Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Using Video Processing for the Full-Field Identification of Backbone Curves in Case of Large Vibrations.

Abstract: Nonlinear modal analysis is a demanding yet imperative task to rigorously address real-life situations where the dynamics involved clearly exceed the limits of linear approximation. The specific case of geometric nonlinearities, where the effects induced by the second and higher-order terms in the strain–displacement relationship cannot be neglected, is of great significance for structural engineering in most of its fields of application—aerospace, civil construction, mechanical systems, and so on. However, this nonlinear behaviour is strongly affected by even small changes in stiffness or mass, e.g., by applying physically-attached sensors to the structure of interest. Indeed, the sensors placement introduces a certain amount of geometric hardening and mass variation, which becomes relevant for very flexible structures. The effects of mass loading, while highly recognised to be much larger in the nonlinear domain than in its linear counterpart, have seldom been explored experimentally. In this context, the aim of this paper is to perform a noncontact, full-field nonlinear investigation of the very light and very flexible XB-1 air wing prototype aluminum spar, applying the well-known resonance decay method. Video processing in general, and a high-speed, optical target tracking technique in particular, are proposed for this purpose; the methodology can be easily extended to any slender beam-like or plate-like element. Obtained results have been used to describe the first nonlinear normal mode of the spar in both unloaded and sensors-loaded conditions by means of their respective backbone curves. Noticeable changes were encountered between the two conditions when the structure undergoes large-amplitude flexural vibrations.

Piecewise Linear Approximation

Abstract: Approximating a complicated function to arbitrary accuracy by “simpler” functions is a basic tool of applied mathematics. We have seen that piecewise polynomials are very useful for this purpose, and that is why approximation by piecewise polynomials plays a very important role in several areas of applied mathematics. For example, the Finite Element Method FEM is an extensively used tool for solving differential equations that is based on piecewise polynomial approximation, see the Chapters FEM for two-point boundary value problems and FEM for Poisson’s equation.

Design and implementation of a uniplanar gradient field coil for magnetic resonance imaging

Abstract: A new approach to design uniplanar gradient coils for magnetic resonance imaging (MRI) is presented. The theoretical formulation involves a constraint cost function between the desired field in a particular region of interest in space and the current-carrying coil plane based on Biot-Savart's integral equation. An appropriate weight function in conjunction with linear approximation functions allows the transformation of the problem formulation into a linear matrix equation in which its iterative solution yields discrete current elements in terms of magnitude and direction within the prescribed coil plane. These current elements can be synthesized into an overall practical wire configuration by suitably adding individual wire loops. Numerical predictions and practical testing for a Gy gradient coil underscore the success of this approach in terms of achieving a highly linear field while maintaining low parasitic fields. © 2004 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 20B: 17–29, 2004

Walsh transforms and cryptographic applications in bias computing

Abstract: Walsh transform is used in a wide variety of scientific and engineering applications, including bent functions and cryptanalytic optimization techniques in cryptography. In linear cryptanalysis, it is a key question to find a good linear approximation, which holds with probability (1+d)/2 and the bias d is large in absolute value. Lu and Desmedt (2011) take a step toward answering this key question in a more generalized setting and initiate the work on the generalized bias problem with linearly-dependent inputs. In this paper, we give fully extended results. Deep insights on assumptions behind the problem are given. We take an information-theoretic approach to show that our bias problem assumes the setting of the maximum input entropy subject to the input constraint. By means of Walsh transform, the bias can be expressed in a simple form. It incorporates Piling-up lemma as a special case. Secondly, as application, we answer a long-standing open problem in correlation attacks on combiners with memory. We give a closed-form exact solution for the correlation involving the multiple polynomial of any weight for the first time. We also give Walsh analysis for numerical approximation. An interesting bias phenomenon is uncovered, i.e., for even and odd weight of the polynomial, the correlation behaves differently. Thirdly, we introduce the notion of weakly biased distribution, and study bias approximation for a more general case by Walsh analysis. We show that for weakly biased distribution, Piling-up lemma is still valid. Our work shows that Walsh analysis is useful and effective to a broad class of cryptanalysis problems.