Narrow Results

In this paper, triality refers to the $S_3$-symmetry of the language of quasigroups, which is related to, but distinct from, the notion of triality as the $S_3$-symmetry of the Dynkin diagram $D_4$. The paper investigates a homogeneous method for rendering the linearization of quasigroups (over a commutative ring) naturally invariant under the action of the triality group, on the basis of an appropriate algebra generated by three invertible, non-commuting coefficient variables that is isomorphic to the group algebra of the free group on two generators. The algebra has a natural quotient given by setting the square of each generating variable to be $-1$. The quotient is an algebra of quaternions over the underlying ring, in a way reminiscent of how symmetric groups appear as quotients of braid groups on declaring the generators to be involutions. The corresponding quasigroups (which are described as quaternionic) are characterized by three equivalent pairs of quasigroup identities, permuted by the triality symmetry. The three pairs of identities are logically independent of each other. Totally symmetric quasigroups (such as Steiner triple systems) are quaternionic.

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.

The considered problem in this paper is a 0-1 Multi-Quadratic Fractional Programming Problem (0-1MQFP) without the restriction of positive denominators. The two important contributions of the paper are: (a) it proposes a linearization technique to solve any type of 0-1MQFP problem; and (b) when applied to the 0-1MQFP problem with the restriction of positive denominators, it requires less number of constraints and variables as compare with the available techniques in the literature. The problem is proved to be NP-hard. The linearization process is summarized with the help of an algorithm and flow chart, and further illustrated by examples. Codes for linearizing 0-1MQFPP is given in the APPENDIX.

In this paper, by introducing a stream function and new coordinates, we transform classical Euler–Boussinesq equations into a vorticity form. We further construct traveling wave solutions and similarity reduction for the vorticity form of Euler–Boussinesq equations. In fact, our similarity reduction provides a kind of linearization transformation of Euler–Boussinesq equations.

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.

In this paper, a multiband low-noise amplifier (LNA) with capability of band tuning is proposed to support the Mobile WiMAX (IEEE 802.16e) standard associated with a large IIP3 at RF frequencies using multi-gated linearization techniques. Proposed LNA architecture has been optimized in terms of linearity based on a comparative analysis between four tunable LNA structures. In addition, a general survey is performed on linearization techniques (multiple-gated transistors, Derivative Superposition (DS), Modified DS). Proposed LNA is constituted of multi-gated linearization technique. Design and simulations have been realized in 0.13 μm RF CMOS process considering a power supply of 1.2 V. Proposed LNA exhibits an IIP3 of +8.8 dBm, +14.3 dBm, and +6.4 dBm at the lower, middle, and upper bands of WIMAX, respectively. It results in a power gain of 10 dB, a noise figure below 2.5 dB and S₁₁ of -11.5 dB, -8.6 dB, and -12.7 dB at the lower, middle, and upper bands. The related power dissipations are 10 mW, 9.9 mW, and 9.4 mW at the lower, middle, and upper bands, respectively.

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.

This work presents a 7–9GHz in-phase/quadrature (I/Q) direct up-conversion mixer employing a linear voltage-to-current converting baseband input stage for 5G new radio frequency range 2 cellular applications. The proposed baseband stage improves both the large-signal handling capability and small-signal linearity at the input stage of the mixer. The designed double-balanced I/Q up-conversion mixer, which is based on Gilbert-cell mixer, consists of a voltage-to-current converting input stage, switching stages, and an RLC tank. Simulated in a 40-nm CMOS process, the up-conversion mixer achieves a conversion gain of 5dB, input-referred third-order intercept point of 9.6dBm, and input 1-dB gain compression point of –0.8dBm. It draws a DC bias current of 15.8mA from a nominal supply voltage of 1.1V, and the active area is 0.129mm².

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.

In this paper we study C¹ linearization and classification of germs of hyperbolic vector fields on Rⁿ. Two sets of algebraic conditions imposed on the eigenvalues of vector fields are given, under either of which strict similarity of the linear parts is necessary and sufficient for C¹ conjugacy of two vector fields. Furthermore, one set of the algebraic conditions together with strict similarity also imply linearization.

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 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.

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.

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.

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.

This paper concerns itself with establishing convergence estimates for a linearized scheme for solving numerically the saturation equation. In a previous paper, error estimates were obtained for the same scheme in L²(0, T₀;L²(Ω)). In this work, we establish error estimates for the linear scheme in L^∞(0, T₀;L²(Ω)) and in L²(0, T₀;H¹(Ω)) (in the discrete norms). Under certain realistic conditions, we show that, if the regularization parameter β and the spatial discretization parameter h are carefully chosen in terms of the time-stepping parameter Δt, the convergence, in these spaces, is at least of order O((Δt)^α) for some determined α > 0, function of a parameter μ > 0 defined in the problem. Examples of possible choices of β and h in terms of Δt are given.

This paper is a natural continuation of our previous paper: "Teleparallel Lagrange geometry and a unified field theory, Class. Quantum Grav.27 (2010) 045005 (29 pp)". In this paper, we apply a linearization scheme on the field equations obtained in the above-mentioned paper. Three important results under the linearization assumption are accomplished. First, the vertical fundamental geometric objects of the EAP-space lose their dependence on the positional argument x. Secondly, our linearized theory in the Cartan-type case coincides with the GFT in the first-order of approximation. Finally, an approximate solution of the vertical field equations is obtained.

In this paper it is shown how one can associate to a finite number of commuting directions in the Lie algebra of upper triangular ℤ × ℤ-matrices an integrable hierarchy consisting of a set of evolution equations for perturbations of the basic directions inside the mentioned Lie algebra. They amount to a tower of differential and difference equations for the coefficients of these perturbed matrices.

The equations of the hierarchy are conveniently formulated in so-called Lax equations for these perturbations. They possess a minimal realization for which it is shown that the relevant evolutions of the perturbation commute.

These Lax equations are shown in a purely algebraic way to be equivalent with zero curvature equations for a collection of finite band matrices, that are the components of a formal connection form. One concludes with the linearization of the hierarchies and the notion of wave matrices at zero, which is the algebraic substitute for a basis of the horizontal sections of the formal connection corresponding to this connection form.

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.