Parameter Epsilon Research Articles

A New Technique to Achieve Torsional Anchor of Fractional Torsion Equation Using Conservation Laws

The main idea in this research is introducing another approximate method to calculate solutions of the fractional Torsion equation, which is one of the applied equations in civil engineering. Since the fractional order is closed to an integer, we convert the fractional Torsion equation to a perturbed ordinary differential equation involving a small parameter epsilon. Then we can find the exact solutions and approximate symmetries for the alternative approximation equation. Also, with help of the definition of conserved vector and the concept of nonlinear self-adjointness, approximate conservation laws(ACL) are obtained without approximate Lagrangians by using their approximate symmetries. In order to apply the presented theory, we apply the Lie symmetry analysis (LSA) and concept of nonlinear self-adjoint Torsion equation, which are very important in mathematics and engineering sciences, especially civil engineering.

A Model of High Latitude Ionospheric Convection Derived From SuperDARN EOF Model Data

AbstractForecasting of the effects of thermospheric drag on satellites will be improved significantly with better modeling of space weather effects on the high‐latitude ionosphere, in particular the Joule heating arising from electric field variability. We use a regression analysis to build a model of the ionospheric convection drift velocity which is driven by relatively few solar and solar wind variables. The model is developed using a solar cycle's worth (1997–2008 inclusive) of 5‐min resolution Empirical Orthogonal Function (EOF) patterns derived from Super Dual Auroral Radar Network (SuperDARN) line‐of‐sight observations of the convection velocity across the high‐latitude northern hemisphere ionosphere. At key stages of development of the model, we use the percentage of explained variance P to see how well the model reproduces the EOF data. The final model is driven by four variables: (a) the interplanetary magnetic field component By, (b) the solar wind coupling parameter epsilon ε, (c) a trigonometric function of day‐of‐year, and (d) the monthly F10.7 index. The model can reproduce the EOF velocities with a characteristic P = 0.7. The model and EOF data compare best around the solar maximum of 2001. is lower around solar minimum, due to occasional limitations in the geographical and temporal coverage of the SuperDARN measurements. This may indicate the need to modify our model around the minimum of the solar cycle. Our model has the potential to be used to forecast the ionospheric electric field using the real‐time solar wind data available from spacecraft located upstream of the Earth.

4D scale-dependent Schwarzschild-AdS/dS black holes: study of shadow and weak deflection angle and greybody bounding

We investigate the shadow, deflection angle, and the greybody bounding of a Schwarzschild-AdS/dS black hole in scale-dependent gravity. We used the EHT data to constrain the parameter epsilon. We have found that within the 1sigma level of uncertainty, epsilon M_0 ranges at 10^{-11} - 10^{-16} orders of magnitude for Sgr. A*, and 10^{-11} - 10^{-17} for M87*. Using these parameters, we explored how the shadow radius behaves as perceived by a static observer. Using the known parameters for Sgr. A* and M87*, we found different shadow cast behavior near the cosmological horizon even if the same scaling parameters were used. We explored the deflection angle {hat{alpha }} in the weak field limit for both massive and null particles, in addition to the effect of finite distances using the non-asymptotically flat version of the Gauss-Bonnet theorem. We have found that the {hat{alpha }} strongly depends on massive particles. Tests on epsilon’s effect using Sgr. A* is only along points of low impact parameters. For M87*, only the scaled BH in AdS type can be tested at high impact parameters at the expense of very low value for {hat{alpha }}. We study the rigorous bound on the greybody factor for scalar field emitted from black holes in the theory and show the bound on the transmission probability. The effects of the scale-dependent gravity on the greybody factors are investigated, and the results indicate that the bound on the greybody factor in this case is less than the bound for the Schwarzschild black hole.

The massless single off-shell scalar box integral — branch cut structure and all-order epsilon expansion

We investigate the single off-shell scalar box integral with massless internal lines in dimensional regularization. A special emphasis is given to higher orders in the dimensional regularization parameter epsilon, its branch cut structure, and kinematic limits. Common representations of the box integral introduce superficial branch cuts, which we eliminate to all orders in the epsilon expansion. In the literature so far a satisfactory solution for this issue only exists up to finite order in the epsilon expansion. However, for certain calculations at NNLO in perturbation theory, higher orders in epsilon of this integral are required. In this paper, we present results to all orders in epsilon in terms of single-valued polylogarithms and explicitly determine the real and imaginary part of the box integral in all kinematic regions.

Pullback dynamics and robustness for the 3D Navier-Stokes-Voigt equations with memory

<abstract><p>The tempered pullback dynamics and robustness of the 3D Navier-Stokes-Voigt equations with memory and perturbed external force are considered in this paper. Based on the global well-posedness results and energy estimates involving memory, a suitable tempered universe is constructed, the robustness is finally established via the upper semi-continuity of tempered pullback attractors when the perturbation parameter epsilon tends to zero.</p></abstract>

Non-trivial class of anisotropic compact stellar model in Rastall gravity

We investigated Rastall gravity, for an anisotropic star with a static spherical symmetry, whereas the matter-geometry coupling as assumed in Rastall Theory (RT) is expected to play a crucial role differentiating RT from General Relativity (GR). Indeed, all the obtained results confirm that RT is not equivalent to GR, however, it produces same amount of anisotropy as GR for static spherically symmetric stellar models. We used the observational constraints on the mass and the radius of the pulsar Her X-1 to determine the model parameters confirming the physical viability of the model. We found that the matter-geometry coupling in RT allows slightly less size than GR for a given mass. We confirmed the model viability via other twenty pulsars’ observations. Utilizing the strong energy condition we determined an upper bound on compactness U_text {max}sim 0.603, in agreement with Buchdahl limit, whereas Rastall parameter epsilon =-0.1. For a surface density compatible with a neutron core at nuclear saturation density the mass-radius curve allows masses up to 3.53 M_odot . We note that there is no equation of state is assumed, however the model fits well with linear behaviour. We split the twenty pulsars into four groups according to the boundary densities. Three groups are compatible with neutron cores while one group fits perfectly with higher boundary density 8times 10^{14}hbox {g/cm}^3 which suggests that those pulsars may have quark-gluon cores.

Modeling of pure visco-qP-wave propagation in attenuating tilted transversely isotropic media based on decoupled fractional Laplacians

The pseudoviscoacoustic anisotropic wave equation is widely used in the oil and gas industry for modeling wavefields in attenuating anisotropic media. Compared to the full viscoelastic anisotropic wave equation, it can greatly reduce the computational cost of wavefield modeling while maintaining the visco-qP-wave kinematics very well. However, even if we place the source in a thin isotropic layer, there will be some unwanted S-wave artifacts in the qP wavefield simulated by the pseudoviscoacoustic anisotropic wave equation due to the stepped approximation of inclined layer interfaces. Furthermore, the wavefield simulated by the pseudoviscoacoustic anisotropic wave equation may suffer from numerical instabilities when the anisotropy parameter epsilon is less than delta. To overcome these problems, we derive a pure-viscoacoustic tilted transversely isotropic (TTI) wave equation in media with anisotropy in velocity and attenuation based on the exact complex-valued phase velocity formula. The pure-viscoacoustic TTI wave equation has decoupled amplitude dissipation and phase dispersion terms, which is suitable for further reverse time migration with Q compensation. For numerical simulations, we adopt the second-order Taylor series expansion to replace the mixed-domain spatially variable fractional Laplacian operator, which guarantees the decoupling of the wavenumber from the space-related fractional order. Then, we use an efficient and stable hybrid finite-difference and pseudospectral method (HFDPSM) to solve the pure-viscoacoustic TTI wave equation. Numerical tests indicate that the simulation results of the newly derived pure-viscoacoustic TTI wave equation are stable, free from S-wave artifacts, and accurate. We further demonstrate that HFDPSM outperforms the pseudospectral method in terms of numerical simulation stability and computing efficiency.

Curved accretion disks around rotating black holes without reflection symmetry

Rotating black holes without equatorial reflection symmetry can naturally arise in effective low-energy theories of fundamental quantum gravity, in particular, when parity-violating interactions are introduced. Adopting a theory-agnostic approach and considering a recently proposed Kerr-like black hole model, we investigate the structure and properties of accretion disk around a rotating black hole without reflection symmetry. In the absence of reflection symmetry, the accretion disk is in general a curved surface in shape, rather than a flat disk lying on the equatorial plane. Furthermore, the parameter epsilon that controls the reflection asymmetry would shrink the size of the prograde innermost stable circular orbits, and enhance the efficiency of the black hole in converting rest-mass energy to radiation during accretion. The retrograde innermost stable circular orbits are stretched but the effects are substantially suppressed. In addition, we find that spin measurements based on the gravitational redshift observations of the disk, assuming a Kerr geometry, may overestimate the true spin values if the central object is actually a Kerr-like black hole with conspicuous equatorial reflection asymmetry. The qualitative results that the accretion disk becomes curved and the prograde innermost stable circular orbits are shrunk turn out to be generic in our model when the reflection asymmetry is small.

Multiple scales and singular limits of perfect fluids

In this article, our goal is to study the singular limits for a scaled barotropic Euler system modeling a rotating, compressible and inviscid fluid, where Mach number =epsilon ^m , Rossby number =epsilon and Froude number =epsilon ^n are proportional to a small parameter epsilon rightarrow 0. The fluid is confined to an infinite slab, the limit behavior is identified as the incompressible Euler system or a damped incompressible Euler system depending on the relation between m and n. For well-prepared initial data, the convergence is shown on the lifespan time interval of the strong solutions of the target system, whereas a class of generalized dissipative solutions is considered for the primitive system. The technique can be adapted to the compressible Navier–Stokes system in the subcritical range of the adiabatic exponent gamma with 1<gamma le frac{3}{2}, where weak solutions are not known to exist.

Well-posed results for nonlocal biparabolic equation with linear and nonlinear source terms

In this paper, we consider the biparabolic problem under nonlocal conditions with both linear and nonlinear source terms. We derive the regularity property of the mild solution for the linear source term while we apply the Banach fixed-point theorem to study the existence and uniqueness of the mild solution for the nonlinear source term. In both cases, we show that the mild solution of our problem converges to the solution of an initial value problem as the parameter epsilon tends to zero. The novelty in our study can be considered as one of the first results on biparabolic equations with nonlocal conditions.

ECR-DBSCAN: An improved DBSCAN based on computational geometry

A new density based clustering algorithm ECR−DBSCAN based on DBSCAN, has been presented in this paper. Computational geometry is applied to develop the modified DBSCAN algorithm. It is well known that the quality of density based clustering depends on its input parameters. However, it is not easy to determine proper values of input parameters for DBSCAN. This paper presents three significant modifications or extensions to DBSCAN related with (i) selection of hyper parameter epsilon (eps) using the radii of empty or voronoi circles (ii) selection of parameter minPoints (mp) for the same epsilon and (iii) redistribution of noise points to suitable clusters using the concept of centroid hinged clustering. ECR−DBSCAN is implemented with PYTHON accompanied by extensive experiment on benchmark data sets. Our experimental results establish the novelty and validity of the proposed clustering method over standard techniques.

Dynamics of charged particles and magnetic dipoles around magnetized quasi-Schwarzschild black holes

In the present paper, we have investigated the motion of charged particles together with magnetic dipoles to determine how well the spacetime deviation parameter epsilon and external uniform magnetic field can mimic the spin of a rotating Kerr black hole. Investigation of charged particle motion has shown that the deviation parameter epsilon in the absence of an external magnetic fields can mimic the rotation parameter of the Kerr spacetime up to a/M approx 0.5. The combination of an external magnetic field and deviation parameter can do even a better job mimicking the rotation parameter up to a/Msimeq 0.93, which corresponds to the rapidly rotating case. Study of the dynamics of the magnetic dipoles around quasi-Schwarzschild black holes in the external magnetic field has shown that there are degeneracy values of the ISCO radius of test particles at epsilon _{cr}>epsilon ge 0.35 which may lead to two different values of the innermost stable circular orbit (ISCO) radius. When the deviation parameter is in the range of epsilon in (-1, 1), it can mimic the spin of a rotating Kerr black hole in the range a/M in (0.0537, 0.3952) for magnetic dipoles with values of the magnetic coupling parameter beta in [-0.25, 0.25] in corotating orbits.

Asymptotic Behaviour of Time Stepping Methods for Phase Field Models

Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error sigma in the limit of small length scale parameter epsilon rightarrow 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity.

A continuously perturbed Dirichlet energy with area-preserving stationary points that \u2018buckle\u2019 and occur in equal-energy pairs

We exhibit a family of convex functionals with infinitely many equal-energy C^1 stationary points that (i) occur in pairs v_{pm } satisfying det nabla v_{pm }=1 on the unit ball B in {mathbb {R}}^2 and (ii) obey the boundary condition v_{pm }=text {id} on partial B. When the parameter epsilon upon which the family of functionals depends exceeds sqrt{2}, the stationary points appear to ‘buckle’ near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps v_{pm }(x) and prove that they are proportional to (epsilon -1/epsilon )ln |x| as x rightarrow 0 in B. The lowest-energy pairs v_{pm } are energy minimizers within the class of twist maps (see Taheri in Topol Methods Nonlinear Anal 33(1):179–204, 2009 or Sivaloganathan and Spector in Arch Ration Mech Anal 196:363–394, 2010), which, for each 0le rle 1, take the circle {xin B: |x|=r} to itself; a fortiori, all v_{pm } are stationary in the class of W^{1,2}(B;{mathbb {R}}^2) maps w obeying w=text {id} on partial B and det nabla w=1 in B.

Rastall gravity extension of the standard Lambda CDM model: theoretical features and observational constraints

We present a detailed investigation of the Rastall gravity extension of the standard Lambda CDM model. We review the model for two simultaneous modifications of different nature in the Friedmann equation due to the Rastall gravity: the new contributions of the material (actual) sources (considered as effective source) and the altered evolution of the material sources. We discuss the role/behavior of these modifications with regard to some low redshift tensions, including the so-called H_0 tension, prevailing within the standard Lambda CDM. We constrain the model at the level of linear perturbations, and obtain the first constraints through a robust and accurate analysis using the latest full Planck cosmic microwave background (CMB) data, with and without including baryon acoustic oscillations (BAO) data. We find that the Rastall parameter epsilon (null for general relativity) is consistent with zero at 68% CL (with a tendency towards positive values, -0.0001< epsilon < 0.0007 (CMB+BAO) at 68% CL), which in turn implies no significant statistical evidence for deviation from general relativity, and also a precision of mathcal {O}(10^{-4}) for the coefficient -1/2 of the term g_{mu nu }R in the Einstein field equations of general relativity (guaranteeing the local energy-momentum conservation). We explore the consequences led by the Rastall gravity on the cosmological parameters in the light of the observational analyses. It turns out that the effective source, with a present-day density parameter Omega _mathrm{X0}=-0.0010pm 0.0013 (CMB+BAO, 68% CL), dynamically screens the usual vacuum energy at high redshifts, but this mechanism barely works due to the opposition by the altered evolution of cold dark matter. Consequently, two simultaneous modifications of different nature in the Friedmann equation by the Rastall gravity act against each other, and do not help to considerably relax the low redshift tensions, including the so-called H_0 tension. Our results may offer a guide for the research community that studies the Rastall gravity in various aspects of gravitation and cosmology.

Fitted Numerical Scheme for Singularly Perturbed Differential Equations Having Two Small Delays

In this paper, singularly perturbed differential equations having delay on the convection and reaction terms are considered. The highest order derivative term in the equation is multiplied by a perturbation parameter epsilon taking arbitrary values in the interval (0; 1]. For small epsilon, the problem involves a boundary layer on the left or right side of the domain depending on the sign of the coefficient of the convective term. The terms involving the delay are approximated using Taylor series approximation. The resulting singularly perturbed boundary value problem is treated using exponentially fitted upwind finite difference method. The stability of the proposed scheme is analysed and investigated using maximum principle and barrier functions for solution bound. The formulated scheme converges independent of the perturbation parameter with rate of convergence O(N−1). Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme to order O(N−2). To validate the theoretical finding, three model examples having boundary layer behaviour are considered. The maximum absolute error and rate of convergence of the scheme are computed. The proposed scheme gives accurate and parameter uniformly convergent result.

Effective boundary conditions at a rough wall: a high-order homogenization approach

Effective boundary conditions, correct to third order in a small parameter epsilon, are derived by homogenization theory for the motion of an incompressible fluid over a rough wall with periodic micro-indentations. The length scale of the indentations is l, and epsilon = l/L ll 1, with L a characteristic length of the macroscopic problem. A multiple scale expansion of the variables allows to recover, at leading order, the usual Navier slip condition. At next order the slip velocity includes a term arising from the streamwise pressure gradient; furthermore, a transpiration velocity {mathcal {O}}(epsilon ^{2}) appears at the fictitious wall where the effective boundary conditions are enforced. Additional terms appear at third order in both wall-tangent and wall-normal components of the velocity. The application of the effective conditions to a macroscopic problem is carried out for the Hiemenz stagnation point flow over a rough wall, highlighting the differences among the exact results and those obtained using conditions of different asymptotic orders.

NNLO zero-jettiness beam and soft functions to higher orders in the dimensional-regularization parameter epsilon

We present the calculation of the next-to-next-to-leading order (NNLO) zero-jettiness beam and soft functions, up to the second order in the expansion in the dimensional regularization parameter epsilon . These higher order terms are needed for the computation of the next-to-next-to-next-to-leading order (hbox {N}^3hbox {LO}) zero-jettiness soft and beam functions. As a byproduct, we confirm the {mathcal {O}}{(epsilon ^0)} results for NNLO beam and soft functions available in the literature by Gaunt et al. (JHEP 4:113, 2014), Gaunt et al. (JHEP 8:20, 2014), Boughezal et al. (Phys Rev D 96:34001, 2017), Monni et al. (JHEP 8:10, 2011) and Kelley et al. (Phys Rev D 84:45022, 2011).

Asymptotic Optimality of a First-Order Approximate Strategy for an Exponential Utility Maximization Problem with a Small Coefficient of Wealth-Dependent Risk Aversion

In Delong [8] we investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. We assume that the insurer’s risk aversion coefficient consists of a constant risk aversion and a small amount of wealth-dependent risk aversion. We apply perturbation theory and expand the equilibrium value function of the optimization problem on the parameter epsilon controlling the degree of the insurer’s risk aversion depending on wealth. We derive a candidate for the first-order approximation to the equilibrium investment strategy. In this paper we formally show that the zeroth-order investment strategy pi _0^* postulated by Delong (Math Methods Oper Res 89:73–113, 2019) performs better than any strategy pi _0 when we compare the asymptotic expansions of the objective functions up to order {mathcal {O}}(1) as epsilon rightarrow 0, and the first-order investment strategy pi _0^*+pi _1^*epsilon postulated by Delong (Math Methods Oper Res 89:73–113, 2019) is the equilibrium strategy in the class of strategies pi ^*_0+pi _1epsilon when we compare the asymptotic expansions of the objective functions up to order {mathcal {O}}(epsilon ^2) as epsilon rightarrow 0, where epsilon denotes the parameter controlling the degree of the insurer’s risk aversion depending on wealth.

Differential Privacy in Practice: Expose your Epsilons!

Differential privacy is at a turning point. Implementations have been successfully leveraged in private industry, the public sector, and academia in a wide variety of applications, allowing scientists, engineers, and researchers the ability to learn about populations of interest without specifically learning about these individuals. Because differential privacy allows us to quantify cumulative privacy loss, these differentially private systems will, for the first time, allow us to measure and compare the total privacy loss due to these personal data-intensive activities. Appropriately leveraged, this could be a watershed moment for privacy.&#x0D; Like other technologies and techniques that allow for a range of instantiations, implementation details matter. When meaningfully implemented, differential privacy supports deep data-driven insights with minimal worst-case privacy loss. When not meaningfully implemented, differential privacy delivers privacy mostly in name. Using differential privacy to maximize learning while providing a meaningful degree of privacy requires judicious choices with respect to the privacy parameter epsilon, among other factors. However, there is little understanding of what is the optimal value of epsilon for a given system or classes of systems/purposes/data etc. or how to go about figuring it out.&#x0D; To understand current differential privacy implementations and how organizations make these key choices in practice, we conducted interviews with practitioners to learn from their experiences of implementing differential privacy. We found no clear consensus on how to choose epsilon, nor is there agreement on how to approach this and other key implementation decisions. Given the importance of these implementation details there is a need for shared learning amongst the differential privacy community. To serve these purposes, we propose the creation of the Epsilon Registry—a publicly available communal body of knowledge about differential privacy implementations that can be used by various stakeholders to drive the identification and adoption of judicious differentially private implementations.