Dilation Equation Research Articles

String equation and quantum cohomology for noncompact symplectic manifolds

We use our Gromov–Witten invariant theory in a previous Note for noncompact geometrically bounded symplectic manifolds to get solutions of the generalized string equation and dilation equation and their variants. More solutions of the WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

On Probability Density for Modeling Video Traffic

Accurate models for variable bit rate (VBR) video traffic need to allow for different frame types present in the video, different activity levels for different frames, and a variable group of pictures (GOP) structure. The temporal as well as the stochastic properties of the trace data need to be captured by any models. We propose some models that capture temporal properties of the data using doubly Markov processes and autoregressive models. We highlight the importance of capturing the stochastic properties of the data accurately, as this leads to significant improvement in the performance of the model. In order to capture the stochastic properties of the traces, the probability density function of the trace data needs to be accurately modeled. Hence, the focus of this paper is on creating autoregressive processes with arbitrary probability densities. We relate this to work in wavelet theory on the solutions to two-scale dilation equations. The performance of our model is evaluated in terms of the stochastic properties of the generated trace as well as using network simulations.

L 2 (R) Solutions of Dilation Equations and Fourier-Like Transforms

We state a novel construction of theFourier transform on L2(R) based on translation and dilation properties which makes use of the multiresolution analysis structure commonly used in the design of wavelets. We examine the conditions imposed by variants of these translation and dilation properties. This allows other characterizations of the Fourier transform to be given, and operators which have similar properties are classified. This is achieved by examining the solution space of various dilation equations, in particular we show that the L2(R) solutions of f (x) = f (2x) + f (2x − 1) are in direct correspondence with L2(±[1, 2)).

On local properties of compactly supported solutions of the two‐coefficient dilation equation

Let a and b be reals. We consider the compactly supported solutions φ : ℝ → ℝ of the two‐coefficient dilation equation φ(x) = aφ(2x) + bφ(2x − 1). In this paper, we determine sets Ba,b, Ca,b, and Za,b defined in the following way: let x ∈ [0, 1]. We say that x ∈ Ba,b (resp., x ∈ Ca,b, x ∈ Za,b) if the zero function is the only compactly supported solution of the two‐coefficient dilation equation, which is bounded in a neighbourhood of x (resp., continuous at x, vanishes in a neighbourhood of x). We also give the structure of the general compactly supported solution of the two‐coefficient dilation equation.

On the existence of irregular solutions of the two-coefficient dilation equation

We prove that for every real a and b such that \( ab \neq 0 \) there exists a very irregular compactly supported solution \( \varphi : {\Bbb R} \rightarrow {\Bbb R} \) of the functional equation¶\( \varphi(x)=a\varphi(2x)+b\varphi(2x-1) \).

Two-dimensional FlR filter design using matrix dilation approach

This paper develops an approach for the optimal design of two-dimensional (2-D) finite impulse response (FIR) filters based on the minimization of a new performance index. The approach gives analytical expressions for the optimal solution and offers a two-parameter family of suboptimal filters. It is shown that the conventional least square solutions is a member of this family. The approach is based on a result in linear algebra, and used in robust control theory, known as the dilation equation. An efficient numerical algorithm for solving the filter design problem using the dilation equation is proposed, and some techniques for choosing the design parameters are discussed. Finally, some examples are shown illustrating the flexibility of the design using the new approach.

Wavelets and differential-dilation equations

It is shown how differential-dilation equations can be constructed using iterations, similar to the iterations with which wavelets and dilation equations are constructed. A continuous-time wavelet is constructed starting from a differential-dilation equation. It has compact support and excellent time domain and frequency domain localization properties. The wavelet is infinitely differentiable and therefore cannot be obtained using digital filter banks. In addition, the wavelet has excellent approximation properties. New sampling and differentiation techniques are also introduced. Results on image interpolation using the solution of the differential-dilation equation are presented. Examples are given, demonstrating the suitability of the new wavelet function for signal analysis.

Characterization of Continuous, Four-Coefficient Scaling Functions via Matrix Spectral Radius

We characterize the existence of continuous solutions of a four-coefficient dilation equation in terms of the usual spectral radius of a matrix. The criteria for the existence of such a solution can be very quickly examined. As a result we give an affirmative answer to a conjecture raised by Colella and Heil in 1992. Moreover, using our criteria we find the smoothest compactly supported four-coefficient orthogonal scaling function and thus the smoothest compactly supported orthonormal wavelet generated by this scaling function.

Dilation symmetries and equations on the lattice

We discuss the role of dilation symmetries for differential difference equations depending on nearest-neighbour interactions. In particular, we show that for a simple class of differential difference equations of this kind, symmetries which depend linearly on time are only compatible with linearizable equations.

Dilation Equations with Exponential Decay Coefficients

Dilation equations with finitely many nonzero coefficients have been discussed by many people, since those equations are related to compactly supported wavelets, which constitute an important family of wavelets. However, some properties required in applications are not compatible with the compactness of the support; e.g., it is impossible to find an orthonormal and compactly supported scaling function with the sampling property except for a trivial case. Based on this fact, Xia and Zhang constructed a cardinal and orthonormal scaling function with exponential decay coefficients. It seems such a class of wavelets is interesting. We shall investigate that in this paper and in particular study the relationship between the exponential decay property of a scaling function and that of the corresponding coefficients.

Arbitrarily Smooth Orthogonal Nonseparable Wavelets in $\R^2$

For each $r\in\N$, we construct a family of bivariate orthogonal wavelets with compact support that are nonseparable and have vanishing moments of order r or less. The starting point of the construction is a scaling function that satisfies a dilation equation with special coefficients and a special dilation matrix M: the coefficients are aligned along two adjacent rows, and $\detwo$. We prove that if $\M^2=\pm 2I$, e.\,g., %%$\M=\ourmtx$ or $\M=\qsymtx$, $M=({0\,\,2 \atop 1\,\,0})$ or $M=({1\,\,\phantom{-}1 \atop 1\,\,-1})$, then the smoothness of the wavelets improves asymptotically by $1-\frac{1}{2}\log_23 \approx 0.2075$ when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough.

Multifractal Measures and a Weak Separation Condition

We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the well-known class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of the two-scale dilation equations. Our main purpose in this paper is to prove the multifractal formalism under such condition.

Inhomogeneous refinement equations

Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement equation $$\phi (t) = \sum\limits_{k \in \mathbb{Z}} {a(k)\phi (2t - k) + F(t)}$$ where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.

On Construction of a Family of Smooth Nonseparable Prewavelets via Infinite Products of Triangularizable Matrices

Infinite products of matrices arise in many areas, such as the study of subdivision and interpolation schemes, Markov chains, and construction of wavelets of compact support. These products are used here to give sufficient conditions for the continuity and differentiability of a class of rectangular compactly supported nonseparable N-dimensional prewavelets or scaling functions. This paper considers the dilation equation $\phi(X)=\sum_K C_K \phi(2X-K)$, where $K\in \{0, \ldots, m\}^N$, $\phi : {\cal R}^N \to {\cal R}$, and $C_K \in {\cal R}$. First, the one-dimensional case is studied, and sufficient conditions on CK, which guarantee a continuous scaling function $\phi(X)$, are given. These conditions are based on simultaneous triangularizability of two special matrices with entries in terms of CK. Then, these results are generalized to N dimensions and applied to the particular case where CK's are obtained by binomial interpolation of their values at the corners of the N-cube, $\{0,m\}^N$. A set of inequalities, based on sums of CK's on the corners of various faces of the N-cube gives sufficient conditions for the existence of smooth solutions to the dilation equation.

Characterization of covergent subdivision schemes

Subdivision schemes provide important techniques for the fast generation of curves and surfaces. A recusive refinement of a given control polygon will lead in the limit to a desired visually smooth object. These methods play also an important role in wavelet analysis. In this paper, we use a rather simple way to characterize the convergence of subdivision schemes for multivariate cases. The results will be used to investigate the regularity of the solutions for dilation equations.

A simple constitutive model for granular soils: Modified stress-dilatancy approach

The dependencies of granular soil behaviour on void ratio and stress are modelled within plasticity theory enriched by a modified stress-dilatancy law. Simplicity has been kept in the development of the constitutive law for easy future implementation in afinite element code. By using a void ratio dependent factor which measures the deviation of the current void ratio from the critical one, Rowe's stress dilatancy equation is modified. This modification indeed corresponds to a new energy dissipation equation for a granular assembly which sustains kinematical constraints under the action of stresses. The proposed constitutive law predicts in a very consistent manner the response of sands in monotonic loading conditions for a large range of initial void ratios and confining pressures without the need to make any adjustments to the material parameters. Well known published data for the drained triaxial compression of Sacramento River sand was successfully modelled.

Relativity on a spreadsheet

A use of spreadsheets to model special relativistic phenomena is presented, based on the connection between electric and magnetic fields in special relativity, suitable for use with A-level students. The time dilation equation is used to carry out transformations between reference frames that show the connection between the fields quantitatively and allow further investigations to be carried out.

Asymptotics and Numerics of Zeros of Polynomials That Are Related to Daubechies Wavelets

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision.

On the existence of a compactly supported $L^{p}$-solution for two-dimensional two-scale dilation equations

On the existence of a compactly supported $L^{p}$-solution for two-dimensional two-scale dilation equations

On Some Sharp Regularity Estimations of $L^2 $-Scaling Functions

Let f be a compactly supported $L^2 $-solution of the two-scale dilation equation and $\alpha $ be the $L^2 $-Lipschitz exponent of f. We prove, in addition to other results, that there exists an integer $k \geq 0$ such that (i) $\frac{1}{{h^{2\alpha } |\ln h|^k }}\int_{ - \infty }^\infty {|f(x + h) - f(x)|^2 dx \approx p(h)} $ as $h \to 0^ + $, where p is a nonzero bounded continuous function with $p(2h) = p(h)$, and (ii) for $s > \alpha $, there exists a nonzero bounded continuous q (depends on s) with $q(2T) = q(T)$ and $\frac{1}{{T^{2(s - \alpha )} (\ln T)^k }}\int_{ - T}^T {|\omega ^s \hat f(\omega )|^2 d\omega \approx q(T)} $ as $T \to \infty $. The above $\alpha $ and k can be calculated through a transition matrix. These improve the previous result of Cohen and Daubechies concerning the Besov space containing f and Villemoes’s result on the Sobolev exponent of $\hat f$.