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Complex Plane (complex + plane)
Selected AbstractsTwo formulations for dynamic response of a cylindrical cavity in cross-anisotropic porous mediaINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 4 2010Hashem Eslami Abstract Two formulations for calculating dynamic response of a cylindrical cavity in cross-anisotropic porous media based on complex functions theory are presented. The basis of the method is the solution of Biot's consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u,w formulation of Biot's equations are solved. Difference of these two solutions refers to use of two various potential functions. Equations for calculating stress, displacement and pore pressure fields of the medium are mentioned based on each two formulations. Copyright © 2009 John Wiley & Sons, Ltd. [source] Survey of quantitative feedback theory (QFT),INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 10 2001Isaac Horowitz QFT is an engineering design theory devoted to the practical design of feedback control systems. The foundation of QFT is that feedback is needed in control only when plant (P), parameter and/or disturbance (D) uncertainties (sets ,,={P}, ,,={D}) exceed the acceptable (A) system performance uncertainty (set ,,={A}). The principal properties of QFT are as follows. (1) The amount of feedback needed is tuned to the (,,, ,,, ,,) sets. If ,, ,exceeds' (,,, ,,), feedback is not needed at all. (2) The simplest modelling is used: (a) command, disturbance and sensor noise inputs, and (b) the available sensing points and the defined outputs. No special controllability test is needed in either linear or non-linear plants. It is inherent in the design procedure. There is no observability problem because uncertainty is included. The number of independent sensors determines the number of independent loop transmissions (Li), the functions which provide the benefits of feedback. (3) The simplest mathematical tools have been found most use ful,primarily frequency response. The uncertainties are expressed as sets in the complex plane. The need for the larger ,,, ,, sets to be squeezed into the smaller ,, set results in bounds on the Li(j,) in the complex plane. In the more complex systems a key problem is the division of the ,feedback burden' among the available Li(j,). Point-by-point frequency synthesis tremendously simplifies this problem. This is also true for highly uncertain non-linear and time-varying plants which are converted into rigorously equivalent linear time invariant plant sets and/or disturbance sets with respect to the acceptable output set ,,. Fixed point theory justifies the equivalence. (4) Design trade-offs are highly transparent in the frequency domain: between design complexity and cost of feedback (primarily bandwidth), sensor noise levels, plant saturation levels, number of sensors needed, relative sizes of ,,, ,, and cost of feedback. The designer sees the trade-offs between these factors as he proceeds and can decide according to their relative importance in his particular situation. QFT design techniques with these properties have been developed step by step for: (i) highly uncertain linear time invariant (LTI) SISO single- and multiple-loop systems, MIMO single-loop matrix and multiple-loop matrix systems; and (ii) non-linear and time-varying SISO and MIMO plants, and to a more limited extent for plants with distributed control inputs and sensors. QFT has also been developed for single- and multiple-loop dithered non-linear (adaptive) systems with LTI plants, and for a special class (FORE) of non-linear compensation. New techniques have been found for handling non-minimum-phase (NMP) MIMO plants, plants with both zeros and poles in the right half-plane and LTI plants with incidental hard non-linearities such as saturation. [source] Asymptotic and spectral properties of operator-valued functions generated by aircraft wing modelMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2004A. V. Balakrishnan Abstract The present paper is devoted to the asymptotic and spectral analysis of an aircraft wing model in a subsonic air flow. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution,convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. More precisely, the generalized resolvent is a finite-meromorphic function on the complex plane having a branch-cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotical approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro-differential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work. Copyright © 2004 John Wiley & Sons, Ltd. [source] Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetryACTA CRYSTALLOGRAPHICA SECTION A, Issue 3 2010A. V. Shutov A class of quasiperiodic tilings of the complex plane is discussed. These tilings are based on ,-expansions corresponding to cubic irrationalities. There are three classes of tilings: Q3, Q4 and Q5. These classes consist of three, four and five pairwise similar prototiles, respectively. A simple algorithm for construction of these tilings is considered. This algorithm uses greedy expansions of natural numbers on some sequence. Weak and strong parameterizations for tilings are obtained. Layerwise growth, the complexity function and the structure of fractal boundaries of tilings are studied. The parameterization of vertices and boundaries of tilings, and also similarity transformations of tilings, are considered. [source] Multiresolution of quasicrystal diffraction spectraACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2009Avi Elkharrat A method for analyzing and classifying two-dimensional pure point diffraction spectra (i.e. a set of Bragg peaks) of certain self-similar structures with scaling factor , > 1, such as quasicrystals, is presented. The two-dimensional pure point diffraction spectrum , is viewed as a point set in the complex plane in which each point is assigned a positive number, its Bragg intensity. Then, by using a nested sequence of self-similar subsets called ,-lattices, we implement a multiresolution analysis of the spectrum ,. This analysis yields a partition of , simultaneously in geometry, in scale and in intensity (the `fingerprint' of the spectrum, not of the diffracting structure itself). The method is tested through numerical explorations of pure point diffraction spectra of various mathematical structures and also with the diffraction pattern of a realistic model of a quasicrystal. [source] The mathematical pendulum from Gauß via Jacobi to RiemannANNALEN DER PHYSIK, Issue 6 2009W. Dittrich Abstract The goal of this article is to introduce double-periodic elliptic functions on the basis of a "simple" mechanical system, that of the mathematical pendulum. Thereby it is not geometry that is in the foreground, as in Gauß's analysis of the lemniscatian curve, but rather the calculation of the specific attributes of elliptic functions with the aid of a periodic integrable system. Not the spatial degree of freedom, but the time variable is continued into the complex plane. This will make it possible for us to not only identify the known real period of the pendulum oscillation, but also to detect a second imaginary period. Only then does the solution of the equation of motion become a Jacobi-type elliptic function. Using the Cauchy integral theorem, which Gauß was already familiar with, as well as the simplest Riemannian surface of the function , we want to calculate the analytic and topological characteristics of the oscillatory motion of a pendulum. Our intent is to show that elliptic functions could have appeared much earlier than 1796 in the literature. Admittedly, for this the field of complex numbers was necessary, as represented in the Gaußian plane of complex numbers. However, Gauß was unwilling to publish his findings because of his "fear of the cry of the Boeotians". [source] Plotting Robust Root Locus For Polynomial Families Of Multilinear Parameter Dependence Based On Zero Inclusion/Exclusion TestsASIAN JOURNAL OF CONTROL, Issue 2 2003Chyi Hwang ABSTRACT The Mapping Theorem by Zadeh and Desoer [17] is a sufficient condition for the zero exclusion of the image or value set of an m -dimensional box B under a multilinear mapping f: Rm , C, where R and C denote the real line and the complex plane, respectively. In this paper, we present a sufficient condition for the zero inclusion of the value set f(B). On the basis of these two conditions and the iterative subdivision of the box B, we propose a numerical procedure for testing whether or not the value set f(B) includes the origin. The procedure is easy to implement and is more efficient than that based on constructing the value set f(B) explicitly. As an application, the proposed zero inclusion test procedure is used along with a homotopy continuation algorithm to trace out the boundary curves of the robust root loci of polynomial families with multilinear parametric uncertainties. [source] | ||||||||||