Showing papers on "Uniform boundedness published in 2001"
TL;DR: In this article, a model describing dynamics of Hopfield neural networks involving variable delays is considered and the existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established.
Abstract: In this article, a model describing dynamics of Hopfield neural networks involving variable delays is considered. Existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established. Further, we derive sufficient criteria of global asymptotic stability (GAS) of the equilibrium point.
110 citations
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TL;DR: This paper introduces a new class of bent functions which they are called hyper-bent functions, which achieve the maximal minimum distance to all the coordinate functions of all bijective monomials.
Abstract: Bent functions have maximal minimum distance to the set of affine functions. In other words, they achieve the maximal minimum distance to all the coordinate functions of affine monomials. In this paper we introduce a new class of bent functions which we call hyper-bent functions. Functions within this class achieve the maximal minimum distance to all the coordinate functions of all bijective monomials. We provide an explicit construction for such functions. We also extend our results to vectorial hyper-bent functions.
90 citations
TL;DR: In this article, a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces was developed, where the distance to the boundary is the distance of the heat semigroup.
Abstract: We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces \(L^q_\delta\), where \(\delta\) is the distance to the boundary. In particular, we prove an optimal \(L^q_\delta-L^r_\delta\) estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems. Namely, under certain assumptions, we prove that any global solution is uniformly bounded for \(t\geq \tau>0\) by a universal constant, independent of the initial data. In all previous results, the bounds for global solutions were depending on the initial data.
50 citations
TL;DR: In this article, it was shown that the maximal operator of the Marczinkiewicz-Fejer meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space Hpto Lp (2/3
Abstract: It is proved that the maximal operator of the Marczinkiewicz-Fejer meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space Hpto Lp (2/3
42 citations
TL;DR: An approach is proposed for making a given stable continuous-time Takagi-Sugeno (TS) fuzzy-system chaotic, by first discretizing it and then using state feedback control of arbitrarily small magnitude.
Abstract: An approach is proposed for making a given stable continuous-time Takagi-Sugeno (TS) fuzzy-system chaotic, by first discretizing it and then using state feedback control of arbitrarily small magnitude. The feedback controller chosen among several candidates is a simple sinusoidal function of the system states, which can lead to uniformly bounded state vectors of the controlled system with positive Lyapunov exponents, and satisfy the chaotic mechanisms of stretching and folding, thereby yielding chaotic dynamics. This approach is mathematically proven for rigorous generation of chaos from a stable continuous-time TS fuzzy system, where the generated chaos is in the sense of Li and Yorke (1975). A numerical example is included to visualize the theoretical analysis and the controller design.
40 citations
01 Apr 2001
TL;DR: A novel adaptive visual feedback scheme is presented to solve the problem of controlling the relative pose between a robot camera and a rigid object of interest by exploiting nonlinear controllability properties and uniform asymptotic stability in the large of the image reference set-point is proved using Lyapunov's direct method.
Abstract: In this paper, a novel adaptive visual feedback scheme is presented to solve the problem of controlling the relative pose between a robot camera and a rigid object of interest. By exploiting nonlinear controllability properties, uniform asymptotic stability in the large of the image reference set-point is proved using Lyapunov's direct method. Moreover, uniform boundedness of the whole state vector is ensured by using an adaptive nonlinear control scheme, in case of unknown object depth. Experimental results with a six-degree-of-freedom robot manipulator endowed with a camera on its wrist validate the framework.
38 citations
TL;DR: A Lyapunov characterization of IOS under a uniform bounded input bounded state (UBIBS) condition is presented, which holds for systems which are not necessarily uniformly bounded.
33 citations
TL;DR: In this article, the authors studied the value distribution and extreme values of eigenfunctions for the "quantized cat map", which is the quantization of a hyperbolic linear map of the torus.
Abstract: We study the value distribution and extreme values of eigenfunctions for the “quantized cat map.” This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map—a commutative group of unitary operators that commute with the map,which we called “Hecke operators.” The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call “Hecke eigenfunctions.”In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the “split primes” for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover, in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weil's theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions.
32 citations
TL;DR: For one-dimensional systems of point charges on the line with a translation invariant distribution, this paper showed that the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length).
Abstract: We present general results for one-dimensional systems of point charges (signed point measures) on the line with a translation invariant distribution μ for which the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length). When the charges are restricted to multiples of a common unit, and their average charge density does not vanish, then the boundedness of the variance implies translation-symmetry breaking—in the sense that there exists a function of the charge configuration that is nontrivially periodic under translations—and hence that μ is not “mixing.” Analogous results are formulated also for one dimensional lattice systems under some constraints on the values of the charges at the lattice sites and their averages. The general results apply to one-dimensional Coulomb systems, and to certain spin chains, putting on common grounds different instances of symmetry breaking encountered there.
31 citations
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TL;DR: In this article, it was shown that a continuous map or a continuous flow with a certain recurrence relation must have a fixed point in a compact set W with the property that the forward orbit of every point in W intersects W, and if the omega limit set of W is nonempty and uniformly bounded, then there is a fixed node in W.
Abstract: We show that a continuous map or a continuous flow on $\R^{n}$ with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in $\R^{n}$ intersects W then there is a fixed point in W. Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point.
29 citations
TL;DR: This work addresses the problem of robust stabilisation of nonlinear systems affected by time-varying uniformly bounded (in time) affine perturbations by showing that under suitable conditions the sliding mode variable can be chosen as the passive output of the perturbed system.
TL;DR: The worst case (information) complexity which is equal to the minimal number of function and derivative evaluations needed to obtain error is studied, and the results of this paper also hold for weighted integration.
TL;DR: This paper presents global stability of the adaptive IIR filter in a nonstationary environment, and it is shown that the filter output is uniformly bounded for all initial conditions.
Posted Content•
TL;DR: In this article, the authors studied the value distribution and extreme values of eigenfunctions for the quantum cat map, which is the quantization of a hyperbolic linear map of the torus.
Abstract: We study the value distribution and extreme values of eigenfunctions for the ``quantized cat map''. This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map - a commutative group of unitary operators which commute with the map, which we called ``Hecke operators''. The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call ``Hecke eigenfunctions''.
In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the ``split primes'' for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weil's theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions.
TL;DR: In this article, the stability of the finite section method for general band-dominated operators on lp spaces over Zk was studied and it was shown that this method is stable if and only if each member of a whole family of operators is invertible and if the norms of these inverses are uniformly bounded.
Abstract: We develop the stability theory for the finite section method for general band-dominated operators on lp spaces over Zk. The main result says that this method is stable if and only if each member of a whole family of operators – the so-called limit operators of the method – is invertible and if the norms of these inverses are uniformly bounded.
TL;DR: In this article, an extension of the Poincare-Melnikov method was used to prove that for weak anisotropy, chaos shows up on the zero-energy manifold, and then a class of isolated periodic orbits and show that the system is nonintegrable.
01 Jan 2001
TL;DR: It is shown that consistent signal extraction is impossible when the errors are distributed according to a density with unbounded support, and the underlying dynamical system admits bomoclinic pairs.
Abstract: The problem of extracting a “signal” xn generated by a dynamical system from a time series yn = xn + en, where en is an observational error, is considered. It is shown that consistent signal extraction is impossible when the errors are distributed according to a density with unbounded support, and the underlying dynamical system admits bomoclinic pairs. It is also shown that consistent signal extraction is possible when the errors are uniformly bounded by a suitable constant and the underlying dynamical system has the “weak orbit separation property”. Simple algorithms for signal recovery are described in the latter case.
01 Jan 2001
TL;DR: A decentralized adaptive control design procedure for large-scale uncertain systems is developed using Single Hidden Layer neural networks, and the proposed adaptive algorithm is implemented in simulation to stabilize an interconnected double inverted pendulum.
Abstract: A decentralized adaptive control design procedure for large-scale uncertain systems is developed using Single Hidden Layer neural networks. The subsystems are assumed to be feedback linearizable and non-affine in the control, and their interconnections bounded linearly by the tracking error norms. Single Hidden Layer neural networks are introduced to approximate the feedback linearization error signal online from available measurements. A robust adaptive signal is required in the analysis to shield the feedback linearizing control law from the interconnection effects. The tracking errors are shown to be uniformly ultimately bounded, and all other signals uniformly bounded. The proposed adaptive algorithm is implemented in simulation to stabilize an interconnected double inverted pendulum.
01 Jan 2001
TL;DR: These notes were produced for Szepessy’s lectures in the summer school on numerical analysis in Durham, July 2000 and proves uniqueness and convergence of uniformly bounded approximations of scalar conservation laws that are consistent with all entropy inequalities and the initial data.
Abstract: These notes were produced for Szepessy’s lectures in the summer school on numerical analysis in Durham, July 2000 The first part gives an introduction to conservation laws In particular it proves uniqueness and convergence of uniformly bounded approximations of scalar conservation laws that are consistent with all entropy inequalities and the initial data Kruzkov’s seminal uniqueness proof is simplified by measure valued solutions to become only one page The convergence proof is then applied to a finite volume method, based on [29]
TL;DR: When the Muntz-Szasz (1953) condition holds, the Mundz-Laguerre filters form a uniformly bounded orthonormal basis in Hardy space, and this has consequences in terms of optimal pole-cancellation schemes and a generalization of Lerch's theorem.
Abstract: When the Muntz-Szasz (1953) condition holds, the Muntz-Laguerre filters form a uniformly bounded orthonormal basis in Hardy space. This has consequences in terms of optimal pole-cancellation schemes, and it also allows for a generalization of Lerch's theorem.
TL;DR: In this paper, complex-valued solutions of the Ginzburg-Landau energy on a smooth bounded simply connected domain Ω of RN, N⩾2 (here e is a parameter between 0 and 1) were considered.
Abstract: We consider complex-valued solutions ue of the Ginzburg–Landau on a smooth bounded simply connected domain Ω of RN, N⩾2 (here e is a parameter between 0 and 1). We assume that ue=ge on ∂Ω, where |ge|=1 and ge is uniformly bounded in H1/2(∂Ω). We also assume that the Ginzburg–Landau energy Ee(ue) is bounded by M0|loge|, where M0 is some given constant. We establish, for every 1⩽p
01 Jan 2001
TL;DR: Simulation results confirm the effectiveness of the proposed repetitive variable structure control approach and show that the filtered tracking error converges to zero on the /spl Lscr/2 norm, and almost perfect periodic tracking is achieved.
Abstract: A repetitive variable structure control (RVSC) approach is originated for the nonlinear system with state-dependent modeling uncertainties and exogenous periodic disturbance. RVSC incorporates repetitive control into VSC. The VSC part ensures the robustness to the uncertain system, and the modeling uncertainties are relaxed to be locally Lipschitz instead of being globally Lipschitz. saturation design to ensures the uniform boundedness of the repetitive control signal. Repetitive control law using directly the VSC signal of the previous cycle for updating is further proposed if a constant bounding function is available. Rigorous proof based on energy function and functional analysis shows that the filtered tracking error converges to zero on the /spl Lscr/2 norm, and almost perfect periodic tracking is achieved. Simulation results confirm the effectiveness of the proposed approach.
TL;DR: Results are given concerning the strong connection between the boundedness of weighted projection onto a subspace and the projection onto its complementary subspace using the inverse weight matrix and explicit bounds are given for the Euclidean norm of the projections.
Abstract: It is known that the norm of the solution to a weighted linear least-squares problem is uniformly bounded for the set of diagonally dominant symmetric positive definite weight matrices. This result is extended to weight matrices that are nonnegative linear combinations of symmetric positive semidefinite matrices. Further, results are given concerning the strong connection between the boundedness of weighted projection onto a subspace and the projection onto its complementary subspace using the inverse weight matrix. In particular, explicit bounds are given for the Euclidean norm of the projections. These results are applied to the Newton equations arising in a primal-dual interior method for convex quadratic programming and boundedness is shown for the corresponding projection operator.
TL;DR: In this paper, a complete characterization of the spectrum of locally square integrable periodically correlated (PC) processes is obtained, which makes use of the author's recent theorem establishing a one to one correspondence between PC processes and a certain class on infinite dimensional stationary processes.
TL;DR: In this article, the uniform boundedness of the weighted Hilbert transform in function spaces associated with a class of even weights on the real line with varying rates of smooth decay near∞ was established.
Abstract: We establish the uniform boundedness of the weighted Hilbert transform in function spaces associated with a class of even weights on the real line with varying rates of smooth decay near∞. We then consider the numerical approximation of the weighted Hilbert transform and to this end we establish convergence results and error estimates which we prove are sharp. Our formulae are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration, augmented by two carefully chosen extra points. Typical examples of weights that are studied are: (a) w α (x) : = exp(− | x|α , α> 1, x ∈ R; (b) w k,β x: =exp (−expk (| x 7verbar;β)), β> 0, k > 1, x ∈ R.
DOI•
11 Nov 2001TL;DR: In this article, the suppression of flow-induced vibration using a simple control algorithm with an assumption that the disturbance as well as the system parameters are bounded variables is presented. But the authors do not consider the effect of variable parameters.
Abstract: We present a study on the suppression of flow-induced vibration using a simple control algorithm with an assumption that the disturbance as well as the system parameters are bounded variables. By introducing three different control signals, we explore three schemes, namely, robust control, sliding mode, and adaptive control. The control schemes are implemented numerically with a few illustrative examples. It is demonstrated that all three schemes can be effectively used for systems with bounded disturbance and variable parameters, which often include the added mass and stiffness as well as viscous shear induced by fluid-structure interactions. Various advantages and disadvantages of different control schemes are illustrated. In general, robust control and adaptive control schemes are (globally) ultimately uniformly bounded, whereas sliding mode scheme is (globally) asymptotically stable. Thus, as we further reduce the integration time step, the residual of robust control and adaptive control schemes will approach a bounded (finite) asymptotic function, and the residual of sliding mode scheme will approach zero. Furthermore, due to self-tuning, the gain of adaptive control scheme is relatively small, yet, the computation cost is higher because of the excessively small time step requirement for the numerical integration. With respect to sliding mode scheme, the control signal is discontinuous due to the sign function and consequently the practical implementation has fast switching fluctuations (chattering).
TL;DR: In this paper, it was shown that for hyperbolic groups, the difference between the metric functions is uniformly bounded, assuming that the ratio of the distances tends to one as the distances grow to infinity.
Abstract: Consider two intrinsic metrics invariant under the same cocompact action of an abelian group. Assume that the ratio of the distances tends to one as the distances grow to infinity. Then it is known (due to D. Burago) that the difference between the metric functions is uniformly bounded. We will prove an analog of this result for hyperbolic groups, as well as a partial generalization of this result for the Heisenberg group: a word metric on the Heisenberg group lies within bounded GH distance from its asymptotic cone.
TL;DR: This work uses inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions).
Abstract: We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions. The algorithms can be implemented in a few lines of high-level language code. In opposition to other black-box algorithms hardly any setup step is required, and thus it is superior in the changing-parameter case.
Book•
01 Jan 2001
TL;DR: In this article, the authors discuss the relationship between single-variable data and linear models of data and present a case study on using regression lines to find the link between education and earnings.
Abstract: CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.
TL;DR: In this paper, a complete marginal analysis of the central optimal solution with respect to both the cost coefficients and the right-hand side components is presented, and the marginal derivatives are uniformly bounded.
Abstract: In this paper we investigate the sensitivity analysis of the parameterized central path. First, a complete marginal analysis of the central optimal solution is developed. This analysis explains the differential properties of the central optimal solution with respect to both the cost coefficients and the right-hand side components. We also show that the marginal derivatives are uniformly bounded. Second, we present three conditions for which the parameterized central path converges. Two of these results allow the difficult situation of simultaneous perturbations in the cost coefficients and right-hand side levels.