Invariant Method Research Articles

Scale Factor Estimation for Quadrotor Monocular-Vision Positioning Algorithms.

Unmanned aerial vehicle (UAV) autonomous navigation requires access to translational and rotational positions and velocities. Since there is no single sensor to measure all UAV states, it is necessary to fuse information from multiple sensors. This paper proposes a deterministic estimator to reconstruct the scale factor of the position determined by a simultaneous localization and mapping (SLAM) algorithm onboard a quadrotor UAV. The position scale factor is unknown when the SLAM algorithm relies on the information from a monocular camera. Only onboard sensor measurements can feed the estimator; thus, a deterministic observer is designed to rebuild the quadrotor translational velocity. The estimator and the observer are designed following the immersion and invariance method and use inertial and visual measurements. Lyapunov’s arguments prove the asymptotic convergence of observer and estimator errors to zero. The proposed estimator’s and observer’s performance is validated through numerical simulations using a physics-based simulator.

Research on the deep learning-based exposure invariant spectral reconstruction method.

The surface spectral reflectance of an object is the key factor for high-fidelity color reproduction and material analysis, and spectral acquisition is the basis of its applications. Based on the theoretical imaging model of a digital camera, the spectral reflectance of any pixels in the image can be obtained through spectral reconstruction technology. This technology can avoid the application limitations of spectral cameras in open scenarios and obtain high spatial resolution multispectral images. However, the current spectral reconstruction algorithms are sensitive to the exposure variant of the test images. That is, when the exposure of the test image is different from that of the training image, the reconstructed spectral curve of the test object will deviate from the real spectral to varying degrees, which will lead to the spectral data of the target object being accurately reconstructed. This article proposes an optimized method for spectral reconstruction based on data augmentation and attention mechanisms using the current deep learning-based spectral reconstruction framework. The proposed method is exposure invariant and will adapt to the open environment in which the light is easily changed and the illumination is non-uniform. Thus, the robustness and reconstruction accuracy of the spectral reconstruction model in practical applications are improved. The experiments show that the proposed method can accurately reconstruct the shape of the spectral reflectance curve of the test object under different test exposure levels. And the spectral reconstruction error of our method at different exposure levels is significantly lower than that of the existing methods, which verifies the proposed method's effectiveness and superiority.

On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.

A class of weighted energy-preserving Du Fort–Frankel difference schemes for solving sine-Gordon-type equations

Recently, invariant energy-quadratization methods (IEQMs) have been introduced by Xiaofeng Yang’s group to develop linear and energy-dissipation-preserving methods for nonlinear energy-dissipation systems. Following their work, two auxiliary functions are firstly introduced to rewrite the sine-Gordon equation (SGE) and coupled sine-Gordon equations (CSGEs) into equivalent systems, respectively. Then, two energy-preserving Du Fort–Frankel-type finite difference methods (EP-DFFT-FDMs) have been suggested for them, respectively. By using the discrete energy methods, the discrete energy conservative laws and convergence rates in the H1-norm have been derived, rigorously. It is worth mentioning that the proposed discrete energy is an approximation to the exact energy of the continuous problem. As hx=O(Δt), hy=O(Δt) and parameter λ≥1/4, the current methods are stable in the H1-norm because numerical solutions obtained by them are bounded in the H1-norm. What is more, as parameter λ≥1/4, the current methods are unconditionally stable in the L2-norm because numerical solutions obtained by them are uniformly bounded in the L2-norm. Moreover, our methods are explicit, and very easy to be implemented. However, a shortcoming of the current methods is that they are conditionally consistent. Namely, Δthx and Δthy tend to zero as time step Δt, spatial mesh sizes hx in x-direction and hy in y-direction tend to zero. Numerical findings support the correctness of theoretical analyses and the performance of the algorithms.

Ellipsoidal Design of Robust Stabilization for Markov Jump Power Systems under Normal and Contingency Conditions

The essential prerequisites for secure customer service are power system stability and reliability. This work shows how to construct a robust switching control for studying power system load changes using an invariant ellipsoid method. Furthermore, the suggested control ensures stability when the system is subjected to random stochastic external disturbances, and functions randomly in two conditions: normal and contingency. The extreme (least) reliability state is chosen as the most severe scenario (corresponding to a transmission line outage). As a two-state Markov random chain, the transition probabilities are utilized to simulate the switching between normal and contingency modes (or processes). To characterize the dynamics of the studied system, a stochastic mathematical model is developed. The effect of stochastic disturbances and random normal/contingency operations is taken into account during the design stage. For a stochastic power system, a novel excitation control is designed. The attractive ellipsoid approach and linear matrix inequalities (LMIs) optimization are used to build the best two-controller gains. Therefore, the proposed modeling/design technique can be employed for the power system under load changes, stochastic topological changes, and random disturbances. Finally, the system’s random dynamics simulation indicates the effectiveness of the designed control law.

Discrete Multiple Second-Order Generalized Integrator With Low-Pass Filters and Frequency-Locked Loop for DC Rejection

The multiple second-order generalized integrator (MSOGI), which is composed of parallel second-order generalized integrators (SOGI), is widely used in grid synchronization and signal extraction, and it usually suffers from dc offset issue. In continuous domain, low-pass filter (LPF) can be added to SOGI to eliminate the dc offset in its quadrature channel. However, in digital implementation, directly discretized LPF-based methods may not be able to eliminate dc offset and ac discretization error simultaneously, where the steady-state error will also be introduced by the conventional frequency-locked loop (FLL). In this article, a modified impulse invariant method is adopted to discretize SOGIs, which presents zero ac errors and good dynamics, but shows dc offset in both output channels. Then, an LPF is introduced to eliminate the dc offset of the discrete SOGI, forming a discrete SOGI-LPF. And, a discrete MSOGI-LPF (DMSOGI-LPF) is formed by paralleling multiple DSOGI-LPFs, which can extract different frequency components. Furthermore, a discrete FLL is proposed to estimate the frequency of the input signal, which suppresses the influence of the dc offset and achieves good dynamics. Finally, the stability analysis of the proposed method based on discrete linear-time periodic (LTP) model is provided. The effectiveness of the proposed method and its discrete LTP model are validated through experimental results.

Invariant subspace method for time-fractional nonlinear evolution equations of the third order

Invariant subspace method for time-fractional nonlinear evolution equations of the third order

C${\cal C}$osmological G${\cal G}$eometric P${\cal P}$hase F${\cal F}$rom P${\cal P}$ure Q${\cal Q}$uantum S${\cal S}$tates: A Study without/ with having Bell's Inequality Violation

AbstractIn this paper, using the concept of Lewis Riesenfeld invariant quantum operator method for finding continuous eigenvalues of quantum mechanical wave functions we derive the analytical expressions for the cosmological geometric phase, which is commonly identified to be the Pancharatnam Berry phase from primordial cosmological perturbation scenario. We compute this cosmological geometric phase from two possible physical situations, (1) In the absence of Bell's inequality violation and (2) In the presence of Bell's inequality violation having the contributions in the sub Hubble region (), super Hubble region () and at the horizon crossing point () for massless field (), partially massless field () and massive/heavy field (), in the background of quantum field theory of spatially flat quasi De Sitter geometry. The prime motivation for this work is to investigate the various unknown quantum mechanical features of primordial universe. To give the realistic interpretation of the derived theoretical results we express everything initially in terms of slowly varying conformal time dependent parameters, and then to connect with cosmological observation we further express the results in terms of cosmological observables, which are spectral index/tilt of scalar mode power spectrum () and tensor‐to‐scalar ratio (r). Finally, this identification helps us to provide the stringent numerical constraints on the Pancharatnam Berry phase, which confronts well with recent cosmological observation.

Dynamics of the free time-dependent effective mass

The consensus is that an object with a large mass will not manifest quantum behavior. Therefore, we expect that the quantumness of a time-dependent effective mass (TDEM) will erase after a long time when the mass profile continuously grows with time. However, the present article depicts that the Wigner quasi-probability distribution (WQD) will manifest an entanglement behavour forever for two spatially separated free TDEM. The time-dependent Schr\"{o}dinger equation for a free particle with TDEM is solved with the help of Lewis-Riesenfeld phase space invariant method. WQD for the system of two identical TDEM with quadratically increasing mass profile shows that the particles are never separated. In particular, their reminiscent is present at the origin of the phase-space forever.

A New Framework for Reconstructing Time Series DMSP-OLS Nighttime Light Data Using the Improved Stepwise Calibration (ISC) Method

Calibration and reconstruction of time series DMSP-OLS nighttime light images are critical for understanding urbanization processes and the evolution of urban spatial patterns from a unique perspective. In this study, we developed an improved stepwise calibration (ISC) method based on numerical constancy to correct and reconstruct the time series of China’s regional nighttime light data, thus eliminating the drawbacks of the invariant target region method. We evaluated the different calibration methods and quantitatively validated the calibrated nighttime light data using gross domestic product (GDP) and electricity consumption (EC) at municipal, provincial, and national scales. The results indicated that the ISC method demonstrated its advantage in screening stable lit pixels and maintaining the temporal variability of multi-year nighttime light variation. The variation curve of reconstructed multi-year nighttime light obtained by the ISC method based on numerical constancy was more consistent with the actual urban development. The ISC method retained the original data’s most abundant and complete information than other calibration methods. Moreover, the significant advantages of this method in the low-light high-variation regions and high-light low-variation regions offered new possibilities for understanding the development of small- and medium-sized nighttime light centers such as towns and villages from a nighttime light perspective. This is an advantage that other calibration methods do not offer. The correlation between the multi-year nighttime light dataset obtained by the ISC method and the socio-economic data was significantly improved. The correlation coefficients with GDP and EC are 0.9695 and 0.9923, respectively. Last but not least, the ISC method is more straightforward to implement. The new framework developed in this study produces a more accurate and reliable long time series nighttime light dataset and provides quality assurance for subsequent research in socio-economic development, urban development, natural disasters, and other fields.

An initial assessment of the sustainability of waterbird harvest in the United Kingdom

Abstract There is a need to assess the sustainability of wild bird harvest in the United Kingdom (UK), and more widely, across Europe. Yet, data on populations and harvest sizes are limited. We used a demographic invariant method (DIM) to estimate potential excess growth (PEG) for populations of UK wintering waterbirds and calculated a sustainable harvest index (SHI) for each. We compared this with population trends and conservation classifications (e.g. Birds of Conservation Concern [BoCC]) to assess the sustainability of harvests and the utility of these classifications. Our approach found evidence for potential overharvest of mallard Anas platyrhynchos, Eurasian teal Anas crecca, gadwall Mareca strepera, Canada geese Branta canadensis, greylag geese Anser anser and woodcock Scolopax rusticola. Whether DIM methods predict overharvest is highly dependent on estimates of maximum population growth rates inferring PEG. We found estimates of maximum population growth to be variable across a range of different methods. We found no relationship between SHI and short‐term wintering trends or conservation classification under the UK's BoCC framework. There was however a positive relationship between SHI and long‐term wintering trends. Policy implications. Our results suggest that UK‐based harvest is unlikely to be a major determinant of population trends for the majority of UK overwintering waterbirds, but harvest rates for some species may exceed that required to maintain stationary population growth. The lack of a relationship between conservation classifications and SHI strongly suggests that such conservation classifications are not an appropriate tool for making decisions about harvest management. Instead, our assessment provides the basis for a framework to make evidence‐based decisions on sustainable harvest levels in the face of incomplete data. There is currently no clear policy instrument in the UK to support such a framework via controls on either harvest effort or mortality of waterfowl. We believe such an instrument is urgently needed to ensure the UK can adapt to changing pressures and ensure the sustainable use of our wildlife populations.

In-medium effects in ϕ meson production in heavy-ion collisions from subthreshold to relativistic energies

We investigate the hidden strange $\phi$ meson production in heavy-ion collisions from subthreshold ($E_{kin}\approx 1$ A GeV) to relativistic ($E_{kin}\approx 21$ A TeV) energies as well as its coupling to the open strange mesons (kaons, antikaons) and their productions. Our study is based on the off-shell microscopic Parton-Hadron-String Dynamics (PHSD) transport approach which is applicable for the dynamical description of strongly interacting hadronic and partonic degrees-of-freedom created in heavy-ion collisions. Implementing novel meson-baryon and meson-hyperon production channels for $\phi$ mesons, calculated within a T-matrix coupled channel approach based on the extended SU(6) chiral effective Lagrangian model, along with the collisional broadening of the $\phi$-meson in-medium spectral function, we find a substantial enhancement of $\phi$ meson production in heavy-ion collisions, especially at sub- and near-thresholds. This allows to describe the experimentally observed strong enhancement of the $\phi/K^-$ ratio at low energies without including hypothetical decays of heavy baryonic resonances to $\phi$ as in alternative approaches. Moreover, we show that in spite of a stronger contribution from enhanced $\phi$ to $K^-$ production, the majority of the experimental data for different A+A systems at low energies favor the scenario with in-medium modifications of the kaon and antikaon properties in the hot and dense environment. Moreover, we study the influence of the final state interactions of $K, \bar K$ mesons on the reconstruction of $\phi$'s by the by invariant mass method.

Quantitative Study of the T-AVIM-Based Simulated IMU Error in Polar Regions.

For solving the problem of polar performance of the inertial navigation system (INS) at mid-low latitudes, the simulation test system constructed by the “attitude and velocity invariant method of trajectory transfer rule based on the transverse coordinate system (T-AVIM)” of the Earth sphere model is used. The test system structure, especially the IMU conversion formula from mid-low latitudes to polar region simulation test, is introduced, and it is proved that the IMU conversion error can be equivalently superimposed on the bias error of the polar simulated IMU. According to the marine estimation formula for the effect of the reference error on the IMU conversion error, the specific influence of the constant error component and the random error component of the reference system on the simulated IMU is analyzed. The calculation method of the simulated IMU error is given with examples and intuitively explained, and the correctness of the theory is verified through simulation experiments.

Lie symmetry analysis and exact solutions of time fractional Black–Scholes equation

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the constructive methods of exact solutions to time fractional Black–Scholes equation. By constructing one-parameter Lie symmetry transformations and their corresponding group generators, time fractional Black–Scholes equation is reduced to a fractional ordinary differential equation and some group-invariant solutions are obtained. Using the invariant subspace method, the analytical representations of two forms of exact solutions of time fractional Black–Scholes equation are given constructively, and the characteristics and differences of the two exact solutions are compared in the sense of geometric figures. In this paper, the form of the equation is generalized, and more group invariant solutions and analytical solutions in the form of separated variables are obtained.

Entanglement in interacting quenched two-body coupled oscillator system

In this work, we explore the effects of a quantum quench on the entanglement measures of a two-body coupled oscillator system having quartic interaction. We use the invariant operator method, under a perturbative framework, for computing the ground state of this system. We give the analytical expressions for the total and reduced density matrix of the system having non-Gaussian, quartic interaction terms. Using this reduced density matrix, we show the analytical calculation of two entanglement measures viz., Von Neumann entanglement entropy using replica trick and Renyi entanglement entropy. Further, we give a numerical estimate of these entanglement measures with respect to the dimensionless parameter $(t/\delta t$) and show its behaviour in the three regimes, i.e; late time behaviour, around the quench point and the early time behaviour. We comment on the variation of these entanglement measures for different orders of coupling strength. The variation of Renyi entropy of different orders has also been discussed.

Lie Symmetry Group, Invariant Subspace, and Conservation Law for the Time-Fractional Derivative Nonlinear Schrödinger Equation

In this paper, a time-fractional derivative nonlinear Schrödinger equation involving the Riemann–Liouville fractional derivative is investigated. We first perform a Lie symmetry analysis of this equation, and then derive the reduced equations under the admitted optimal-symmetry system. Moreover, with the invariant subspace method, several exact solutions for the equation and their figures are presented. Finally, the new conservation theorem is applied to construct the conservation laws of the equation.

Elastic analysis of irregularly or sparsely sampled curves.

We provide statistical analysis methods for samples of curves in two or more dimensions, where the image, but not the parameterization of the curves, is of interest and suitable alignment/registration is thus necessary. Examples are handwritten letters, movement paths, or object outlines. We focus in particular on the computation of (smooth) means and distances, allowing, for example, classification or clustering. Existing parameterization invariant analysis methods based on the elastic distance of the curves modulo parameterization, using the square-root-velocity framework, have limitations in common realistic settings where curves are irregularly and potentially sparsely observed. We propose using spline curves to model smooth or polygonal (Fréchet) means of open or closed curves with respect to the elastic distance and show identifiability of the spline model modulo parameterization. We further provide methods and algorithms to approximate the elastic distance for irregularly or sparsely observed curves, via interpreting them as polygons. We illustrate the usefulness of our methods on two datasets. The first application classifies irregularly sampled spirals drawn by Parkinson's patients and healthy controls, based on the elastic distance to a mean spiral curve computed using our approach. The second application clusters sparsely sampled GPS tracks based on the elastic distance and computes smooth cluster means to find new paths on the Tempelhof field in Berlin. All methods are implemented in the R-package "elasdics" and evaluated insimulations.

Novel, linear, decoupled and unconditionally energy stable numerical methods for the coupled Cahn–Hilliard equations

This paper uses a novel numerical approach to approximate the coupled Cahn–Hilliard equations, which are a highly nonlinear system depicting the phase separation of the homopolymer and copolymer mixtures. The new method is named 3S-IEQ, and its construction and calculation are more straightforward than the invariant energy quadratization and scalar auxiliary variable methods. Notably, we only need to solve two linear decoupled constant-coefficient equations at each time step. Numerical simulations are shown

An Analysis of Household Income in Poland and Slovakia Based on Selected Income Models

Abstract Research background: Studies of income distributions may focus on describing empirical distributions by theoretical models. This approach can be useful for many reasons. When choosing a statistical model, it is important both to find out a theoretical distribution function that would characterize empirical frequency distribution and to choose suitable methods to calculate the parameters of the model. In the literature, there are proposals for various types of mathematical functions. Very high accuracy with empirical distribution is characteristic for Dagum and log-normal models. In 2010 Zenga proposed a new three-parameter model for economic size distribution which possesses interesting statistical properties. Purpose: The aim of this paper is to apply and compare Dagum, log-normal and Zenga models to income distributions in Poland and Slovakia. Research methodology: The studies are based on research data from the European Quality of Life Surveys (EQLS), whose purpose is to measure both objective and subjective indicators of the standard of living of citizens and their households. The estimation parameters were obtained by means of the maximum likelihood method and D’Addario’s invariants method. Results: The results of the conducted approximations, presented in the paper confirmed the good consistency of the Dagum and Zenga distributions with the empirical income distribution of households in Poland and Slovakia. Novelty: The Zenga distribution was used for the first time to describe the income distribution of the Slovak population and it allows the best fit to the empirical data of this country.

Distributed Consensus Tracking Control of a Second-Order Nonlinear Multiagent System via Immersion and Invariance Method

In this article, a consensus tracking control problem is investigated for a second-order nonlinear multiagent system (MAS). Based on the immersion and invariance (I&I) theory, a distributed I&I consensus tracking control strategy is proposed with known dynamics first. On the basis of graph theory, an error system is established. The tracking problem of an MAS is transformed into a stabilizing issue. An implicit manifold is constructed, and an improved form of the off-the-manifold coordinate, with a regulating factor, is introduced. Further, a consensus tracking control problem for an MAS with unknown dynamics is studied. A distributed I&I consensus tracking control strategy is designed in combination with an adaptive I&I estimator. The error between the real value and its estimation is selected as an I&I implicit manifold, and an off-the-manifold coordinate is then defined. An adaptive update law is designed for guaranteeing the invariance and attractiveness of this manifold. Finally, two illustrative examples corroborate the effectiveness of the obtained theoretical results.