Narrow Results

This paper proposes a new method to improve accuracy and real-time performance of inertial joint angle estimation for upper limb rehabilitation applications by modeling body acceleration and adding low-cost markerless optical position sensors. A method based on a combination of the 3D rigid body kinematic equations and Denavit-Hartenberg (DH) convention is used to model body acceleration. Using this model, body acceleration measurements of the accelerometer are utilized to increase linearization order and compensate for body acceleration perturbations. To correct for the sensor-to-segment misalignment of the inertial sensors, position measurements of a low-cost markerless position sensor are used. Joint angles are estimated by Extended Kalman Filter (EKF) and compared with Unscented Kalman Filter (UKF) in terms of performance. Simulations are performed to quantify the existing error and potential improvements achievable by the proposed method. Experiments on a human test subject performing an upper limb rehabilitation task is used to validate the simulation results in realistic conditions.

In this paper, to obtain accurate numerical solutions of coupled nonlinear Schrödinger–Korteweg-de Vries (KdV) equations a Haar wavelet collocation method is proposed. An explicit time stepping scheme is used for discretization of time derivatives and nonlinear terms that appeared in the equations are linearized by a linearization technique and space derivatives are discretized by Haar wavelets. In order to test the accuracy and reliability of the proposed method $L_2$, $L_{\infty }$ error norms and conserved quantities are used. Also obtained results are compared with previous ones obtained by finite element method, Crank–Nicolson method and radial basis function meshless methods. Error analysis of Haar wavelets is also given.

We study topological properties of Ind-groups $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ and $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of $\mathrm{\text{Aut}}(\mathrm{\text{Aut}}(A))$, where $A$ is the polynomial or free associative algebra over the base field $K$. We prove that all Ind-scheme automorphisms of $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ are inner for $n\ge 3$, and all Ind-scheme automorphisms of $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ are semi-inner. As an application, we prove that $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ cannot be embedded into $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov–Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.

Digital recognitions are playing a vital function in the current era of technological advancements. Hence, they offer more possible ways of performing handwritten character recognition (HCR). Generally, recognizing the Tamil handwritten texts is highly complicated, in comparison to the Western scripts. Nevertheless, many researchers have presented several real-time approaches to achieve Tamil character recognition (TCR) in offline mode. This paper introduces a new handwritten TCR (HTCR) approach with two phases: (1) pre-processing and (2) classification. Primarily, the scanned document in the Tamil language is pre-processed via the steps like RGB to grayscale conversion, binarization with thresholding, image complementation, application of morphological operations and linearization. The pre-processed images are then classified using an optimized convolutional neural network (CNN) model. Further, the fully connected layer (FCL) and the weights are tuned optimally via a new sea lion with self-adaptiveness (SL-SA) algorithm. Lastly, the adopted model is evaluated using various measures to prove its supremacy over the existing schemes.

The dynamics of nonlinear reaction–diffusion systems is dominated by the onset of patterns, and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlevé singularity structure analysis singles out a special case (m=2) as integrable. More interestingly, a Bäcklund transformation is shown to give rise to a linearizing transformation for the integrable case. A Lie symmetry analysis again separates out the same m=2 case as the integrable one and hence we report several physically interesting solutions via similarity reductions. Thus we give a group theoretical interpretation for the system under study. Explicit and numerical solutions for specific cases of nonintegrable systems are also given. In particular, the system is found to exhibit different types of traveling wave solutions and patterns, static structures and localized structures. Besides the Lie symmetry analysis, nonclassical and generalized conditional symmetry analysis are also carried out.

Shape Memory Alloys (SMAs) are now widely used as a damping element into the isolation systems. The pre-stressed SMAs exhibit hysteretic damping through a nonlinear flag-shaped hysteresis loop. Many nonlinear models of the SMA are available to depict such behavior. The nonlinear models require a lot of effort and computational time for the analysis of base-isolated structures. Therefore, the codes recommend that a nonlinear model can be replaced by an equivalent linear model in the analysis. Linearization is a method to convert the nonlinearity of a system into a system with analogues linear parameters. This paper proposes an empirical equation for a damping ratio to get a linear damping coefficient of the SMAs which can be used in the seismic analysis of base-isolated structures.

The evaluation of any damping ratio using the traditional system identification method does not give precise solutions due to variation in hysteretic parameters and the unpredictable nature of an earthquake. The empirical equation is proposed using a set of optimal statistical data obtained from the seismic analysis of a base isolated structure. Moreover, analysis of the base isolated structure using the newly modified equivalent elastic-viscous SMA model gives comparable and conservative results with a nonlinear SMA model as compared to the existing elastic-viscous SMA model. Since the hysteresis parameters are used to derive the empirical equation for the damping ratio, this equation is also applicable for any type of structure.

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalent of the Lie conditions for such systems. We explicitly solve this branch of the linearization problem by point transformations in the case of a square system of two second-order ODEs. Necessary and sufficient conditions for linearization to the simplest system by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a rederivation of Lie's linearization conditions for a single second-order ODE and sheds light on more recent results for them. In particular we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover these results contain the quadratic case as a special case. Examples are given to illustrate our results.

We study the class of the ordinary differential equations of the form ẍ + a₂(t, x)ẋ² + a₁(t, x)ẋ + a₀(t, x) = 0, that admit v = ∂_x as λ-symmetry for some λ = α(t, x)ẋ + β(t, x). This class coincides with the class of the second-order equations that have first integrals of the form C(t) + 1/(A(t, x)ẋ + B(t, x)). We provide a method to calculate the functions A, B and C that define the first integral. Some relationships with the class of equations linearizable by local and a specific type of nonlocal transformations are also presented.

Nonlinear behavior is present in the operating conditions of many mechanical systems, especially if nonsmall oscillations are considered. In these cases, in order to improve vibration control performance, a common engineering practice is to design the control system on a set of linearized models, for given operating conditions. The well-known gain-scheduling technique allows the parameters of the control law to be changed according to the current working condition, also increasing system stability. However, more recently new control logics directly applicable to the systems in nonlinear form have been developed. The aim of this paper is to study, both numerically and experimentally, the dynamic of a mechanical system (a 3-link flexible manipulator) comparing the performance of a fully nonlinear control (the sliding-mode control) and a standard linearized approach.

In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.

This contribution is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers [1–5] in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas and are presented from a new viewpoint.

This paper considers the design of robust logarithmic $\mu$-law companding quantizers for the use in analog-to-digital converters (ADCs) in communication system receivers. The quantizers are designed for signals with the Gaussian distribution, since signals at the receivers of communication systems can be very well modeled by this type of distribution. Furthermore, linearization of the logarithmic $\mu$-law companding function is performed to simplify hardware implementation of the quantizers. In order to reduce energy consumption, low-resolution quantizers are considered (up to 5 bits per sample). The main advantage of these quantizers is high robustness — they can provide approximately constant SNR in a wide range of signal power (this is very important since the signal power at receivers can vary in wide range, due to fading and other transmission effects). Using the logarithmic $\mu$-law companding quantizers there is no need for using automatic gain control (AGC), which reduces the implementation complexity and increases the speed of the ADCs due to the absence of AGC delay. Numerical results show that the proposed model achieves good performances, better than a uniform quantizer, especially in a wide range of signal power. The proposed low-bit ADCs can be used in MIMO and 5G massive MIMO systems, where due to very high operating frequencies and a large number of receiving channels (and consequently a large number of ADCs), the reduction of ADC complexity and energy consumption becomes a significant goal.

We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of 𝕂^k, where 𝕂 is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two- and three-dimensional classes of polynomial or rational maps. In particular, we find the global periodic cases for several Lyness-type recurrences.

We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover, we provide a construction of minimal linear representations for the inverse of nonzero elements.

The high field limit for the semiconductor Boltzmann equation with Pauli exclusion terms is investigated. The limit problem is shown to have a unique solution for every given density. The proof relies on a linearization procedure together with a continuation argument. The density is finally proven to converge in the high field limit towards the solution of a nonlinear hyperbolic equation.

Necessary and sufficient conditions which allow a second-order stochastic ordinary differential equation to be transformed to linear form are presented. The transformation can be chosen in a way so that all but one of the coefficients in the stochastic integral part vanish. The linearization criteria thus obtained are used to determine the general form of a linearizable Langevin equation.

A linearized system of hydrodynamics for ideal compressible fluids is considered. It is shown that in the case when the sound speed in the medium is equal to zero the characteristics change their multiplicity. For this case the asymptotic solution of the Cauchy problem with high frequency initial data is constructed. We prove that in this case the linearized system is unstable with respect to the high frequency perturbations.

Lie’s method of converting a nonlinear second order scalar ordinary differential equation (ODE) to linear form with point transformations has already been extended to higher order ODEs and systems of ODEs. For 2^nd order linearizable ODEs a unique equivalence class (with 8 infinitesimal symmetry generators) exists, whereas for the third order there are three classes with 4, 5 and 7 generators. For 2-d systems of second order ODEs there are five classes with 5, 6, 7, 8 and 15 dimensional Lie point symmetry algebras. A complex procedure (explained in §2) has been adopted to linearize a class of 2-d systems of second order ODEs, which is shown to possess 6, 7 and 15 dimensional algebras. Here we use the complex procedure for the symmetry group classification of those 2-d systems of 3^rd order ODEs that correspond to scalar linearizable complex third order ODEs. Five equivalence classes of such systems with Lie algebras of dimension 8, 9, 10, 11 and 13, are proved to exist.