A CHARACTERIZATION OF ORTHONORMAL WAVELET FAMILIES IN SOBOLEV SPACES

In this paper, a characterization of orthonormal wavelet families in Sobolev spaces H s （R） is established.

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincare map of the system is constructed. Using the Poincare map and the Gram Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.

MULTIDIMENSIONAL MULTI-KNOTS PIECEWISE LINEAR SPECTRAL SEQUENCES

Motivated by representing multidimensional periodic nonlinear and non-stationary signals （functions）, we study a class of orthonormal exponential basis for L2（Id） with I ：= [0, 1）, whose exponential parts are piecewise linear spectral sequences with p-knots. It is widely applied in time-frequency analysis.

AN MRA YIELDS AN ORTHONORMAL WAVELET BASIS——AN OPERATOR THEORETIC PROOF

In this paper we show how to construct a scaling function and an orthonor-mal wavelet basis from a multiresolution approximation using an operator theoreticmethod.

Comparative Analysis of Quantum based Receivers with Linear and Nonlinear Adaptive Receivers

In this paper,we propose a new linear multiuser receiver for synchronous code-division multiple-access (CDMA) systems,referred to as the orthogonal multiuser(OMU) receiver.Unlike the linear minimum mean-squared error(MMSE) receiver,the OMU receiver depends only on the signature vectors and does not require knowledge of the received amplitudes or the channel signal-to-noise ratio(SNR).Here we develop methods that construct an optimal set of vectors with a specified inner product structure,from a given set of vectors in a complex Hilbert space.The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors.An algorithm has been developed using the principles of quantum parameters and some of its axioms and constraints.In place of the classical matched filter(MF) receiver we propose a modified receiver.This approach assumes that improving the accuracy will necessarily result in im- proved performance.The simulation results provided here suggests that in certain cases the OMU and POMU receivers can significantly increase the probability of correct detection with low error rate over the MF receiver.

Inversion Method for Sound Velocity Profile of Eddy in Deep Ocean

The modal wave number tomography approach is used to obtain sound speed profile of water column in deep ocean. The approach consists of estimation of the local modal eigenvalues from complex pressure field and use of these data as input to modal perturbative inversion method for obtaining the local sound speed profile. The empirical orthonormal function (EOF) is applied to reduce the parameter search space. The ocean environment used for numerical simulations includes the Munk profile as the unperturbed background speed profile and a weak Gaussian eddy as the sound speed profile perturbation. The results of numerical simulations show the method is capable of monitoring the oceanic interior structure.

What can routh table offer in addition to stability?

Routh stability test is covered in almost all undergraduate control texts. It determines the stability or, a little beyond, the number of unstable roots of a polynomial in terms of the signs of certain entries of the Routh table constructed from the coefficients of the polynomial. The use of the Routh table, as far as the common textbooks show, is only limited to this function. We will show that the Routh table can actually be used to construct an orthonormal basis in the space of strictly proper rational functions with a common stable denominator. This orthonormal basis can then be used for many other purposes, including the computation of the H2 norm, the Hankel singular values and singular vectors, model reduction, H∞ optimization, etc. Keywords Routh stablity criterion - Orthonormal basis - Root-mean-squared value - Hankel operater - Nehari problem - Model reduction This work was supported by the Hong Kong Research Grants Council.

Kautz Function Based Continuous-Time Model Predictive Controller for Load Frequency Control in a Multi-Area Power System

A continuous-time Model Predictive Controller was proposed using Kautz function in order to improve the performance of Load Frequency Control(LFC).A dynamic model of an interconnected power system was used for Model Predictive Controller(MPC)design.MPC predicts the future trajectory of the dynamic model by calculating the optimal closed loop feedback gain matrix.In this paper,the optimal closed loop feedback gain matrix was calculated using Kautz function.Being an Orthonormal Basis Function(OBF),Kautz function has an advantage of solving complex pole-based nonlinear system.Genetic Algorithm(GA)was applied to optimally tune the Kautz function-based MPC.A constraint based on phase plane analysis was implemented with the cost function in order to improve the robustness of the Kautz function-based MPC.The proposed method was simulated with three area interconnected power system and the efficiency of the proposed method was measured and exhibited by comparing with conventional Proportional and Integral(PI)controller and Linear Quadratic Regulation(LQR).

Three Levels of Tensor Theory

There are many forms of tensor theory which are quite different. Discussions are put forward and their relationships are found. The differences between them depend on whether there is metric on the space and the basis is orthonormal.

A Modified Homotopy Analysis Method for Solving Boundary Layer Equations

A new modification of the Homotopy Analysis Method (HAM) is presented for highly nonlinear ODEs on a semi-infinite domain. The main advantage of the modified HAM is that the number of terms in the series solution can be greatly reduced;meanwhile the accuracy of the solution can be well retained. In this way, much less CPU is needed. Two typical examples are used to illustrate the efficiency of the proposed approach.

Representation of Functions in <i>L</i>¹<sub style="margin-left:-6px"><i>μ</i></sub>Weighted Spaces by Series with Monotone Coefficients in the Walsh Genrealized System

Let is the Walsh generalized system. In the paper constructed a weighted space , and series in the Walsh generalized system with monotonically decreasing coefficient such that for each function in the space one can find a subseries that converges to in the weighted and almost everywhere on [0,1].

Construction of nonseparable orthonormal compactly supported wavelet bases for L^2(R^d)

Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively.Two methods for constructing nonseparable Ω-filter banks from M-filter banks and N-filter banks are presented,where Ω is a（r＋s） ×（r＋s） dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Ω-T Zr＋s/Zr＋s areΓΩ={[γT h,ζ T q] T：γh∈ΓM,ζq∈ΓN}.Specially,Ω can be [MΘ0N],whereΘis a r×s integer matrix with M-1Θbeing also an integer matrix.Moreover,if the constructed filter bank satisfies Lawton＇s condition,which can be easy to verify,then it generates an orthonormal nonseparable Ω-wavelet basis for L2（Rr＋s）.Properties,including Lawton＇s condition,vanishing moments and regularity of the new Ω-filter banks or new Ω-wavelet basis are discussed then.Finally,a class of nonseparable Ω-wavelet basis for L2（Rr＋1） are constructed and three other examples are given to illustrate the results.In particular,when M=N=2,all results obtained in this paper appeared in[1].

An SAV Method for Imaginary Time Gradient Flow Model in Density Functional Theory

In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure system.To handle the orthonormality constraint on those wave functions,two kinds of penalty terms are introduced in designing the modified energy functional in SAV,i.e.,one for the norm preserving of each wave function,another for the orthogonality between each pair of different wave functions.A numerical method consisting of a designed scheme and a linear finite element method is used for the discretization.Theoretically,the desired unconditional decay of modified energy can be obtained from our method,while computationally,both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical experiments.More importantly,numerical results show that the orthonormality among those wave functions can be automatically preserved,without explicitly preserving orthogonalization operations.This implies the potential of our method in large-scale simulations in density functional theory.

Pointwise Convergence and Uniform Convergence of Wavelet Frame Series

Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.

Characterization of Compact Support of Fourier Transform for Orthonormal Wavelets of L^2(R^d)

Let{ψμ} be an orthonormal wavelet of L^2（R^d） and the support of a whole of its Fourier transform be Uμsupp{ψμ}=Пi=1^d[Ai, Di]-Пi=1^d（Bi, Ci）, Ai≤Bi≤Ci≤Di. Under the weakest condition that each │ψμ│, is continuous for ω ∈ δ（Пi=1^d[Ai, Di]）, a characterization of the above support of a whole is given.

Constructing super Gabor frames:the rational time-frequency lattice case

For a time-frequency lattice Λ = A Z d × B Z d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det( AB) | = 1 L . The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles R d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.

Construction of compactly supported orthonormal wavelets with beautiful structure

In this paper, a new method of constructing symmetric (antisymmetric) scal-ing and wavelet filters is introduced, and we get a new type of wavelet system that hasvery beautiful structure. Using this kind of wavelet system, we can achieve filters withthe properties: rational, symmetric or antisymmetric, the lengths of the filters are shorterand the corresponding functions have higher smoothness, so they have good prospect inapplications.

Optimal representation of multivariate functions or data in visualizable low-dimensional spaces

It is intended to find the best representation of high-dimensional functions or multivariate data in L2(W) with fewest number of terms, each of them is a combination of one-variable function. A system of nonlinear integral equations has been derived as an eigenvalue problem of gradient operator in the said space. It proved that the complete set of eigenfunctions generated by the gradient operator constitutes anorthonormal system, and any function of L2(W) can beexpanded with fewest terms and exponential rapidity of convergence. It is also proved as a corollary, thegreatest eigenvalue of the integral operators hasmultiplicity 1 if the dimension of the underlying space nn = 2, 4 and 6.

On IGMRES: An incomplete generalized minimal residual method for large unsymmetric linear systems

The truncated version of the generalized minimal residual method (GMRES), the incomplete generalized minimal residual method (IGMRES), is studied. It is based on an incomplete orthogonalization of the Krylov vectors in question, and gives an approximate or quasi_minimum residual solution over the Krylov subspace. A convergence analysis of this method is given, showing that in the non_restarted version IGMRES can behave like GMRES once the basis vectors of Krylov subspace generated by the incomplete orthogonalization are strongly linearly independent. Meanwhile, some relationships between the residual norms for IOM and IGMRES are established. Numerical experiments are reported to show convergence behavior of IGMRES and of its restarted version IGMRES( m ).

A Characterization of Multidimensional Multi-knot Piecewise Linear Spectral Sequence and Its Applications

We characterize a class of piecewise linear spectral sequences. Associated with the spectral sequence, we construct an orthonormal exponential bases for L2（[0,1）d）, which are called generalized Fourier bases. Moreover, we investigate the convergence of Bochner-Riesz means of the generalized Fourier series.