Invariant Approach Research Articles

Invariance and Ibragimov approach with Lie algebra of a nonlinear coupled elastic wave system

The propagation of elastic waves in hyperelastic materials is described by a nonlinear system of partial differential equations (PDEs) governing the material’s motion. Hyperelastic materials are characterized by a strain–energy density function that correlates material deformation with stored elastic energy. For simplicity, we focus on the one-dimensional case where the displacement field, denoted as ϕ(x,t), signifies material deformation along the x-direction at position x and time t. The governing equation for elastic wave propagation in hyperelastic materials is derived from the strain–energy density function and associated stress–strain relations. In this study, Lie symmetry analysis is used to examine a nonlinear system of such equations relevant to the propagation of elastic waves in hyperelastic materials. The resulting equations are solved by applying Lie’s invariance criterion, yielding a ten-dimensional Lie algebra. An optimal system is derived from this algebra, allowing for the identification of invariant solutions under certain conditions. Additionally, the multiplier approach and Ibragimov’s new conservation laws are utilized to obtain the conservation laws for this coupled system of elastic waves. The outcome presented here is innovative and suggests utilizing the Lie symmetry method for investigating hyperelastic materials.

Invariant approach to the driven Jaynes-Cummings model

We investigate the dynamics of the driven Jaynes-Cummings model, where a two-level atom interacts with a quantized field and both, atom and field, are driven by an external classical field. Via an invariant approach, we are able to transform the corresponding Hamiltonian into the one of the standard Jaynes-Cummings model. Subsequently, the exact analytical solution of the Schrödinger equation for the driven system is obtained and employed to analyze some of its dynamical variables.

A Note on Invariant Description of SU(2)–Structures in Dimension 5

We develop an invariant approach to SU(2)–structures on spin 5–manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to SU(2). Moreover, we show how to induce quaternionic structure on the contact distribution of the considered SU(2)–structure. We show the invariance of certain components of the covariant derivative ∇φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla \\varphi $$\\end{document}, where φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} is any spinor field defining SU(2)–structure. This shows, as expected, that (at least some of) the intrinsic torsion modules can be derived invariantly with the spinorial approach. We conclude with the explicit description of the intrinsic torsion and the characteristic connection.

Towards invariant extended Kalman filter-based resilient distributed state estimation for moving robots over mobile sensor networks under deception attacks

This paper studies the resilient distributed state estimation problem for moving robots over wireless sensor networks (WSNs) under deception attacks. The state of a moving robot consists of its position and orientation, evolving according to a nonlinear motion model, and the WSNs are mobile. These settings make the problem more challenging than most existing researches on the resilient distributed state estimation which recover stationary or linear states by static WSNs. To achieve the resilient distributed state estimation for the robot over a mobile sensor network (MSN), we propose a novel invariant extended Kalman filter-based resilient distributed state estimation approach. Specifically, based on the mappings between a Lie group and its Lie algebra, we first define a new type of estimation errors and linearize it. Utilizing the linearization results, the calculation of the current prior estimate based on the last posterior one is achieved by each sensor, which avoids the estimation biases induced by the computation of Jacobian matrices. Then, sensors in the MSN fuse the local prior estimates of their neighbors according to the covariance intersection fusion rule, obtaining more accurate estimates than their local prior ones. Finally, measurements collected by the sensors are used to correct the estimates. In the correction process, an adaptive gain is designed for each sensor to restrict the negative effects of the attacks on the estimates, which is crucial for the accomplishment of the resilient estimation task. We illustrate the performance of the approach by convergence analysis of the estimation errors based on the boundedness of the covariance matrix estimates computed in the state estimation process and verify it by simulation and experimental results.

Modeling 3D transverse elasto-plastic damage of unidirectional fiber-reinforced polymer composites using a smeared crack approach

This work presents a novel formulation of a 3D smeared crack model for unidirectional fiber-reinforced polymer composites based on a stress invariant approach for transverse yielding and failure initiation, and on a continuum damage approach. This formulation is developed to facilitate the implementation in an implicit solver, increasing solution robustness and computational efficiency in quasi-static and long duration analyses. The performance of the model is evaluated using monotonic and non-monotonic damage evolution, verified with single element tests to demonstrate the consistency of the proposed formulation. Additional benchmark examples regarding off-axis tension and compression tests are simulated and compared with the experimental results, showing good agreement for the plastic response, failure load and failure strain.

A contact formulation using local frame of SE(3) group for corotational beam against rigid solid via moving signed distance field

In this paper, we present a contact formulation for corotational beam and rigid body based on body-attached signed distance field (SDF). All formulations are based on the unified SE(3) framework, whose prominent advantage lies in its frame invariance property. For solid body, the geometry surface can be implicitly described by a moving SDF. Consequently, the interaction between beams and rigid solids can be viewed as immersion in a field-described force. The contact force is formulated using SDF interpolation, and exact tangent matrices are provided to facilitate implicit iteration. Notably, this approach eliminates the need for contact detection during simulation, because the SDF database is constructed before the simulation. To enhance the spatial adaptivity of the SDF algorithm, we introduce an octree refinement strategy. Besides, we incorporate the concept of contact potential volume into the subdivided stopping criterion. Compared to the uniform SDF, this approach demonstrates higher generation speed and smaller errors. Several examples are used to verify the accuracy, convergence and efficiency of the proposed formulation. It is observed that the invariance property of SE(3) description leads to enhanced computational performance. Both frame invariance and SDF-based approach are expected to improve efficiency for large-scale problems. The presented method is easily applicable to other types of flexible bodies.

Saturated control for uncertain nonlinear impulsive systems with non-uniformly distributed packet loss

This article aims to provide a unified theoretical framework for studying the dynamic behavior of uncertain impulsive systems involving saturation structure and packet loss phenomenon. The concept named average packet loss interval (APLI) is developed to characterize the distribution of packet loss intervals. In this scenario, the traditional restrictions on the number, the proportion, or the upper bound of packet loss intervals can be significantly relaxed. By modeling the original systems as the asynchronous impulsive switched systems, some sufficient criteria are derived via the set invariant approach, addressing how to integrate the effects of impulsive and external disturbances into input-to-state stability (ISS) performance. It is also shown that the robustness of system performance against packet loss can be guaranteed ideally. Then, an optimization algorithm is designed with the hope of getting the maximum estimate of the domain of initial values. Finally, two examples are discussed to demonstrate the effectiveness of our proposed results.

Large Order-Invariant Bayesian VARs with Stochastic Volatility

Many popular specifications for Vector Autoregressions (VARs) with multivariate stochastic volatility are not invariant to the way the variables are ordered due to the use of a lower triangular parameterization of the error covariance matrix. We show that the order invariance problem in existing approaches is likely to become more serious in large VARs. We propose the use of a specification which avoids the use of this lower triangular parameterization. We show that the presence of multivariate stochastic volatility allows for identification of the proposed model and prove that it is invariant to ordering. We develop a Markov chain Monte Carlo algorithm which allows for Bayesian estimation and prediction. In exercises involving artificial and real macroeconomic data, we demonstrate that the choice of variable ordering can have non-negligible effects on empirical results when using the nonorder invariant approach. In a macroeconomic forecasting exercise involving VARs with 20 variables we find that our order-invariant approach leads to the best forecasts and that some choices of variable ordering can lead to poor forecasts using a conventional, non-order invariant, approach.

A generalized data envelopment analysis approach for fixed cost allocation with preference information

Exploring a commonly acceptable fixed cost allocation (FCA) scheme is a challenging yet significant task, since one may contend with conflicting objectives. In this paper, we propose a new data envelopment analysis based FCA technique. The proposed approach aims to balance individual efficiency guarantees and collective preference objectives simultaneously. Particularly, a preference value loss (PVL) index is explicitly constructed to quantify the impact of preference considerations. Moreover, our generalized FCA strategy guarantees a minimal PVL and provides a reasonable and realizable lexicographic evaluation procedure to single out the final FCA scheme. Interestingly, the efficiency invariance and efficiency variance approaches from the prior literature are special cases of our proposed approach. Finally, the proposed approach is validated with a numerical example and an empirical application to bank maintenance fee distribution.

The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems

We consider the FitzHugh-Nagumo system equipped with boundary condition of Dirichlet type on some two/three-dimensional domains. This system describes the signal transmission across the axonal membrane in neurophysiology. It is a semilinear parabolic PDE for the voltage variable coupled with a first-order ODE of space-time type for the recovery variable. We prove that there exists a finite-dimensional global manifold in the case of the fast recovery variable. Since the manifold is uniformly attracting, it gives geometric insight into the global long-time dynamics of the solutions. The proof is based on an abstract invariant manifold theorem for dynamical systems on a Banach space.

An efficient linearly implicit and energy‐conservative scheme for two dimensional Klein–Gordon–Schrödinger equations

AbstractThe Klein–Gordon–Schrödinger equations describe a classical model of interaction of nucleon field with meson field in physics, how to design the energy conservative and stable schemes is an important issue. This paper aims to develop a linearized energy‐preserve, unconditionally stable and efficient scheme for Klein–Gordon–Schrödinger equations. Some auxiliary variables are utilized to circumvent the imaginary functions of Klein–Gordon–Schrödinger equations, and transform the original system into its real formulation. Based on the invariant energy quadratization approach, an equivalent system is deduced by introducing a Lagrange multiplier. Then the efficient and unconditionally stable scheme is designed to discretize the deduced equivalent system. A numerical analysis of the proposed scheme is presented to illustrate its uniquely solvability and convergence. Numerical examples are provided to validate accuracy, energy and mass conservation laws, and stability of our proposed method.

The modeling and identification of walking beam type slab reheating furnace based on immersion and invariance disturbance estimation

This paper presents the modeling and identification of a walking-beam-type slab reheating furnace in one of the biggest Iron and Steel Industry in Turkey. Firstly, the dynamical model of the furnace is established as it is running with 12 control zones based on energy balance equations in the zones. Secondly, a disturbance observer is designed to estimate the disturbances that represents unmeasurable energy losses and the effects of unknown dynamics of the system based on Immersion and Invariance (I&I) approach. Then, model is identified with real data collected from the running furnace. The model and real furnace outputs are compared zone by zone. The results demonstrate the validity of the model.

Symmetries, associated first integrals and successive reduction of Schrödinger type and other second order difference equations

In this article, we perform an invariance analysis of a class of the nonlinear Schrödinger second order difference equation where the discrete version is obtained by some discretization procedure. Firstly, the invariance approach adopted is the Lie symmetry one and, thereafter, we proceed to construct first integrals of the discrete equation. Furthermore, we check if there exists an ‘association’ between the symmetries and first integrals. Reductions to obtain solutions are then presented.

Neutrino phenomenology, W -mass anomaly, and muon ( g−2 ) in a minimal type-III seesaw model using a T′ modular symmetry

In this study, we attempt to introduce a model to illustrate neutrino phenomenology by incorporating two right-handed fermion triplet superfields, i.e., $\Sigma_{R_j}$, in the presence of the modular symmetry $\Gamma_3^\prime \simeq A_4^\prime$, a double cover of the $A_4$ modular symmetry. The motivation in utilizing double cover is, so far only even modular forms were considered for constructing modular invariant models, but, in this case, it is possible to extend the modular invariance approach to general integral weight modular forms, i.e., the odd weight modular forms. Hence, this type of amalgamation between $T^\prime$ modular symmetry and minimally extending the seesaw can correctly explain the neutrino phenomenology. Additionally, we have made an attempt to accommodate the most recent measurement of the $W$ boson mass, published by the CDF-II collaboration and shed some light on the recent results of muon $(g-2)$. Finally, we have discussed lepton flavor violation in order to establish a constraint on the mass of right-handed fermion.

Dynamics of the Cl- + CH3I reaction on a high-level abinitio analytical potential energy surface.

We have developed a full-dimensional analytical abinitio potential energy surface (PES) for the Cl- + CH3I reaction using the Robosurfer program system. The energy points have been computed using a robust composite method defined as CCSD-F12b + BCCD(T) - BCCD with the aug-cc-pVTZ(-PP) basis set and have been fitted by the permutationally invariant polynomial approach. Quasi-classical trajectory simulations on the new PES reveal that two product channels are open in the collision energy (Ecoll) range of 1-80 kcal/mol, i.e., SN2 leading to I- + CH3Cl and iodine abstraction (above ∼45 kcal/mol) resulting in ICl- + CH3. Scattering angle, initial attack angle, product translational energy, and product internal energy distributions show that the SN2 reaction is indirect at low Ecoll and becomes direct-rebound-back-side (CH3-side) attack-type, as Ecoll increases. Iodine abstraction mainly proceeds with direct stripping mechanism with side-on/back-side attack preference. Comparison with crossed-beam experiments and previous direct dynamics simulations shows quantitative or qualitative agreement and also highlights possible theoretical and/or experimental issues motivating further research.

Why Full, Partial, or Approximate Measurement Invariance Are Not a Prerequisite for Meaningful and Valid Group Comparisons

It is frequently stated in the literature that measurement invariance is a prerequisite for the comparison of group means or standard deviations of the latent variable in factor models. This article argues that measurement invariance is not necessary for meaningful and valid comparisons across groups. There is unavoidable ambiguity in how researchers can define comparisons if measurement invariance is violated. Moreover, there is no support for preferring the partial invariance approach over competing approaches, such as invariance alignment, robust linking, or Bayesian approximate invariance. Furthermore, we also argue why an intentionally misspecified multiple-group factor model with invariant item parameters can be justified if measurement invariance is violated.

Existence proofs for strange attractors in piecewise-smooth nonlinear Lozi-Hénon and Belykh maps

ABSTRACT Extensive research has previously been carried out on the existence of strange attractors in 2D piecewise linear maps, including the renowned Lozi map. However, the rigorous analysis of strange attractors in 2D nonlinear maps remains an underdeveloped area of study. In this paper, we introduce a 2D map with a single piecewise-smooth nonlinear function that is suitable for analytic studies of its strange attractor. This 2D piecewise-smooth nonlinear map represents a broad range of chaotic maps, including a hybrid Lozi-Hénon map and the Belykh map. To prove the existence of a strange attractor in the 2D map, we (i) construct its trapping region that contains all limit sets and (ii) demonstrate that the invariant set's trajectories have negative and positive Lyapunov exponents. We develop an invariant cone approach to establish the latter property, which involves constructing expanding and contracting cones as bounds for the eigenvectors of the variational equations along the chaotic trajectories. We apply our approach to analyze the chaotic hybrid Lozi-Hénon map, the original Lozi map, and the Belykh map.

Robust model predictive tracking control for the wheeled mobile robot with boundary uncertain based on linear matrix inequalities

AbstractIn this study, a robust model predictive controller is designed for the trajectory tracking problem of non‐holonomic constrained wheeled mobile robot based on an elliptic invariant set approach. The controller is based on a time‐varying error model of robot kinematics and uses linear matrix inequalities to solve the robust tracking problem taking uncertainties into account. The uncertainties are modelled by linear fractional transform form to contain both parameter perturbations and external disturbances. The control strategy consists of a feedforward term that drives the centre of the ellipse to the reference point and a feedback term that converges the uncertain system state error to the equilibrium point. The strategy stabilises the nominal system and ensures that all states of the uncertain system remain within the ellipsoid at each step, thus achieving robust stability of the uncertain system. Finally, the robustness of the algorithm and its resistance to disturbances are verified by simulation and experiment.

Edmonds' problem and the membership problem for orbit semigroups of quiver representations

A central problem in algebraic complexity, posed by J. Edmonds [16] , asks to decide if the span of a given l -tuple W = ( W 1 , … , W l ) of N × N complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing W as a representation of the l -Kronecker quiver K l , Edmonds' problem can be rephrased as asking to decide if there exists a semi-invariant on the representation space ( C N × N ) l of weight ( 1 , − 1 ) that does not vanish at W . In other words, Edmonds' problem is asking to decide if the weight ( 1 , − 1 ) belongs to the orbit semigroup of W . Let Q be an arbitrary acyclic quiver and W a representation of Q . We study the membership problem for the orbit semi-group of W by focusing on the so-called W -saturated weights. We first show that for any given W -saturated weight σ , checking if σ belongs to the orbit semigroup of W can be done in deterministic polynomial time. Next, let ( Q , R ) be an acyclic bound quiver with bound quiver algebra A = K Q / 〈 R 〉 and assume that W satisfies the relations in R . We show that if A / Ann A ( W ) is a tame algebra then any weight σ in the weight semigroup of W is W -saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.

Learning rotation equivalent scene representation from instance-level semantics: A novel top-down perspective

This paper focuses on rotation variant scene recognition. Different from existing rotation invariant recognition approaches which learn from either rotated images or rotated convolutional filters in a bottom-up manner, a new top-down perspective by learning is explored from instance-level semantic representation. The goal is to eliminate the convolutional feature differences in bottom-up feature propagation caused by the rotation sensitive nature of convolution operation. Our rotation equivalent convolutional neural network (RE-CNN) scheme consists of three components. Firstly, our key instance selection module highlights the instances strongly related to the scene scheme regardless of their orientation. Secondly, our key instance aggregation module builds a scene representation invariant to the position change of each instance caused by rotation. Finally, our semantic fusion module allows the framework to be organized as a whole and implements rotation regularization. Notably, our RE-CNN scheme can be adapted to existing CNNs in a plug-in-and-play manner. Extensive experiments on rotation variant scene recognition benchmarks from four domains demonstrate the state-of-the-art performance and generalization capability of the proposed RE-CNN.