to efficiently and accurately solve the Allen–Cahn equation in a convex polygonal domain in Rd(d=1,2,3). The method combines the benefits of deep learning and splitting strategies, leveraging the strengths of both approaches. Essentially, a second-order splitting method is employed to split the Allen–Cahn equation into two simpler linear and non-linear sub-problems. While the nonlinear sub-problem can be solved analytically, the deep neural network is utilized to approximate the linear sub-problem. By integrating deep learning into the splitting strategy, we achieve a more efficient and accurate solution for the Allen–Cahn equation, demonstrating promising results. We also derive an error estimate for the proposed hybrid method. Modified space adaptivity and transform learning techniques are employed to enhance the efficiency of the neural network.

Biexciton in Prolate Ellipsoidal Quantum Dot: Optical-Magnetic Properties

In this study, we present a theoretical framework for assessing the ground state energy of a biexciton within an ellipsoidal quantum dot subjected to an external magnetic field. Our method involves employing the variational method and the geometrical adiabatic approximation. We calculate the biexciton energy and binding energy, considering their dependence on the magnetic field strength and the small geometrical parameter of the ellipsoidal quantum dot. Our findings reveal an increase in magnetobiexciton energy with the strength of the magnetic field. Additionally, we analyze the biexciton’s magnetization concerning the external magnetic field and the small semiaxis of the ellipsoidal quantum dot. Moreover, we provide an estimation of the magnetobiexciton recombination radiative lifetime in the ellipsoidal quantum dot, which falls within the nanosecond range.