Vector Difference Research Articles

Accuracy Simulation Method Based on Assembly Process Information Mapping

Assembly process information has a direct impact on product manufacturing accuracy. In order to accurately predict product accuracy, a simulation method based on assembly process information mapping was proposed. The assembly deviation information was expressed as the in-part deviation reference, feature deviation Vector direction, feature deviation vector size, deviation transmission connection form between the parts and the deviation transmission direction, and so on. The mapping between the assembly process information and the deviation transmission information was established. Then the assembly accuracy simulation algorithm including the main algorithm and assembly deviation transfer model building algorithm was established. Taking C919 internal flap assembly as an example, the effectiveness of this method had been verified.

Sensitivity of the Eisenberg--Noe Clearing Vector to Individual Interbank Liabilities

We quantify the sensitivity of the Eisenberg--Noe clearing vector to estimation errors in the bilateral liabilities of a financial system. The interbank liabilities matrix is a crucial input to the computation of the clearing vector. However, in practice central bankers and regulators must often estimate this matrix because complete information on bilateral liabilities is rarely available. As a result, the clearing vector may suffer from estimation errors in the liabilities matrix. We quantify the clearing vector's sensitivity to such estimation errors and show that its directional derivatives are, like the clearing vector itself, solutions of fixed point equations. We describe estimation errors utilizing a basis for the space of matrices representing permissible perturbations and derive analytical solutions to the maximal deviations of the Eisenberg--Noe clearing vector. This allows us to compute upper bounds for the worst case perturbations of the clearing vector. Moreover, we quantify the probability of observing clearing vector deviations of a certain magnitude, for uniformly or normally distributed errors in the relative liability matrix. Applying our methodology to a dataset of European banks, we find that perturbations to the relative liabilities can result in economically sizeable differences that could lead to an underestimation of the risk of contagion. Our results are a first step towards allowing regulators to quantify errors in their simulations.

A transverse local feedback linearization approach to human head movement control

In the mid-nineteenth century, Donders had proposed that for every human head rotating away from the primary pointing direction, the rotational vectors in the direction of the corresponding axes of rotation, is restricted to lie on a surface. Donders’ intuition was that under such a restriction, the head orientation would be a function of its pointing direction. In this paper, we revisit Donders’ Law and show that indeed the proposed intuition is true for a restricted class of head-orientations satisfying a class of quadratic Donders’ surfaces, if the head points to a suitable neighborhood of the frontal pointing direction. Moreover, on a suitably chosen subspace of the 3D rotation group SO(3), we describe a head movement dynamical system with input control signals that are the three external torques on the head provided by muscles. Three output signals are also suitably chosen as follows. Two of the output signals are coordinates of the frontal pointing direction. The third signal measures deviation of the state vector from the Donders’ surface. We claim that the square system is locally feedback linearizable on the subspace chosen, and the linear dynamics is decomposed into parts, transverse and tangential to the Donders’ surface. We demonstrate our approach by synthesizing a tracking and path-following controller. Additionally, for different choices of the Donders’ surface parameters, head gaits are visualized by simulating different movement patterns of the head-top vector, as the head-pointing vector rotates around a circle.

Multivariable integrated evaluation of model performance with the vector field evaluation diagram

Abstract. This paper develops a multivariable integrated evaluation (MVIE) method to measure the overall performance of climate model in simulating multiple fields. The general idea of MVIE is to group various scalar fields into a vector field and compare the constructed vector field against the observed one using the vector field evaluation (VFE) diagram. The VFE diagram was devised based on the cosine relationship between three statistical quantities: root mean square length (RMSL) of a vector field, vector field similarity coefficient, and root mean square vector deviation (RMSVD). The three statistical quantities can reasonably represent the corresponding statistics between two multidimensional vector fields. Therefore, one can summarize the three statistics of multiple scalar fields using the VFE diagram and facilitate the intercomparison of model performance. The VFE diagram can illustrate how much the overall root mean square deviation of various fields is attributable to the differences in the root mean square value and how much is due to the poor pattern similarity. The MVIE method can be flexibly applied to full fields (including both the mean and anomaly) or anomaly fields depending on the application. We also propose a multivariable integrated evaluation index (MIEI) which takes the amplitude and pattern similarity of multiple scalar fields into account. The MIEI is expected to provide a more accurate evaluation of model performance in simulating multiple fields. The MIEI, VFE diagram, and commonly used statistical metrics for individual variables constitute a hierarchical evaluation methodology, which can provide a more comprehensive evaluation of model performance.

Special vortex in relativistic hydrodynamics

An exact solution of the Euler equations governing the flow of a compressible fluid in relativistic hydrodynamics is found and studied. It is a relativistic analogue of the Ovsyannikov vortex (special vortex) investigated earlier for classical gas dynamics. Solutions are partially invariant of Defect 1 and Rank 2 with respect to the rotation group. A theorem on the representation of the factor-system in the form of a union of a non-invariant subsystem for the function determining the deviation of the velocity vector from the meridian, and invariant subsystem for determination of thermodynamic parameters, the Lorentz factor and the radial velocity component is proved. Compatibility conditions for the overdetermined non-invariant subsystem are obtained.A stationary solution of this type is studied in detail. It is proved that its invariant subsystem reduces to an implicit differential equation. For this equation, the manifold of branching of solutions is investigated, and a set of singular points is found.

SBRT in the Post-Prostatectomy Setting: Analysis of Interfraction Variability

Our institution has a phase 1 clinical trial evaluating acute toxicity in the setting of SBRT to the prostate fossa following prostatectomy. Patients enrolled on this trial are simulated and treated with fiducial markers in the prostate fossa, a rectal balloon and a daily ultrasound (US) of the bladder for immobilization and target visualization. This study reports the interfraction motion analysis based on the described immobilization technique. At this point, the study has enrolled 12 patients onto dose levels 1 and 2 (3.6Gy/fraction x 15 fractions and 4.7Gy/fraction x 10 fractions, respectively). All patients had a rectal balloon for consistency in rectal volume and immobilization of the prostate bed. Patients were treated with a full bladder that was within 20% of the bladder volume on the day of CT simulation as verified by daily US. All patients were implanted with three gold fiducial markers into the prostate bed. KV and CBCT were performed on day 1 to verify fiducial placement and bladder and rectal volume. Daily kV and weekly CBCT were used for daily alignment based on fiducial markers. For 10 out of 12 patients, the patients’ shift data were gathered and a shift vector was calculated. The mean shifts for this cohort of the patients was -0.2 mm, 0.4 mm, -1.1 mm in anterior and posterior, superior and inferior, and left and right, respectively. The standard deviation was 4.0 mm, 3.7 mm, -3.4 mm in anterior and posterior, superior and inferior and left and right, respectively. It was demonstrated that the current bladder filling and IGRT protocol resulted in minimal systematic shift (small mean shifts). Most shifts were random which were reflected in the standard deviation. The mean and standard deviation of the shift vector was 5.3 mm and 3.9 mm, respectively. The calculated percentage bladder volume compared to bladder volume during the CT simulation had a mean of 3.6% and standard deviation of 8.9%. No correlation was found between magnitudes of patient shifts and bladder filling for this cohort of patients. With the current rectal/bladder immobilization and IGRT protocol, this cohort of the patients had a mean shift of no more than 1.1 mm but a relatively large standard deviation of 4.0 mm in each direction. Setup variation did not follow a specific pattern based on bladder or rectal filling. For the subset of the patients who have reasonable CBCT volumetric information, dosimetric analysis will be performed by deforming planning CT and dose to CBCT using VelocityTM.

Sensitivity of the Eisenberg-Noe Clearing Vector to Individual Interbank Liabilities

We quantify the sensitivity of the Eisenberg-Noe clearing vector to estimation errors in the bilateral liabilities of a financial system. The interbank liabilities matrix is a crucial input to the computation of the clearing vector. However, in practice central bankers and regulators must often estimate this matrix because complete information on bilateral liabilities is rarely available. As a result, the clearing vector may suffer from estimation errors in the liabilities matrix. We quantify the clearing vector's sensitivity to such estimation errors and show that its directional derivatives are, like the clearing vector itself, solutions of fixed point equations. We describe estimation errors utilizing a basis for the space of matrices representing permissible perturbations and derive analytical solutions to the maximal deviations of the Eisenberg-Noe clearing vector. This allows us to compute upper bounds for the worst case perturbations of the clearing vector. Moreover, we quantify the probability of observing clearing vector deviations of a certain magnitude, for uniformly or normally distributed errors in the relative liability matrix. Applying our methodology to a dataset of European banks, we find that perturbations to the relative liabilities can result in economically sizeable differences that could lead to an underestimation of the risk of contagion. Importantly, our results allow regulators to bound the error of their simulations.

Stability of the line preserving flows

We examine the equations that are used to describe flows which preserve field lines. We study what happens if we introduce perturbations to the governing equations. The stability of the line preserving flows in the case of the magneto-fluids permeated by magnetic fields is strictly connected to the non-null magnetic reconnection processes. In most of our study we use the Euler potential representation of the external magnetic field. We provide general expressions for the perturbations of the Euler potentials that describe the magnetic field. Similarly, we provide expressions for the case of steady flow as well as we obtain certain conditions required for the stability of the flow. In addition, for steady flows we formulate conditions under which the perturbations of the external field are negligible and the field may be described by its initial unperturbed form. Then we consider the flow equation that transforms quantities from the laboratory coordinate system to the related external field coordinate system. We introduce perturbations to the equation and obtain its simplified versions for the case of a steady flow. For a given system, use of this method allows us to simplify the considerations provided that some part of the system may be described as a perturbation. Next, to study regions favourable for the magnetic reconnection to occur we introduce a deviation vector to the basic line preserving flows condition equation. We provide expressions of the vector for some simplifying cases. This method allows us to examine if given perturbations either stabilise the system or induce magnetic reconnection. To illustrate some of our results we study two examples, namely a simple laboratory plasma flow and a simple planetary magnetosphere model.

Fixed-pattern noise correction method based on improved moment matching for a TDI CMOS image sensor.

In this paper, an improved moment matching method based on a spatial correlation filter (SCF) and bilateral filter (BF) is proposed to correct the fixed-pattern noise (FPN) of a time-delay-integration CMOS image sensor (TDI-CIS). First, the values of row FPN (RFPN) and column FPN (CFPN) are estimated and added to the original image through SCF and BF, respectively. Then the filtered image will be processed by an improved moment matching method with a moving window. Experimental results based on a 128-stage TDI-CIS show that, after correcting the FPN in the image captured under uniform illumination, the standard deviation of row mean vector (SDRMV) decreases from 5.6761 LSB to 0.1948 LSB, while the standard deviation of the column mean vector (SDCMV) decreases from 15.2005 LSB to 13.1949LSB. In addition, for different images captured by different TDI-CISs, the average decrease of SDRMV and SDCMV is 5.4922/2.0357 LSB, respectively. Comparative experimental results indicate that the proposed method can effectively correct the FPNs of different TDI-CISs while maintaining image details without any auxiliary equipment.

KCC theory and its application in a tumor growth model

In this study, we consider the stability of tumor model by using the standard differential geometric method that is known as Kosambi‐Cartan‐Chern (KCC) theory or Jacobi stability analysis. In the KCC theory, we describe the time evolution of tumor model in geometric terms. We obtain nonlinear connection, Berwald connection and KCC invariants. The second KCC invariant gives the Jacobi stability properties of tumor model. We found that the equilibrium points are Jacobi unstable for positive coordinates. We also discussed the time evolution of components of deviation tensor and the behavior of deviation vector near the equilibrium points.

Nasal Root Deviation in Unicoronal Craniosynostosis: A Craniometric Analysis of Early and Late Postoperative Outcomes.

Current operative techniques for correcting unicoronal craniosynostosis (UCS) leave the nasal bones untouched, resulting in an unclear long-term impact on nasal root deviation. The purpose of this study is to quantify nasal root deviation in the preoperative and late postoperative setting in patients who have undergone conventional single-staged UCS correction. The authors performed a retrospective, craniometric analysis of nasal root deviation comparing preoperative computed tomography scans, with those of the early, and late postoperative period. Three vectors were analyzed to measure nasal root deviation, one extending from the nasion to the rhinion (nasal bone vector), the second from the rhinion to the anterior nasal spine (nasal aperture vector), and the third from the nasion to the anterior nasal spine (nasal longitudinal vector). Twenty-five subjects were included in the study. Average ages at the time of preoperative, early, and late postoperative imaging were 0.6 ± 0.3, 0.9 ± 0.6, and 9.3 ± 2.7 years, respectively. Improvement of angular deviation of both the nasal aperture vector and nasal longitudinal vector was observed. Mean angular deviation of the nasal aperture vector was 6.0 ± 1.9 degrees preoperatively, 6.0 ± 2.1 degrees early postoperatively (P = 0.952), and 2.4 ± 2.1 in the late postoperative period (P = 0.013). Mean angular deviation of the nasal longitudinal vector was 5.7+2.0 degrees preoperatively, 5.8 ± 2.3 degrees early postoperatively (P = 0.948), and 3.7 ± 1.6 degrees in the late postoperative period (P = 0.019). Nasal root deviation decreased significantly only in the late postoperative period, lending credence to the notion that though UCS correction does not directly address nasal root deviation, this pathology improves significantly over time.

Tidal forces in Kiselev black hole

The aim of this paper is to examine the tidal forces occurring in a Kiselev black hole surrounded by radiation and dust fluids. It is noted that the radial and angular components of the tidal force change the sign between event and Cauchy horizons. We solve the geodesic deviation equation for radially free-falling bodies toward Kiselev black holes. We explain the geodesic deviation vector graphically and point out the location of the event and Cauchy horizons for specific values of the radiation and dust parameters.

Deviation and precession effects in the field of a weak gravitational wave

Deviation and precession effects of a bunch of spinning particles in the field of a weak gravitational plane wave are studied according to the Mathisson-Papapetrou-Dixon (MPD) model. Before the passage of the wave the particles are at rest with associated spin vector aligned along a given direction with constant magnitude. The interaction with the gravitational wave causes the particles to keep moving on the 2-plane orthogonal to the direction of propagation of the wave, with the transverse spin vector undergoing oscillations around the initial orientation. The transport equations for both the deviation vector an spin vector between two neighboring world lines of such a congruence are then solved by a suitable extension of the MPD model off the spinning particle's world line. In order obtain measurable physical quantities a "laboratory" has been set up by constructing a Fermi coordinate system attached to a reference world line. The {\it exact} transformation between TT coordinates and Fermi coordinates is derived too.

Fuzzy Adaptive Predictive Inverse for Nonlinear Transient Heat Transfer Process

For nonlinear transient heat transfer system, a fuzzy adaptive predictive inverse method (FAPIM) is proposed to inverse transient boundary heat flux. The influence relationship matrix is utilized to establish time-varying linear prediction model of the temperatures at measurement point. Then, the predictive and measurement temperatures are used to inverse the heat flux at current moment by rolling optimization. A decentralized fuzzy inference (DFI) mechanism is established. The deviation vector of the predictive temperature is adopted to conduct decentralized inference by a set of fuzzy inference units, and then, the influence relationship matrix is updated online to guarantee the adaptive ability of the prediction model by weighting fuzzy inference components. FAPIM is utilized to inverse the unknown heat flux of a heat transfer system with temperature-dependent thermal properties, which has shown that the inverse method has better adaptive ability for the inverse problems of nonlinear heat transfer system.

Jacobi stability analysis of modified Chua circuit system

In this paper, we analyze the nonlinear dynamics of the modified Chua circuit system from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. We reformulate the modified Chua circuit system as a set of two second-order nonlinear differential equations and obtain five KCC-invariants which express the intrinsic geometric properties. The deviation tensor and its eigenvalues are obtained, that determine the stability of the system. We also obtain the condition for Jacobi stability and discuss the behavior of deviation vector near equilibrium points.

Jacobi Stability Analysis of Rössler System

In this paper, Rössler system has been studied by using differential geometry method i.e. with KCC-theory. We obtained the deviation tensor and its eigenvalue which determine the stability of Rössler system. We also consider the time evolution of the components of deviation tensor and deviation vector near the equilibrium points.

Sensitivity analysis in inventory models by means of ergodicity coefficients

We investigate the sensitivity of a class of (s,S) inventory models with respect to the perturbation of the demand distribution in terms of ergodicity coefficients. Ergodicity coefficients can be regarded as matrix norms that are useful to study qualitative as well as quantitative properties of some classes of stochastic systems. We obtain estimations of the absolute deviation of the stationary vector of the underlying Markov chain subject to perturbations. The particular structure of the transition matrix of this class of models allows us to derive simple closed form formulae for the computation of the ergodicity coefficients as well as the perturbation bounds. The perturbation bounds obtained can help us to decide whether or not the model remains an acceptable representation of the real system and thus decide whether or not it can be trusted for real life applications. Under perturbation, e.g. estimation errors or approximations, an optimal solution for the mathematical model may not be optimal for the real system and if implemented, the real performance measures may deteriorate considerably and deviate from the targeted values. Numerical examples are given to illustrate how the perturbation of the demand distribution may have a considerable impact on the optimal inventory policy and the performance measures in some cases. In a practical setting, the understanding of the sensitivity results for inventory models can help us to identify the parameters that have to be estimated with most attention and how to build models that are robust and as close as possible to the real systems they represent.

Optimization of a Diaphragm for a Micro-Shock Tube-Based Drug Delivery Method.

This paper presents the design optimization of diaphragms for a micro-shock tube-based drug delivery device. The function of the diaphragm is to impart the required velocity and direction to the loosely held drug particles on the diaphragm through van der Waals interaction. The finite element model-based studies involved diaphragms made up of copper, brass and aluminium. The study of the influence of material and geometric parameters serves as a vital tool in optimizing the magnitude and direction of velocity distribution on the diaphragm surface. Experiments carried out using a micro-shock tube validate the final deformed shape of the diaphragms determined from the finite element simulation. The diaphragm yields a maximum velocity of 335 m/s for which the maximum deviation of the velocity vector is 0.62°. Drug particles that travel to the destination target tissue are simulated using the estimated velocity distribution and angular deviation. Further, a theoretical model of penetration helps in the prediction of the drug particle penetration in the skin tissue like a target, which is found to be 0.126 mm. The design and calibration procedure of a micro-shock tube device to alter drug particle penetration considering the skin thickness and property are presented.

Geometric characteristics of quantum evolution: curvature and torsion

We study characteristics of quantum evolution which can be called curvature and torsion. The curvature shows a deviation of the state vector in quantum evolution from the geodesic line. The torsion shows a deviation of state vector from the plane of evolution (a two-dimensional subspace) at a given time.

A Nonlinear H-Infinity Approach to Optimal Control of PEM Fuel Cells

A new nonlinear H-infinity control approach is applied to PEM fuel cells. First, the dynamic model of the PEM fuel cells undergoes approximate linearisation, through Taylor series expansion, round local operating points which are defined at each time instant by the present value of the system's state vector and the last value of the control input that was exerted on it. The linearisation procedure requires the computation of Jacobian matrices at the aforementioned operating points. The modelling error, which is due to the truncation of higher order terms in the Taylor series expansion is perceived as a perturbation that should be compensated by the robustness of the control loop. Next, for the linearized equivalent model of the PEM fuel cells, an H-infinity feedback control loop is designed. This approach, is based on the concept of a differential game that takes place between the control input (which tries to minimize the deviation of the state vector from the reference setpoints) and the disturbance input (that tries to maximize it). In such a case, the computation of the optimal control input requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. The known robustness properties of H-infinity control enable compensation of model uncertainty and rejection of the perturbation terms that affect the PEM fuel cells system. The stability of the control loop is proven through Lyapunov analysis. Actually, it is shown that H-infinity tracking performance is succeeded, while conditionally the asymptotic stability of the control loop is also demonstrated. The efficiency of the proposed control scheme for the PEM fuel cells model is further confirmed through simulation experiments.