Orthogonal Coordinates Research Articles

General Fractional Vector Calculus

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

Diffractive Sensor Elements for Registration of Long-Term Instability at Writing of Computer-Generated Holograms.

The research and development of methods using of the specialized diffractive microstructure sensors embedded in the pattern of computer-generated holograms (CGH) manufactured on circular and X-Y laser writing systems is discussed. These microstructures consist of two parts: one of which is written before the CGH in the field of future hologram and the second one is written during the long-term writing of the CGH. The shift between the first and second part of the microstructure is the trace of the writing errors and allows one to determine and calculate the error of CGH fabrication along both orthogonal coordinates. The developed method is based on the principle of diffraction-based overlay with 1D and 2D built-in diffractive microstructure-sensors. Mathematical modeling and results of experimental test writings of such diffractive microstructure sensors are described. The efficiency of using these types of build-in sensors for the writing errors estimation for CGHs is demonstrated.

An Interpolation From Sol to Hyperbolic Space

We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with canonical left-invariant metrics, becomes an interpolation from a model of the Sol geometry to a model of Hyperbolic Space, with a stop at . These Lie groups are also Bianchi groups of Type VI with orthogonal coordinates. As a continuation of joint work with Richard Schwartz on Sol, we primarily analyze those Lie groups in our interpolation with some positive sectional curvature. Our main result is a characterization of the cut locus at the identity of the group that maximizes scalar curvature.

Intuitive Spatial Tactile Feedback for Better Awareness about Robot Trajectory during Human-Robot Collaboration.

In this work, we extend the previously proposed approach of improving mutual perception during human–robot collaboration by communicating the robot’s motion intentions and status to a human worker using hand-worn haptic feedback devices. The improvement is presented by introducing spatial tactile feedback, which provides the human worker with more intuitive information about the currently planned robot’s trajectory, given its spatial configuration. The enhanced feedback devices communicate directional information through activation of six tactors spatially organised to represent an orthogonal coordinate frame: the vibration activates on the side of the feedback device that is closest to the future path of the robot. To test the effectiveness of the improved human–machine interface, two user studies were prepared and conducted. The first study aimed to quantitatively evaluate the ease of differentiating activation of individual tactors of the notification devices. The second user study aimed to assess the overall usability of the enhanced notification mode for improving human awareness about the planned trajectory of a robot. The results of the first experiment allowed to identify the tactors for which vibration intensity was most often confused by users. The results of the second experiment showed that the enhanced notification system allowed the participants to complete the task faster and, in general, improved user awareness of the robot’s movement plan, according to both objective and subjective data. Moreover, the majority of participants (82%) favoured the improved notification system over its previous non-directional version and vision-based inspection.

Food quality evaluation in drying: Structuring of measurable food attributes into multi-dimensional fuzzy sets

Food quality is a fuzzy category, which could be evaluated using fuzzy logic. Our approach to food quality evaluation is based on mapping food quality attributes into a fuzzy domain as a multi-dimensional fuzzy sets. First, the data representing quality attributes are mapped into orthogonal coordinates using PCA to reduce dimensionality. Second, subtractive clustering (SC) is applied to determine a representative number of clusters. Each point in the dataset is associated with each cluster by credibilistic fuzzy C-means clustering (CFCM). After data organized in fuzzy clusters, an artificial neural network (ANN) is trained to associate each point in the dataset with its membership degree in each cluster. Trained ANN serves as a predictive model to convert real-time data stream into the multi-dimensional fuzzy domain. The application of this methodology is illustrated for real-time quality evaluation in shrimp batch drying. In this study 27 quality attributes have been merged into 9 orthonormal vectors, which have been clustered into 10 fuzzy sets. This structuring of the experimental fuzzy domain allowed the development of a multi-dimensional kinetic model, which improved the quality of shrimp drying. The computational time for quality identification in the fuzzy domain is below 1 sec, which is satisfactory for most real-time applications. This data-driven algorithm is completely automated and has unlimited potential for real-time fuzzy control and optimization. Highlights Multi-dimensional fuzzy sets are unique identifiers of food quality Extracting principal information in orthonormal coordinates Using the artificial neural network for predicting membership functions A multi-dimensional fuzzy kinetics model was developed Structuring of fuzzy domain decreased computational time for fuzzy control

Fractional solutions for the inextensible Heisenberg antiferromagnetic fow and solitonic magnetic flux surfaces in the binormal direction

Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute Lorentz force equations associated with the magnetic b-lines in the binormal direction. Evolution equations of magnetic b-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model.

Approximate solutions for the inextensible Heisenberg antiferromagnetic flow and solitonic magnetic flux surfaces in the normal direction in Minkowski space

Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute fractional Lorentz force equations associated with the magnetic n-lines in the normal direction in Minkowski space. Fractional evolution equations of magnetic n-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their approximate solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model in Minkowski space.

Open-geometry modal method based on transverse electric and transverse magnetic mode expansion for orthogonal curvilinear coordinates.

A vectorial modal method is presented based on transverse magnetic and transverse electric mode expansion, which significantly simplifies the evaluation of the operator matrix. The method, which features a true open boundary condition, is introduced for an orthogonal curvilinear coordinate system with the specific examples of circular and elliptical geometries presented. We validate the method by considering challenging problems, such as the calculation of spontaneous emission rates, of modal reflection coefficients and of the effect of the emitter spatial misalignment on the spontaneous emission β factor. Results are compared with literature.

Quantifying Climate Signals: Spicity, Orthogonality, and Distance

AbstractThe variability of water masses has been analyzed using temperature and salinity or density and various forms of spicity/spiciness. The objective way of comparing these spicity/spiciness functions is not the difference between these definitions; instead, it is through the application of these functions in tracing water masses and climate signal analysis. The major advantage of the spicity function introduced by Huang et al. Journal of Geophysical Research: Oceans, is that this function in combination with density gives rise to an orthogonal coordinate, in which the signals are separated into two independent components. In addition, a new quantity, distance, is introduced, that can be used to trace water masses both along and across isopycnal surfaces. Spicity can serve as an appropriate tool to extract globally varying water mass properties as a supplement to variations of density alone.

Stresses and deformations of an eccentric cylindrical cam during hydro-joining

The stresses and deformations of an eccentric cylindrical cam during the process of hydro-joining have been calculated in orthogonal curvilinear coordinates according to the mechanical model of the cam and governing equations in terms of appropriate complex potentials with suitable boundary conditions. The radial and shearing stress coefficients determined for an example cam are far less than that of the tangential stress during hydro-joining. The tangential stress coefficient of the outer surface of the cam is greater than that of the internal surface when the polar angle exceeds a particular value. The position of the maximum value of the radial stress coefficient is located on the internal surface of the cam, and the maximum shear stress coefficient is located between the inside and outside surfaces of the cam. The cam deformations on the internal and external surfaces under internal pressure respectively attain maximum values at particular angles. The maximum values of the radial and y-directional deformations are located at the position of the minimum wall thickness. The radial deformations determined for an example cam are far larger that the tangential deformations during hydro-joining. The errors between the theoretical and numerical solutions for the tangential stress and the y-directional deformation are both very small.

Some thoughts on counts in sequencing studies.

Measurements in sequencing studies are mostly based on counts. There is a lack of theoretical developments for the analysis and modelling of this type of data. Some thoughts in this direction are presented, which might serve as a seed. The main issues addressed are the compositional character of multinomial probabilities and the corresponding representation in orthogonal (isometric) coordinates, and modelling distributions for sequencing data taking into account possible effects of amplification techniques.

Optimal Path Planning and Speed Control Integration Strategy for UGVs in Static and Dynamic Environments

In this paper, we present an optimal path planning and speed control strategy for motion planning of unmanned ground vehicles to follow the desired path in various driving environments. The novelty of this work is that it integrates the longitudinal speed and lateral motion control by maximizing the longitudinal progression and minimizing the lateral path tracking errors and formulates a standard quadratic optimization problem. Firstly, a dynamic programming approach is introduced to convexify the corridor constraints in path pre-selection and replanned for obstacle avoidance. The idea is to find a convex feasible set for the optimization problem using the convex constraints. And then a standard quadratic convex problem is formulated to optimize the longitudinal motion and steering angle control in prediction and control horizon. The longitudinal progression and lateral tracking errors are decomposed in orthogonal curvilinear coordinates. The centerline as well as road borders are also described by using cubic spline curves. Some challenge scenarios including static and moving participating obstacle vehicles are tested and validated for the proposed motion planning strategy. The feasibility and performance of the proposed strategy are demonstrated by numerical experiments in terms of the stability and maneuverability of unmanned ground vehicles in different driving conditions.

System of Orthogonal Curvilinear Coordinates on the Isentropic Surface behind a Detached Bow Shock Wave

Three-dimensional ideal gas flows behind a detached bow shock wave near the nonsymmetrical nose part of a bluff body in a uniform supersonic flow are investigated. The vector lines of the vector product of the velocity and entropy gradient are considered on isentropic surfaces, which are stream surfaces starting on closed lines located on the shock wave and covering the leading point of the shock wave. It is shown that if these vector lines encircle the isentropic surface they are closed. This means that an orthogonal curvilinear coordinate system in which the coordinate lines coincide with the streamlines may be constructed on each isentropic surface.

A New Fast Factorized Back-Projection Algorithm with Reduced Topography Sensibility for Missile-Borne SAR Focusing with Diving Movement

Time-domain algorithms have significant performance advantages for missile-borne synthetic aperture radar (SAR) focusing with diving movement. However, due to the diving curve trajectory of the missile platform, the range and angular histories of the target become very sensitive to unknown tomography, which provides difficulties for SAR algorithm development. To address this problem, we have proposed a new fast factorized back-projection (FFBP) algorithm with reduced topography sensibility for missile-borne SAR focusing. The new algorithm was designed based on an orthogonal cylindrical coordinate (OCC) system, in which the cross section of a cylinder in the coordinate system is approximately orthogonal to the diving curve trajectory. Owing to the acquisition symmetry of the OCC system, the range and the angular histories of the grid in the OCC geometry become less dependent of the topography in every recursion of FFBP implementation, which can dramatically reduce the adverse effects of unknown topography and achieve high focusing performance. In the simulation, echo signal based on a set of typical parameters from a missile-borne SAR system is generated with unknown tomography. Promising results with 1 m resolution are finally achieved, which demonstrates the performance of the proposed algorithm. The limitation of the algorithm is also discussed in the final part, which will facilitate the development of raw data processes in practical application.

Carrera unified formulation (CUF) for the micropolar plates and shells. I. Higher order theory

Starting from the variational principle of virtual power for the 3-D equations of the micropolar theory of elasticity in orthogonal curvilinear coordinates and using generalized series in terms of the plate thickness coordinates a new higher order model of orthotropic micropolar plates and shells have been developed here. Following Carrera Unified Formulation (CUF), the stress and strain tensors, as well as the vectors of displacements and rotation, have been expanded into series in terms of the shell thickness coordinates. Then, all the equations of the micropolar theory of elasticity (including generalized Hooke’s law) have been transformed to the corresponding equations for the coefficients of the series expansion on the plate thickness coordinates. A system of differential equations in terms of the displacements and rotation vectors and natural boundary conditions for the coefficients of the series expansion of the shell thickness coordinates have been obtained in the same way as in the classical theory of elasticity. All equations for the higher order theory of micropolar plates and shells have been developed and presented here. The obtained equations can be used for calculating the stress-strain and for modeling thin walled structures in macro, micro, and nanoscale when taking into account micropolar couple stress and rotation effects.

Methods of forecasting flows in computer networks on the basis of Рad approximation

An important component of a large corporate computer network management system is the network status forecasting unit. Forecasting should take into account the characteristics of the processes and flows circulating in the network. This will produce optimal control signals to control the network or its segment. It is shown that the flows in such networks (in particular, traffic) have a significant heterogeneity, ie the presence of significant emissions against the background of a small average value. Given these and other characteristics, flows in large corporate computer networks can be considered non-stationary. A method for predicting nonstationary time series using the Padé approximation, a powerful and accurate method for estimating the parameters of random processes, is proposed. This method can be used especially successfully in the presence of nonstationaries of the most various nature. To ensure the stability of the method and the stability of the results, it is proposed to force the poles of the approximating function into the stability zone - a single circle of the z-plane with the rules of conformal transformation: transformation of linear dimensions and preservation of angles between orthogonal coordinates. ). It is shown that in compliance with the conformity of the proposed transformation, the dynamic characteristics of the estimation and forecasting system are preserved. Numerical methods for finding Padé approximations are analyzed. The requirements to the Padé approximation algorithm are determined. It is shown that the algorithm must indicate at the output that the approximation that is calculated is degenerate according to the accepted criterion, ie to include a reliable degeneracy test. Al-algorithm should also be effective, but efficiency is not as important as reliability and resilience. The accuracy of numerical calculations is of paramount importance, because the information that allows the Padé approximation to carry out the analytical continuation of the function far beyond the circle of convergence, is enclosed in far decimal places of the data of the series coefficients.

A fast pressure-correction method for incompressible flows over curved walls

We have developed an explicit and direct pressure-correction method for simulating incompressible flows over curved walls. In order to integrate the Navier Stokes (NS) equations in time, we have developed an explicit, three-stage, third-order Runge-Kutta based projection-method (FastRK3) which requires solving the Poisson equation for pressure only once per time step. We have chosen to discretize the incompressible NS equations written in the orthogonal coordinates rather than the general formulation in curvilinear coordinates because the former does not contain cross-derivatives in the advection, diffusion, Laplacian, and gradient operators. Thus, the computational cost of solving the NS equations is substantially reduced and the numerical stencils of the finite difference approximations to these operators mirror that of the Cartesian formulation. This property also allows us to develop an FFT-based Poisson solver for pressure (FastPoc) for those cases where the components of the metric tensor are independent of one spatial direction: surfaces of linear translation (e.g., curved ramps and bumps) and surfaces of revolution (e.g., axisymmetric ramps). We have verified and validated FastRK3 and we have applied FastRK3 for simulating separated flows over ramps and a bump. Finally, our results show that the new FFT-based Poisson solver, FastPoc, is thirty to sixty times faster than the multigrid-based linear solver (depending on the tolerance set for the multigrid solver), and the new flow solver, FastRK3, is overall four to seven times faster when using FastPoc rather than multigrid. In summary, given that the computational mesh satisfies the property of orthogonality, FastRK3 can simulate flows over curved walls with second-order accuracy in space.

Orthonormal balances as a means of characterizing dietary exposure

Evaluating the effect of nutritional components on health outcomes is hampered by technical difficulties that are closely related to the compositional nature of dietary data. Compositional data consist of parts of a whole and, on the basis of the assumption that any relevant information they provide is contained in the ratios between them, procedures based on log-ratio transformations have been proposed for their analysis. It has been previously suggested that nutrient compositions can be analyzed using isometric log-ratio transformations which allow the compositions to be represented by a system of orthogonal coordinates corresponding to the log-contrasts between the compositional parts. One particular expression of these contrasts is called balances, and the aim of this note is to provide a step-by-step description of how to construct orthonormal balances and show how they can be applied to dietary data by means of an example based on data coming from an Italian population-based study. Nutrient balances are new variables that can be included as covariates in regression models. After fitting a linear regression model using estimates of relative fat mass as a response variable, scores indicating the obesogenic potential of foods can be defined on the basis of their nutritional contents and the estimated effect of the balances on relative fat mass. This approach to evaluating the relationship between diet and health outcomes provides insights into the roles of specific nutrient ratios and foods within a comprehensive and integrated framework that complies with the essentially compositional nature of dietary data.

On Algebraic-Geometric Methods for Constructing Submanifolds with Flat Normal Bundle and Holonomic Net of Curvature Lines

In this paper we propose a generalization of Krichever’s algebraic-geometric construction of orthogonal coordinate systems in a flat space. In the theory of integrable systems of hydrodynamic type a fundamental role is also played by orthogonal coordinates in some special nonflat spaces. The most important class of such spaces is given by metrics of submanifolds in flat spaces that have flat normal bundle and holonomic net of curvature lines, which defines orthogonal coordinates on the submanifold. We propose a method for constructing such submanifolds from algebraic-geometric data. Explicit examples are presented.

A NEW CONTRIBUTION IN STOCHASTIC HYDRODYNAMIC LUBRICATION FOR ARBITRARY BIO-SURFACES

This paper shows a recent progress described in curvilinear, orthogonal coordinates of the method of estimation of stochastic bio-hydrodynamic lubrication parameters. Here are discussed real arbitrary movable, non-rotational living biological surfaces coated with phospholipid bi-layers and lubricated with biological liquids. Non-rotational, curvilinear cooperating biological surfaces take the place in various biological nods for example in sacra bone, femoral bone, knee cap, calf bone and hip joint, elbow joint, knee joint, jump joint. Moreover are assumed biological non-rotational friction nods between human skin and tightly sport dress lubricated with the sweat. The main focus of the paper was to demonstrate the influence of expected values variations and standard deviation of the human joint gap height on the hydrodynamic lubrication parameters occurring during the friction process. It is very important to notice that the random gap height variations imply on the apparent dynamic viscosity of biological fluid or synovial fluid.