Radial Derivative Research Articles

Fractional powers of Clifford d-bar and radial derivatives

We introduce new non-local derivative operators in the context of Clifford analysis, by giving meaning to fractional powers of the d-bar and radial derivatives in this setting. The definitions arise from binomial-type series constructions, and we justify them by proving natural properties including semigroup/composition properties and the new operators' action on special functions acting on paravectors.

On the Product of Weighted Composition Operators and Radial Derivative Operators from the Bloch-Type Space into the Bers-Type Space on the Fourth Loo-Keng Hua Domain

Let HEIV be the fourth Loo-Keng Hua domain. We study the boundedness of the product of the weighted composition operator and the radial derivative operator from the Bloch-type space Bα(HEIV) into the Bers-type space Aβ(HEIV) and provide the necessary and sufficient conditions for their boundedness.

A novel phase measuring deflectometry based on polar coordinate

Phase measuring deflectometry (PMD) is a method measure the surface of a mirror. However, when measuring convex mirrors, Cartesian coordinate fringes experience extreme compression. To address this issue, this paper proposes a novel PMD based on polar coordinate. This new method defines two linearly independent phase modulation directions, using the orthogonal basis of polar coordinate. It establishes polar coordinate fringes with rotational symmetry and radial pre-modulation, effectively reducing the impact of extreme compression at the edges. To correct the phase errors in polar coordinate fringes, a phase error compensation algorithm based on greyscale gradient is introduced. The algorithm calculates the influence factor of the eight connected domains around the error points to be compensated, utilising the phase grey gradient. The wavefront gradient data obtained through polar coordinate fringes are radial and tangential. Hence, a surface reconstruction method is proposed based on the Zernike partial derivative polynomial based on polar coordinate. In this method, the tangential partial derivatives and radial partial derivatives of the first 36 terms of Zernike in polar coordinate to construct Gram matrix equations. As a result, linearly independent Zernike recovery coefficients are obtained from coupled aliasing gradient data. Compared to the Cartesian coordinate system, the proposed method significantly reduces the fitting coefficient errors. Experimental measurements of convex mirrors with radii of curvature of 200 mm and 100 mm were conducted. The results demonstrate that compared to traditional PMD, this technique not only effectively suppresses extreme compression and increases the measurement area but also improves measurement accuracy by six times.

Scaling of the geomagnetic secular variation timescale

SUMMARY The ratio of the magnetic power spectrum and the secular variation spectrum measured at the Earth’s surface provides a timescale $\tau _{\rm sv}(l)$ as a function of spherical harmonic degree l. $\tau _{\rm sv}$ is often assumed to be representative of timescales related to the dynamo inside the outer core and its scaling with l is debated. To assess the validity of this surmise and to study the time variation of the geomagnetic field $\dot{\boldsymbol {B}}$ inside the outer core, we introduce a magnetic timescale spectrum $\tau (l,r)$ that is valid for all radius r above the inner core and reduces to the usual $\tau _{\rm sv}$ at and above the core–mantle boundary (CMB). We study $\tau$ in a numerical geodynamo model. At the CMB, we find that $\tau \sim l^{-1}$ is valid at both the large and small scales, in agreement with previous numerical studies on $\tau _{\rm sv}$. Just below the CMB, the scaling undergoes a sharp transition at small l. Consequently, in the interior of the outer core, $\tau$ exhibits different scaling at the large and small scales, specifically, the scaling of $\tau$ becomes shallower than $l^{-1}$ at small l. We find that this transition at the large scales stems from the fact that the horizontal components of the magnetic field evolve faster than the radial component in the interior. In contrast, the magnetic field at the CMB must match onto a potential field, hence the dynamics of the radial and horizontal magnetic fields are tied together. The upshot is $\tau _{\rm sv}$ becomes unreliable in estimating timescales inside the outer core. Another question concerning $\tau$ is whether an argument based on the frozen-flux hypothesis can be used to explain its scaling. To investigate this, we analyse the induction equation in the spectral space. We find that away from both boundaries, the magnetic diffusion term is negligible in the power spectrum of $\dot{\boldsymbol {B}}$. However, $\dot{\boldsymbol {B}}$ is controlled by the radial derivative in the induction term, thus invalidating the frozen-flux argument. Near the CMB, magnetic diffusion starts to affect $\dot{\boldsymbol {B}}$ rendering the frozen-flux hypothesis inapplicable. We also examine the effects of different velocity boundary conditions and find that the above results apply for both no-slip and stress-free conditions at the CMB.

Turunan Radial dari Anomali Gravitasi untuk Identifikasi Injeksi Air pada Reservoir

Time Lapse micro-gravity or 4D microgravity has been developed and applied for various purposes related to the dynamics change beneath the earth's surface. The principle is a repeated and periodic measurement to determine the change in gravity value at time interval as a yield of the dynamics changes in the density change of the subsurface. This method can be used for monitoring and identify water or gas injections in reservoirs. Mass density taht changes due to gas injection provide changes in the values of gravity over time. This change allows us to identify the dynamics of gas movement in reservoir. A reservoir model has been made in this research with gas injected in reservoir bulk. The time lapse gravity respons value is calculated numerically. Oil-gas contact is an edge between gas injection front with oil which move direction away from injection point. The first and second radial derivatives of 4D gravity can provide as a front of the gas movement in the reservoir. A radial derivatives is a horizontal derivative of gravity anomaly along radial direction from a point which selected as a entre of the anomaly. The second radial derivative is derivative of radial deriative. Radial derivatives value of this model response have range of 20 microGals/m and second radial derivative has a values of 0.05 microGals/m2. Graphics of absolute value of radial derivative and second radial derivative of 4D gravity anomaly seemingly correlated with the oil-gas contact of reservoir models. Radial derivative and second radial derivative has been proven as new available method wich has ability to shows a boundary of a horizontal layer with vertical density contrast. Keywords: Radial derivatives, reservoir, gas injection.

Forward Modeling of Gravity Anomalies for Identification of Burried Cylindrical Body Using Radial Derivative

Radial Derivative Forward Modeling of Gravity Anomalies for Identification of Cylindrical Geological Features. The gravity method is a geophysical method with exploration costs that are quite cheap compared to other geophysical methods. This method is based on the density contrast of the target body with the surrounding. The cylindrical body is one of the targets among various other geological features. This research was conducted to test the ability of radial derivatives of gravity anomalies for targets in the form of cylindrical body. Radial derivatives consist of a first derivative and a second derivative. Forward modeling of cylindrical geological features is carried out analytically and with finite elements. Both calculations were carried out with a computer program based on Matlab. The results show that there is no difference in results either analytically or finite element wise. This method has been proven to be able to provide clear boundary positions on cylindrical geological features.

A gravitational eye: a method for extracting maximum information from gravitational potentials

Gravity measurements have uses in a wide range of fields including geological mapping and mine-shaft inspection. The specific application under consideration sets limits on the survey and the amount of information that can be obtained. For example, in a conventional gravity survey at the Earth’s surface a gravimeter is translated on a two-dimensional planar grid taking measurements of the vertical component of gravity. If, however, the survey points cannot be chosen so freely, for example if the gravimeter is constrained to operate in a tunnel where only a one-dimensional line of data could be taken, less information will be obtained. To address this situation, we investigate an alternative approach, in the form of an instrument which rotates around a central point measuring the gravitational potential or its radial derivative on the boundary of a sphere. The ability to record additional components of gravity by rotating the gravimeter will give more information than obtained with a single measurement traditionally taken at each point on a survey, consequently reducing ambiguities in interpretation. We term a device which measures the potential, or its radial derivatives, around the surface of a sphere a gravitational eye. In this article we explore ideas of resolution and propose a thought experiment for comparing the performance of diverse types of gravitational eye. We also discuss radial analytic continuation towards sources of gravity and the resulting resolution enhancement, before finally discussing the possibility of using cold-atom gravimetry and gradiometry to construct a gravitational eye. If realised, the gravitational eye will offer revolutionary capability enabling the maximum information to be obtained about features in all directions around it.

A dynamical systems formulation for inhomogeneous LRS-II spacetimes

We present a dynamical system formulation for inhomogeneous LRS-II spacetimes using the covariant 1+1+2 decomposition approach. Our approach describes the LRS-II dynamics from the point of view of a comoving observer. Promoting the covariant radial derivatives of the covariant dynamical quantities to new dynamical variables and utilizing the commutation relation between the covariant temporal and radial derivatives, we were able to construct an autonomous system of first-order ordinary differential equations along with some purely algebraic constraints. Using our dynamical system formulation we found several interesting features in the LRS-II phase space with dust, one of them being that the homogeneous solutions constitute an invariant submanifold. For the particular case of LTB, we were also able to recover the previously known result that an expanding LTB tends to Milne in the absence of a cosmological constant, providing a potential validation of our formalism.

Higher-order optimality conditions with separated derivatives and sensitivity analysis for set-valued optimization

In this paper, we establish optimality conditions and sensitivity analysis of set-valued optimization problems in terms of higher-order radial derivatives. First, we obtain the optimality conditions with separated derivatives for a set-valued optimization problem, here separated derivatives means the derivatives of objective and constraint functions are different. Then, some duality theorems for a mixed type of primal-dual set-valued optimization problem are gained. Finally, several results concerning higher-order sensitivity analysis are presented. The main results of this paper are illustrated by some concrete examples.

Higher-Order Radial Hadamard Directional Derivatives and Applications to Set-Valued Equilibrium Problems

In this paper, we first introduce the higher-order lower and upper radial Hadamard directional derivatives for set-valued maps, which can capture the global information of concerning maps, and discuss the relationships with other existed higher-order derivatives. Second, based on the introduced derivatives, we establish optimality conditions for Henig efficient solutions of a set-valued equilibrium problem with constraints. Particularly, the optimality conditions hold in the case where the derivatives of objective and constraint functions are separated. Finally, we give some duality theorems for a mixed type of primal-dual set-valued equilibrium problem. The main results of this paper are illustrated by several concrete examples.

A simple analytical model for turbulent kinetic energy dissipation for self-similar round turbulent jets

This work presents a simple analytical model for the streamwise and radial variations of turbulent kinetic energy dissipation in an incompressible round turbulent jet. The key assumptions in the model are: similarity in the axial velocity profile with a Gaussian shape, axisymmetric flow and the dominance of radial derivatives of the mean velocity over axial direction derivatives (similar to boundary layer theory). Initially, a simplified eddy-viscosity relation for turbulent stresses is derived using the algebraic stress model by Gatski & Speziale (J. Fluid Mech., vol. 254, 1993, pp. 59–78). Subsequently, with this eddy-viscosity relation, the relation for variations of turbulent kinetic energy dissipation is formulated using the conservation of turbulent kinetic energy. To extract the necessary constants of the model, experimental velocity statistics for round jets are obtained through particle image velocimetry measurements. The experimental results of the mean entrainment coefficient for turbulent jets are also analysed. When comparing the radial variation of turbulent kinetic energy dissipation from the model with experimental results at Reynolds number $1.4\times 10^5$ and numerical results at Reynolds number $1200$ from the available literature, we observe a maximum error of $10\,\%$ and $15\,\%$ , respectively. Finally, using the validated model, we analyse the impact of mean velocity evolution parameters on the behaviour of turbulent kinetic energy dissipation and discuss its potential significance in future studies.

A fast convergent solution of wave propagation for multilayer inhomogeneous cylindrical dielectric waveguides using a semianalytical method

Abstract This paper presents an accurate and efficient semianalytical method based on the Galerkin procedure for solving electromagnetic wave propagation problems in multilayer inhomogeneous cylindrical dielectric waveguides. The method represents the field in each inhomogeneous layer by a linear combination of eigenfunctions with unknown coefficients, which are expressed using the inner products of a series of basis functions, following the Galerkin procedure. The continuity of the field and its radial derivative is enforced at the interface between adjacent layers. By applying this procedure to all inhomogeneous layers, the Helmholtz equations are transformed into linear algebraic equations with expanded coefficients in matrix form, allowing the complicated wave propagation problem in a multilayer inhomogeneous waveguide to be solved as a matrix eigenvalue problem. The method is validated by providing detailed propagation characteristics for various multilayer inhomogeneous cylinders with different permittivity profiles. The accuracy and efficiency of the proposed method are demonstrated through comparisons with results obtained using other numerical techniques.

δARD loss for low-contrast medical image segmentation

Medical image segmentation is essential to image-based disease analysis and has proven to be significantly helpful for doctors to make decisions. Due to the low-contrast of some medical images, the accurate segmentation of medical images has always been a challenging problem. The experiment found that UNet with current loss functions cannot capture subtle information in target contours or regions in low-contrast medical images, which are crucial for subsequent disease diagnosis. We propose a robust loss by incorporating the difference in average radial derivative (ARD), length and region area to further help the network to achieve more accurate segmentation results. We evaluated the proposed loss function using UNet as the base segmentation network compared to five conventional loss functions on one private and four public medical image datasets. Experimental results illustrate that UNet with the proposed loss function can achieve the best segmentation performance, even better than the outstanding deep learning models with original loss functions. Furthermore, three representative datasets were chosen to validate the effectiveness of the proposed δARD loss function with seven different models. The experiments revealed δARD loss’s plug-and-play feature and its robustness over multiple models and datasets.

Numerical computation of quasinormal modes in the first-order approach to black hole perturbations in modified gravity

We present a novel approach to the numerical computation of quasi-normal modes, based on the first-order (in radial derivative) formulation of the equations of motion and using a matrix version of the continued fraction method. This numerical method is particularly suited to the study of static black holes in modified gravity, where the traditional second-order, Schrödinger-like, form of the equations of motion is not always available. Our approach relies on the knowledge of the asymptotic behaviours of the perturbations near the black hole horizon and at spatial infinity, which can be obtained via the systematic algorithm that we have proposed recently. In this work, we first present our method for the perturbations of a Schwarzschild black hole and show that we recover the well-know frequencies of the QNMs to a very high precision. We then apply our method to the axial perturbations of an exact black hole solution in a particular scalar-tensor theory of gravity. We also cross-check the obtained QNM frequencies with other numerical methods.

The effect of the diamagnetic drift on geodesic acoustic modes in tokamak plasmas

The effect of the radial derivative of the equilibrium particle distribution, i.e., the diamagnetic drift effect, on geodesic acoustic modes is taken into account in this work. The effect was routinely neglected in the previous studies on the geodesic acoustic mode since the dominant mode number is m/n=0/0. However, for finite electron temperatures, small m ≠ 0 side bands are present and the diamagnetic drift effect enters through these side bands. In this work, we find that the mode frequencies increase with the particle density gradient. The temperature ratio between electrons and ions, i.e., τ=Te/Ti, is a key parameter influencing this effect. The effect is more prominent for higher τ values. Another effect is the symmetry breaking of the propagating direction of the sideband potentials. In contrast to the pure standing wave form, the potential perturbation consists of a standing wave superimposed with a small amplitude traveling wave, which is nearly proportional to the density gradient.

In horizon penetrating coordinates: Kerr black hole metric perturbation, construction and completion

We investigate the Teukolsky equation in horizon-penetrating coordinates to study the behavior of perturbation waves crossing the outer horizon. For this purpose, we use the null ingoing/outgoing Eddington–Finkelstein coordinates. The first derivative of the radial equation is a Fuchsian differential equation with an additional regular singularity to the ones the radial one has. The radial functions satisfy the physical boundary conditions without imposing any regularity conditions. We also observe that the Hertz-Weyl scalar equations preserve their angular and radial signatures in these coordinates. Using the angular equation, we construct the metric perturbation for a circularly orbiting perturber around a black hole in Kerr spacetime in a horizon-penetrating setting. Furthermore, we completed the missing metric pieces due to the mass M and angular momentum J perturbations. We also provide an explicit formula for the metric perturbation as a function of the radial part, its derivative, and the angular part of the solution to the Teukolsky equation. Finally, we discuss the importance of the extra singularity in the radial derivative for the convergence of the metric expansion.

Bounded below composition operators on the space of Bloch functions on the unit ball of a Hilbert space

Let B_E be the open unit ball of a complex finite or infinite dimensional Hilbert space E and consider the space mathcal {B}(B_E) of Bloch functions on B_E. Using Lipschitz continuity of the dilation map on B_E given by x mapsto (1-Vert xVert ^2) mathcal {R}f(x) for x in B_E, where mathcal {R}f denotes the radial derivative of f in mathcal {B}(B_E), we study when a composition operator on mathcal {B}(B_E) is bounded below.

Approximation on hexagonal domains by Taylor-Abel-Poisson means

Approximative properties of the Taylor-Abel-Poisson linear summation method of Fourier series are considered for functions of several variables, periodic with respect to the hexagonal domain, in the integral metric. In particular, direct and inverse theorems are proved in terms of approximations of functions by the Taylor-Abel-Poisson means and K-functionals generated by radial derivatives. Bernstein type inequalities for L1-norm of radial derivatives of arbitrary order of the Poisson kernel are also obtained.

Dark matter fluid constraints from galaxy rotation curves

Galaxy rotation curves are considered to be convincing evidence for dark matter or some dynamically equivalent alternative mechanism. Starting only from the rotation curve data, we present a model independent approach of testing a general hypothesis that dark matter has the properties of a barotropic fluid. It is shown how the speed of sound squared can be expressed in terms of rotation curve data and their radial derivatives and how model independent constraints can be obtained from the requirements that it is confined between 0 and c^2. Using the Milky Way rotation curve data available in the literature, we obtain the constraints on the barotropic fluid speed of sound and illustrate the potential of this approach. Technical challenges, limitations and possible future extensions and improvements of the proposed approach are discussed.

On the energy limit of compact isochronous cyclotrons

Existing analytical models for transverse beam dynamics in isochronous cyclotrons are often not valid or not precise for relativistic energies. The main difficulty in developing such models lies in the fact that cross-terms between derivatives of the average magnetic field and the azimuthally varying components cannot be neglected at higher energies. Taking such cross-terms rigorously into account results in an even larger number of terms that need to be included in the equations. In this paper, a method is developed which is relativistically correct and which provides results that are practical and easy to use. We derive new formulas, graphs and tables for the radial and vertical tunes in terms of the flutter, its radial derivatives, the spiral angle and the relativistic gamma. Using this method, we study the 2νr = N structural resonance (N is number of sectors) and provide formulas and graphs for its stopband. Combining those equations with the new equation for the vertical tune, we find the stability zone and the energy limit of compact isochronous cyclotrons for any value of N. We confront the new analytical method with closed orbit simulations of the IBA C400 cyclotron for hadron therapy.