Suitable Linear Transformation Research Articles

Analytical and Numerical Approximations to Some Coupled Forced Damped Duffing Oscillators

In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other.

Periodicity-enhanced sparse representation for rolling bearing incipient fault detection

Incipient fault detection of rolling bearings is a challenging task since the weak fault features are disturbed by heavy background noise. This paper develops a periodicity-enhanced sparse representation method to address this issue. Firstly, periodicity-enhanced basis pursuit denoising (PBPD) is proposed by the theoretical derivation. Fault proportion is defined to quantify the single fault severity of sparse signals, then a periodicity-decision criterion for determining the optimal potential fault period is designed to periodically filter the last sparse signal. Secondly, the suitable linear transformation for PBPD is investigated in comparison and maximal overlapping discrete wavelet packet transform is adopted eventually. Thirdly, adaptive selection strategies are developed for the key parameters of PBPD. Simulations and experimental verifications demonstrate PBPD’s excellent performance in rolling bearing incipient fault detection.

The required acoustic parameters simplification of invisibility cloaks and concentrators using the impedance-tunable coordinate transformation

Transformation acoustics, as an unconventional theory, provides a powerful tool to design various kinds of acoustic devices with excellent functionalities. However, the required ideal parameters, which are prescribed by the method, are both complex and hard to implement-even using acoustic metamaterials. Furthermore, simplified parameter materials are generally favored in transformation-acoustic design due to its easier realization with artificial structures. In this letter, we propose a coordinate transformation methodology for achieving simplified parameters by tuning the impedance distribution in the geometric limit, where the transformation media parameters can be derived by setting tunable impedance functions in the original space and a combination of suitable linear or nonlinear coordinate transformation. Based on this approach, both two-dimensional acoustic cloak and concentrators are designed with different sets of simplified parameters. Numerical simulations indicate good performance of these devices with minimized scattering at higher frequencies. The proposed method provides more opportunities to realize the designed acoustic devices experimentally, and can also be used for other transformation-acoustic designs including 3D cases.

A new solution method for a class of large dimension rank-two nonconvex programs

Abstract Low-rank problems are nonlinear minimization problems in which the objective function, by means of a suitable linear transformation of the variables, depends on very few variables. These problems often arise in quantitative management science applications, for example, in location models, transportation problems, production planning, data envelopment analysis and multiobjective programs. They are usually approached by means of outer approximation, branch and bound, branch and select and optimal level solution methods. The paper studies, from both a theoretical and an algorithmic point of view, a class of large-dimension rank-two nonconvex problems having a polyhedral feasible region and $f(x)=\phi (c^Tx+c_0,d^Tx+d_0)$ as the objective function. The proposed solution algorithm unifies a new partitioning method, an outer approximation approach and a mixed method. The results of a computational test are provided to compare these three approaches with the optimal level solutions method. In particular, the new partitioning method performs very well in solving large problems.

Joint Optimization of Dimension Assignment and Compression in Distributed Estimation Fusion

This paper studies linear distributed estimation of an unknown random parameter vector in a bandwidth-constrained multisensor network. To meet the bandwidth limitations, each sensor converts its observation into a low-dimensional datum via a suitable linear transformation. Then, the fusion center estimates the parameter vector by linearly combining all the received low-dimensional data, aiming at minimizing the estimation mean square error. The main purpose of this paper is to jointly determine the compression dimension of each sensor (referred to as dimension assignment) and design the corresponding compression matrix when the total compression dimensions is limited. Such a joint design problem can be formulated as a rank-constrained optimization problem and it is shown to be NP-hard for the first time. In addition, successive quadratic upper-bound minimization (SQUM), SQUM-block coordinate descent (SQUM-BCD) and nuclear norm regularization (NNR) methods are developed to solve it approximately. Furthermore, we show that any accumulation point of the sequence generated by the SQUM method satisfies the Karush-Kuhn-Tucker conditions of the rank-constrained optimization problem, and the Phase II algorithm of the SQUM-BCD and NNR methods (both are two-phase algorithms and have the same Phase II algorithm) guarantees convergence at least to a stationary point. Numerical experiments illustrate the advantages of the proposed methods compared with the existing method.

Computation of Invariant Curves and Identifying the Type of Critical Point

This paper considers two dimensional systems which have purely imaginary eigenvalues. In order to obtain more propositions of invariant curves, the authors transform the real systems into complex differential systems by using a suitable linear transformation. The authors also propose an algorithm to compute exponential factors. An improved method of constructing integrating factor by using all invariant curves is presented and can be used in determining the type of the equilibrium points.

Soret and Dufour Effects on Viscoelastic Radiative and Heat Absorbing Nanofluid Driven by a Stretched Sheet with Inclined Magnetic Field

An investigation has been performed to analyze the impacts of Soret and Dufour on natural convective and heat absorbing flow of viscoelastic radiative nanofluid driven by a linearly stretched sheet considering inclined magnetic field. By making use of suitable linear transformations, the mathematical equations of problem are changed into the extremely non-linear coupled system of ordinary differential equations. Further, solutions of these differential equations are obtained by implementing GFEM (Galerkin finite element method). The consequence of various controlling pertinent parameters on nanofluid velocity, solutal concentration, temperature and nanofluid concentration are illustrated by means of various graphs while from engineering aspect numerical values of the shear stress, wall temperature gradient, solutal and nanoparticles concentration rate at the stretched sheet are presented in different tables. The numerical results are compared for mono and double diffusive nanofluids which yield that the aligned magnetic field, viscoelasticity, solutal and Brownian diffusivity have significant impacts on the flow field. The reliability of implemented method is authenticated by comparing our results with the previously published results under certain conditions, which signifies the correctness of the implemented method. The present investigation is applicable in several industrial processes such as coolant application, nano-drug delivery, cooling of microchip, heat exchanger technology, biological fluid movement and oceanography etc.Keywords: Magnetic field; Viscoelastic nanofluid; Thermal radiation; Heat absorption

New limit results related to the coupon collector’s problem

We study point processes associated with coupon collector’s problem, that are defined as follows. We draw with replacement from the set of the first n positive integers until all elements are sampled, assuming that all elements have equal probability of being drawn. The point process we are interested in is determined by ordinal numbers of drawing elements that didn’t appear before. The set of real numbers is considered as the state space. We prove that the point process obtained after a suitable linear transformation of the state space converges weakly to the limiting Poisson random measure whose mean measure is determined. We also consider rates of convergence in certain limit theorems for the problem of collecting pairs.

Functional analysis of variance for Hilbert-valued multivariate fixed effect models

This paper presents new results on functional analysis of variance for fixed effect models with correlated Hilbert-valued Gaussian error components. The geometry of the reproducing kernel Hilbert space of the error term is considered in the computation of the total sum of squares, the residual sum of squares, and the sum of squares due to the regression. Under suitable linear transformation of the correlated functional data, the distributional characteristics of these statistics, their moment generating and characteristic functions, are derived. Fixed effect linear hypothesis testing is finally formulated in the Hilbert-valued multivariate Gaussian context considered.

Polynomials with constant Hessian determinants in dimension three

In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n≤3 over a field K of characteristic zero. We do this by extending the following result for n≤2 by F. Dillen to n≤3: If f is a polynomial of degree larger than 2 in n≤3 variables such that the Hessian determinant of f is constant, then after a suitable linear transformation (replacing f by f(Tx) for some T∈GLn(K)), the Hessian matrix of f becomes zero below the anti-diagonal. The result does not hold for larger n.The proof of the case det⁡Hf∈K⁎ is based on the following result, which in turn is based on the already known case det⁡Hf=0: If f is a polynomial in n≤3 variables such that det⁡Hf≠0, then after a suitable linear transformation, there exists a positive weight function w on the variables such that the Hessian determinant of the w-leading part of f is nonzero. This result does not hold for larger n either (even if we replace ‘positive’ by ‘nontrivial’ above).In the last section, we show that the Jacobian conjecture holds for gradient maps over the reals whose linear part is the identity map, by proving that such gradient maps are translations (i.e. have degree 1) if they satisfy the Keller condition. We do this by showing that this problem is the polynomial case of the main result of [13]. For polynomials in dimension n≤3, we generalize this result to arbitrary fields of characteristic zero.

Deriving amplitude equations via evolutionary $\Gamma$-convergence

We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary $\Gamma$-convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show $\Gamma$-convergence of the associated energies in suitable function spaces.  &nbsp The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in $L^2$, while for the case of a quadratic nonlinearity we need to impose weak convergence in $H^1$. However, we do not need well-preparedness of the initial conditions.

Stability of heteroclinic cycle connecting hyperbolic saddles in higher dimensional space

In this paper, the stability of nontwisted heteroclinic loop to 2 hyperbolic saddles is considered in arbitrarily finite dimensional spaces. Under the condition that the Poincar′e map is well-defined, the criterion is given for the asymptotic stability of the heteroclinic loop confined in its partial neighborhood. The stability results for 3-dimensional system are extened to m + n + 2 dimensional space, where m 0, n 0. By taking a suitable linear transformation, we get the first normal form, by a coordinate change to straighten the local stable manifold and the local unstable manifold, we establish the second normal form. Then, in the small neighborhood of the saddle P1, P2, we select two cross sections transversal to the heteroclinic orbit Γ respectively, and construct the Poincar′e map by two steps: in the small neighborhood of the saddle, we build the main part of the singular flow map by the linearly approximate system, in the regular tubular neighborhood of Γ, we use a differential homeomorphism to express the regular flow map. The Poincar′e map is achieved by composing the singular flow map and the regular flow map. At last, by estimating the modules of some vectors rather skilled, we obtain the ratio of the distance between the first recurrent point and the heteroclinic point to the distance between the initial point and the heteroclinic point. As a direct consequence, we derive two quite concise stability criteria for the non-resonant heteroclinic cycle to hyperbolic saddles.

Mathematical identification of diffuse and localized instabilities in fluid‐saturated sands

SUMMARYThe mathematical properties of diffuse and localized failure modes in fluid‐saturated sands are investigated. The granular medium is modeled as an elastoplastic solid, and a recently proposed set of scalar indices, here referred to as moduli of instability, is used to identify the onset of potential bifurcations of the incremental response. First, the analytical properties of these moduli are discussed, stressing their dependence on the kinematic constraints associated with the imposed deformation modes. Then, by using an elastoplastic model for sands, drained and undrained loading paths are simulated under axisymmetric, plane‐strain and simple shear conditions. For each deformation mode, the instability moduli are computed and monitored throughout the simulations, with the purpose of elucidating the consequences of changes in control conditions. In addition, it is illustrated that suitable linear transformations allow the same strategy to be used to perform drained or undrained shear band analyses and predict the interval of possible band inclinations. The final comparison against literature experiments on loose Hostun sand shows that the instability moduli are indicators of the loss of resistance against specific modes of deformation. As a result, they can be used to identify and explain a number of failure mechanisms that can be commonly observed in experiments. Copyright © 2013 John Wiley & Sons, Ltd.

THE LOCAL STOCHASTIC STABILITY FOR COMPLEX NETWORKS UNDER PINNING CONTROL

The local stochastic stability of complex networks under pinning control is studied, with stochastic perturbations to the coupling strengths. The nodes of complex network are modeled as second-order differential equations subject to stochastic parametric excitations. The complex network is first linearized at its trivial solution, and the resulting equations are reduced to independent subsystems by using a suitable linear transformation. Then the condition of the stochastic stability can be determined by Lyapunov exponents of the subsystems. The largest Lyapunov exponent of the subsystem can be expressed analytically, as a function of the eigenvalue of a matrix associated with the coupling matrix and pinning control matrix for a given system of parameters. And the stability region with respect to the eigenvalue can be obtained. It is pointed out that the local stochastic stability of the network is finally determined by the maximal and minimum eigenvalues of the matrix. Numerical results with positive and negative damping coefficients are given to illustrate the criterion. Moreover, for the positive damping coefficient case, the pinning control may destabilize the network; and for the negative damping coefficient case, the pinning control may stabilize the network.

Environmental robust speech and speaker recognition through multi-channel histogram equalization

Feature statistics normalization in the cepstral domain is one of the most performing approaches for robust automaticspeech and speaker recognition in noisy acoustic scenarios: feature coefficients are normalized by using suitable linear or nonlinear transformations in order to match the noisy speech statistics to the clean speech one. Histogram equalization (HEQ) belongs to such a category of algorithms and has proved to be effective on purpose and therefore taken here as reference. In this paper the presence of multi-channel acoustic channels is used to enhance the statistics modeling capabilities of the HEQ algorithm, by exploiting the availability of multiple noisy speech occurrences, with the aim of maximizing the effectiveness of the cepstra normalization process. Computer simulations based on the Aurora 2 database in speech and speaker recognition scenarios have shown that a significant recognition improvement with respect to the single-channel counterpart and other multi-channel techniques can be achieved confirming the effectiveness of the idea. The proposed algorithmic configuration has also been combined with the kernel estimation technique in order to further improve the speech recognition performances.

A Finite Element Model for Direction-Dependent Mechanical Response to Nanoindentation of Cortical Bone Allowing for Anisotropic Post-Yield Behavior of the Tissue

A finite element model was developed for numerical simulations of nanoindentation tests on cortical bone. The model allows for anisotropic elastic and post-yield behavior of the tissue. The material model for the post-yield behavior was obtained through a suitable linear transformation of the stress tensor components to define the properties of the real anisotropic material in terms of a fictitious isotropic solid. A tension-compression yield stress mismatch and a direction-dependent yield stress are allowed for. The constitutive parameters are determined on the basis of literature experimental data. Indentation experiments along the axial (the longitudinal direction of long bones) and transverse directions have been simulated with the purpose to calculate the indentation moduli and the tissue hardness in both the indentation directions. The results have shown that the transverse to axial mismatch of indentation moduli was correctly simulated regardless of the constitutive parameters used to describe the post-yield behavior. The axial to transverse hardness mismatch observed in experimental studies (see, for example, Rho et al. [1999, "Elastic Properties of Microstructural Components of Human Bone Tissue as Measured by Nanoindentation," J. Biomed. Mater. Res., 45, pp. 48-54] for results on human tibial cortical bone) can be correctly simulated through an anisotropic yield constitutive model. Furthermore, previous experimental results have shown that cortical bone tissue subject to nanoindentation does not exhibit piling-up. The numerical model presented in this paper shows that the probe tip-tissue friction and the post-yield deformation modes play a relevant role in this respect; in particular, a small dilatation angle, ruling the volumetric inelastic strain, is required to approach the experimental findings.

Interactions with hidden complementarities

We consider a finite population simultaneous move game with heterogeneous interaction modes across different pairs of players. We allow for general interaction patterns, but restrict our analysis to games whose pure strategy Nash equilibrium conditions boil down to a set of piece-wise linear conditions, so that an equilibrium is a solution to a linear complementarity problem. We introduce a new class of games for which a suitable linear transformation of the original interaction matrix induces a game with complementarities. We provide general moderation conditions on the interaction matrix such that a game in this class has a unique Nash equilibrium, that we are able to characterize by means of a closed-form expression involving a generalized version of the Katz network measure of node centrality.

Finite Horizon H∞ Preview Control

The paper focuses on the H ∞ preview control problem in the finite horizon. Starting with the traditional idea, we found the sticking point and used a suitable linear transformation to eliminate it. Finally, we obtained a sufficient and necessary condition and a simple control-law for the problem. Furthermore, a numerical example is also provided to illustrate that the controller can effectively improve the closed-loop performance.

ON THE LOCAL STOCHASTIC STABILITY OF NONLINEAR COMPLEX NETWORKS

The local stochastic stability of nonlinear complex networks is studied, subject to stochastic perturbations to the coupling strengths and stochastic parametric excitations to the nodes. The complex network is first linearized at its trivial solution and then the linearized network is reduced to N independent subsystems by using a suitable linear transformation, where N is the size of the network. The largest Lyapunov exponent for each subsystem is then calculated and all the approximate analytical solutions are evaluated for some specific cases. It is found that the largest Lyapunov exponent among all subsystems is the one associated with the subsystem that has the largest or the smallest eigenvalue of the configuration matrix of the network. Finally, an example is given to demonstrate the validity and accuracy of the theoretical analysis.

Linear Transformations and the k-Means Clustering Algorithm

Functional data can be clustered by plugging estimated regression coefficients from individual curves into the k-means algorithm. Clustering results can differ depending on how the curves are fit to the data. Estimating curves using different sets of basis functions corresponds to different linear transformations of the data. k-means clustering is not invariant to linear transformations of the data. The optimal linear transformation for clustering will stretch the distribution so that the primary direction of variability aligns with actual differences in the clusters. It is shown that clustering the raw data will often give results similar to clustering regression coefficients obtained using an orthogonal design matrix. Clustering functional data using an L2 metric on function space can be achieved by clustering a suitable linear transformation of the regression coefficients. An example where depressed individuals are treated with an antidepressant is used for illustration.