Noncommutative analysis of Hermite expansions

An image watermarking scheme that combines Hermite functions expansion and space/spatial-frequency analysis is proposed. In the first step, the Hermite functions expansion is employed to select busy regions for watermark embedding. In the second step, the space/spatial-frequency representation and Hermite functions expansion are combined to design the imperceptible watermark, using the host local frequency content. The Hermite expansion has been done by using the fast Hermite projection method. Recursive realization of Hermite functions significantly speeds up the algorithms for regions selection and watermark design. The watermark detection is performed within the space/spatial-frequency domain. The detection performance is increased due to the high information redundancy in that domain in comparison with the space or frequency domains, respectively. The performance of the proposed procedure has been tested experimentally for different watermark strengths, i.e., for different values of the peak signal-to-noise ratio (PSNR). The proposed approach provides high detection performance even for high PSNR values. It offers a good compromise between detection performance (including the robustness to a wide variety of common attacks) and imperceptibility.

The purpose of this paper is to study coefficient multipliers of the Hardy spaces \(H^p({\mathbb R})\) associated with Hermite expansions. The main results are that, if a sequence \(\{\lambda _n\}_{n=0}^{\infty }\) satisfies the condition \(\sum _{k=n}^{2n}|\lambda _k|^q=O\left( n^{\frac{q}{2}\left( \frac{7}{6}-\frac{1}{p}\right) }\right) \), then \(\{\lambda _n\}\) is a multiplier of \(H^p({\mathbb R})\) into the sequence space \(\ell ^q\) associated with Hermite expansions for (i) \(p=1\), \(2\le q 1\) and for \(f\in H^p({\mathbb R})\), \(0<p\le 1\), the coefficients \(a_n(f)\) of its Hermite expansion satisfy \(\sum _{k=1}^{\infty }n_k^{\frac{7}{6}-\frac{1}{p}}|a_{n_k}(f)|^2<\infty \). The results in the paper are proved in a more general case, that is, for the generalized Hermite functions which are defined by \({\mathcal {H}}_{2k}^{(\lambda )}(x)=c_kL_k^{(\lambda -1/2)}(x^2)e^{-\frac{x^2}{2}}|x|^{\lambda }\), \({\mathcal {H}}_{2k-1}^{(\lambda )}(x)=c_k k^{-1/2}xL_{k-1}^{(\lambda +1/2)}(x^2)e^{-\frac{x^2}{2}}|x|^{\lambda }\), where \(c_k=\left( 2k!/\Gamma (k+\lambda +1/2)\right) ^{1/2}\). Note that \({\mathcal {H}}_n(x)={\mathcal {H}}_n^{(0)}(x)\) (\(n\ge 0\)) are the usual Hermite functions.

Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the Triebel-Lizorkin and Besov spaces on ℝ d induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite-Triebel-Lizorkin and Besov spaces are, in general, different from the respective classical spaces.

In this study, a new filter-matrix lattice Boltzmann (FMLB) model is proposed and extended to include incompressible thermal flows. A new equilibrium solution is found in the improved FMLB model, which is derived from the Hermite expansion. As a result, the velocity-dependent pressure is removed, which is an inherent defect of Somers's FMLB model. In addition, the improved model is extended to include incompressible thermal flows by introducing a class of temperature-distribution function for evaluating the temperature field. Two different temperature-distribution functions are discussed. The improved FMLB model and the temperature-evaluation equation are combined into one coupled model. Numerical simulations are performed on the two-dimensional (2D) lid-driven square cavity flow and the 2D natural convection flow in a square cavity using the improved FMLB model and the two coupled models, respectively. The numerical results of the 2D lid-driven square cavity flow show that the improved FMLB model is superior to the lattice Bhatnagar-Gross-Krook (LBGK) model in terms of both accuracy and stability. When compared with the multi-relaxation-time (MRT) model, the similar accuracy and slightly enhanced stability can be obtained by the improved model. The advantage of the improved model is that it no longer relies on difficult selection of the free parameters requested by the MRT model; in addition, the force term is already included in the collision operator of the improved model. In the case of 2D natural convection flow, the numerical results of the two present models are almost the same, and both exhibit good agreement with the benchmark solution.

Owing to its high accuracy, the radial basis function (RBF) is gaining popularity in function interpolation and for solving partial differential equations (PDEs). The implementation of RBF methods is independent of the locations of the points and the dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is mainly affected by the basis function and the node distribution. If the shape parameter has a small value, then the RBF becomes accurate but unstable. Several approaches have been proposed in the literature to overcome the instability issue. Changing or expanding the radial basis function is one of the most commonly used approaches because it addresses the stability problem directly. However, the main issue with most of those approaches is that they require the optimization of additional parameters, such as the truncation order of the expansion, to obtain the desired accuracy. In this work, the Hermite polynomial is used to expand the RBF with respect to the shape parameter to determine a stable basis, even when the shape parameter approaches zero, and the approach does not require the optimization of any parameters. Furthermore, the Hermite polynomial properties enable the RBF to be evaluated stably even when the shape parameter equals zero. The proposed approach was benchmarked to test its reliability, and the obtained results indicate that the accuracy is independent of or weakly dependent on the shape parameter. However, the convergence depends on the order of the truncation of the expansion. Additionally, it is observed that the new approach improves accuracy and yields the accurate interpolation, derivative approximation, and PDE solution.

then f(x) = ce-x. In ?2 a new proof is given using Liouville's theorem. So far as I am aware, the method of proof does not apply to any essentially more general problems, depending as it does on unique properties of the analytic function ez. However, I have obtained a few similar results by other methods, and to establish a sufficiently broad framework to contain all cases, it is convenient to pose a somewhat indefinite general problem: PROBLEM A. Let C1 be a linear space of functions; let T be an operator on Ci to C1; and let .| be a norm defined on C1. When does fEC1 and II TkfII _1, k=O, 1, 2, ,imply that fC C2 where C2 is a finite-dimensional subspace of C1? A narrower framework will also be considered: PROBLEM B. Let T be a differential operator and let g be a function such that Tg -g (or +g). When does I Tkf| <gi, k=O, 1, 2, imply f = cg? A very simple result of type A is obtained in ?3 by taking T to be a self-adjoint operator on a Hilbert space. Differential operators are natural choices for T, and some special cases are spelled out using Fourier and Hermite expansions. A corollary of type B is given. In ?4, I consider the possibility of generalizing results for particular operators T to results for conjugate operators of the form STS-1. For Problem B a natural choice for S is an operator which acts in effect to change the variables of the differential equation Tf = -f. By employing this device, results are obtained for example for T=pl(x)D +po(x) and for T=AD2+BD+C using respectively Tagamlitzki's theorem and a Fourier expansion.

Reading Interest Rate and Bond Futures Options' Smiles: How PIBOR and Notional Operators Appreciated the 1997 French Snap Election

Propagating waves in a binary gas mixture from a boundary of variable temperatures and velocities are discussed on the basis of two Boltzmann equations. Variable temperatures on the boundary could be gotten by using a thin thermal absorbent with fine holes heated by laser lights of variable powers, and variable velocities on a boundary could be gotten by moving gases through a porous media. In this paper, the Hermite expansion method (Grad [1]) of the distribution function would be extended to a binary gas-mixture, i.e. the Herimite expansion method is used to solve two distribution functions for both gases. We have two Boltzmann equations expanded by Hermite functions. The Galerkin method is applied to solve two Boltzmann equations, i.e. the scalar products of Hermite functions and two expanded Boltzmann equations are executed. Thus, a system of partial differential equations for the expansion coefficients, which are functions of position and time, is obtained. The Hermite expansion is taken up to the third order, so that we have twenty coefficients and a system of twenty partial differential equations. Here, the one dimensional flow in space is considered, so that the number of expansion coefficients and differential equations would reduce from twenty to eight. The intermolecular force consisting of an inverse fifth- and inverse third- order power of intermolecular distance. We have applied this method to the relaxation phenomena of the binary gas mixture with different temperatures and velocities, which are both uniform in space ([2]and [3]). In this paper, we are going to discuss propagating waves in the gas mixture. Temperatures, number densities and velocities at x = 0 could be given. We calculate numerically the time developments of velocity, temperature and number density. The initial conditions of \( v_1^{(i)}(x,0), T^{(i)}(x,0), n^{(i)}(x,0) \) etc. are taken as uniform in 0 ≤ x ≤ ∞ . The boundary conditions at x = 0 are taken as \( T^{(i)}(0,t)=T_b^{(i)}(t),\> v_1^{(i)}(0,t)=u_b^{(i)}(t), \> n^{(i)}(0,t)=n_b^{(i)}(t) \). Time developments of temperatures, mass velocities and number densities etc. have been calculated.

In this paper, we introduce the neutral Hermite product and study products of tempered distributions based on the Hermite expansions. In particular, we consider products of distributions x n and δ(m)(x) by using the neutral Hermite product and the results are identical with the classical theory of Schwartz distributions. Finally, the recurrence relations about the Hermite expansion coefficients of x ±n δ(m)(x) are obtained for the numerical computation.

The likelihood functions for discretely observed nonlinear continuous time models based on stochastic differential equations are not available except for a few cases. Various parameter estimation techniques have been proposed, each with advantages, disadvantages and limitations depending on the application. Most applications still use the Euler–Maruyama discretization, despite many proofs of its bias. More sophisticated methods, such as Kessler’s Gaussian approximation, Ozaki’s local linearization, Aït–Sahalia’s Hermite expansions or MCMC methods, might be complex to implement, do not scale well with increasing model dimension or can be numerically unstable. We propose two efficient and easy-to-implement likelihood-based estimators based on the Lie–Trotter (LT) and the Strang (S) splitting schemes. We prove that S has Lp convergence rate of order 1, a property already known for LT. We show that the estimators are consistent and asymptotically efficient under the less restrictive one-sided Lipschitz assumption. A numerical study on the 3-dimensional stochastic Lorenz system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on precision and computational speed compared to the state-of-the-art.

We present new HST/STIS observations of Centaurus A. [SIII] 9533A was used to study the kinematics in the nuclear region with a 0.1" spatial resolution. The STIS data and the VLT/ISAAC spectra by Marconi et al. (2001) provide independent and consistent measures of the BH mass, which are in agreement with our previous estimate based on the ISAAC data alone: MBH=(1.1+/-0.1) 10^8 Msun for a disk inclination of i=25deg or or MBH=(6.5+/-0.7) 10^7 Msun for i=35deg, the largest i value allowed by the data. We find that the choice of the intrinsic surface brightness distribution, a crucial element in the modeling, has no effects on MBH but has a large impact on the gas velocity dispersion. A mismatch between the observed and model velocity dispersion is not necessarily an indication of non-circular motions or kinematically hot gas, but is as easily due to an inaccurate computation arising from too course a model grid, or the adoption of an intrinsic brightness distribution which is too smooth. The observed velocity dispersion, line profiles and the higher order moments in the Hermite expansion, h_3 and h_4, are consistent with emission from a rotating disk. Results from gas kinematical estimate are in good agreement with a recent stellar dynamical estimate of MBH. The BH mass in Centaurus A agrees with the correlation with infrared luminosity and mass of the host spheroid but is not in disagreement with the stellar velocity dispersion if one takes into account the intrinsic scatter of the MBH-sigma correlation. Finally, using HST data we can constrain the size of any cluster of dark objects alternative to a BH to r<0.035" (~0.6pc). Thus Centaurus A ranks among the best cases for supermassive Black Holes in galactic nuclei. (ABRIDGED)

Two three-dimensional (3D) lattice Boltzmann models in the framework of coupled double-distribution-function approach for compressible flows, in which specific-heat ratio and Prandtl number can be adjustable, are developed in this paper. The main differences between the two models are discrete equilibrium density and total energy distribution function. One is the D3Q25 model obtained from spherical function, and the other is the D3Q27 standard lattice model obtained from Hermite expansions of the corresponding continuous equilibrium distribution functions. The two models are tested by numerical simulations of some typical compressible flows, and their numerical stability and precision are also analysed. The results indicate that the two models are capable for supersonic flows, while the one from Hermite expansions is not suitable for compressible flows with shock waves.

The delta expansion for the transition density of diffusion models

We apply the Wiener Hermite (WH) expansion to the non-linear evolution of\nLarge-Scale Structure, and obtain an approximate expression for the matter\npower spectrum in full order of the expansion. This method allows us to expand\nany random function in terms of an orthonormal bases in space of random\nfunctions in such a way that the first order of the expansion expresses the\nGaussian distribution, and others are the deviation from the Gaussianity. It is\nproved that the Wiener Hermite expansion is mathematically equivalent with the\n$\\Gamma$-expansion approach in the Renormalized Perturbation Theory (RPT).\nWhile an exponential behavior in the high-$k$ limit has been proved for the\nmass density and velocity fluctuations of Dark Matter in the RPT, we reprove\nthe behavior in the context of the Wiener Hermite expansion using the result of\nStandard Perturbation Theory (SPT). We propose a new approximate expression for\nthe matter power spectrum which interpolates the low-$k$ expression\ncorresponding to 1-loop level in SPT and the high-$k$ expression obtained by\ntaking a high-$k$ limit of the WH expansion. The validity of our prescription\nis specifically verified by comparing with the 2-loop solutions of the SPT. The\nproposed power spectrum agrees with the result of $N$-body simulation with the\naccuracy better than 1% or 2% in a range of the BAO scales, where the wave\nnumber is about $k= 0.2 \\sim 0.4$ $h{\\rm Mpc^{-1}}$ at $z=0.5-3.0$. This\naccuracy is comparable to or slightly less than the ones in the Closure Theory\nwhose the fractional difference from N body result is within 1%. A merit of our\nmethod is that the computational time is very short because only single and\ndouble integrals are involved in our solution.\n

Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\al\in[-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validate that the known version of Hardy's inequality for the Hermite functions is sharp.