AN MRA YIELDS AN ORTHONORMAL WAVELET BASIS AN OPERATOR THEORETIC PROOF

Abstract. In this paper we show how to construct a scaling function and an orthonormal wavelet basis from a multiresolution approximation using an operator theoretic method.     

Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications

A maximum principle is proved for the weak solutions of the telegraph equation in space dimension three u_tt−Δ_xu+cu_t+λu=f(t,x), when c>0, λ∈(0,c²/4] and (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u_tt−Δ_xu+cu_t=F(t,x,u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given.     

Gaps and Fatou theorems for series in Schauder basis of holomorphic functions

We prove a gap theorem and the “Fatou change-of-sign theorem” [Fatou, P., 1906, Sèries trigonométriques e séries de Taylor. Acta Mathematica, 39, 335–400] for expansions in common Schauder basis of holomorphic functions.     

A greedy algorithm for partition of unity collocation method in pricing American options

A greedy algorithm in combination with radial basis functions partition of unity collocation (GRBF‐PUC) scheme is used as a locally meshless method for American option pricing. The radial basis function partition of unity method (RBF‐PUM) is a localization technique. Because of having interpolation matrices with large condition numbers, global approximants and some local ones suffer from instability. To overcome this, a greedy algorithm is added to RBF‐PUM. The greedy algorithm furnishes a subset of best nodes among the points X. Such nodes are then used as points of trial in a locally supported RBF approximant for each partition. Using of greedy selected points leads to decreasing the condition number of interpolation matrices and reducing the burdensome in pricing American options.     

An extension of almost sure central limit theorem for order statistics

Let {X _n ,n ≥ 1} be a sequence of i.i.d. random variables. Let M _n and m _n denote the first and the second largest maxima. Assume that there are normalizing sequences a _n  > 0, b _n and a nondegenerate limit distribution G, such that . Assume also that {d _k ,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have

A note on the existence of positive bounded solutions for an epidemic model

In this short note, we establish an existence and uniqueness theorem about a positive bounded solution for a nonlinear infinite delay integral equation, which arises in some epidemic problems. As one can see, our main result can deal with some cases, to which many previous results cannot be applied. In addition, we show that our main result can also be applied to a Lasota–Wazewska model.     

An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function (x;α,h)≡exp(-[α²/h²]x²) where α is a user-choosable constant. Unfortunately, computing the coefficients of (x-jh;α,h) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C_j(x;α,h) is defined by the set of linear combinations of the Gaussians such that C_j(kh)=1 when k=j and C_j(kh)=0 for all integers . We show that the cardinal functions for the uniform grid are C_j(x;h,α)=C(x/h-j;α) where C(X;α)≈(α²/π)sin(πX)/sinh(α²X). The relative error is only about 4exp(-2π²/α²) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h→0, but only to an “error saturation” proportional to exp(-π²/α²). Because the error in our approximation to the master cardinal function C(X;α) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f(x). The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.     

An almost sure central limit theorem for products of sums of partial sums under association

Let be a strictly stationary positively or negatively associated sequence of positive random variables with EX₁=μ>0, and VarX₁=σ²<∞. Denote , and γ=σ/μ the coefficient of variation. Under suitable conditions, we show that

     

A note on the computation of an orthonormal basis for the null space of a matrix

A highly regarded method to obtain an orthonormal basis,Z, for the null space of a matrixA ^T is theQR decomposition ofA, whereQ is the product of Householder matrices. In several optimization contextsA(x) varies continuously withx and it is desirable thatZ(x) vary continuously also. In this note we demonstrate that thestandard implementation of theQR decomposition doesnot yield an orthonormal basisZ(x) whose elements vary continuously withx. We suggest three possible remedies. Work supported in part by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.     

An almost fourth order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction-diffusion equations

We consider a system of M(≥2) singularly perturbed equations of reaction-diffusion type coupled through the reaction term. A high order Schwarz domain decomposition method is developed to solve the system numerically. The method splits the original domain into three overlapping subdomains. On two boundary layer subdomains we use a compact fourth order difference scheme on a uniform mesh while on the interior subdomain we use a hybrid scheme on a uniform mesh. We prove that the method is almost fourth order ε-uniformly convergent. Furthermore, we prove that when ε is small, one iteration is sufficient to get almost fourth order ε-uniform convergence. Numerical experiments are performed to support the theoretical results.     

Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree

In this paper,we study the strong law of large numbers and Shannon-McMillan (S-M) theorem for Markov chains indexed by an infinite tree with uniformly bounded degree.The results generalize the analogous results on a homogeneous tree.     

An alternative approach to the uniqueness of unconditional basis of for

We give an alternative and much simpler proof of the uniqueness of unconditional basis (up to equivalence and permutation) in the quasi-Banach spaces ℓ_p(c₀) for 0<p<1 and its complemented subspaces with unconditional basis. The new approach uses the fact that the Banach envelope of these spaces is not sufficiently Euclidean with the lattice structure induced by its unconditional basis.     

An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables

Let X,X₁,X₂,… be a sequence of independent and identically distributed positive random variables with EX=μ>0. In this paper we show that the almost sure central limit theorem for self-normalized products of sums holds only under the assumptions that X belongs to the domain of attraction of the normal law.     

Empirical Gramian-based spatial basis functions for model reduction of nonlinear distributed parameter systems

Correct selection of spatial basis functions is crucial for model reduction for nonlinear distributed parameter systems in engineering applications. To construct appropriate reduced models, modelling accuracy and computational costs must be balanced. In this paper, empirical Gramian-based spatial basis functions were proposed for model reduction of nonlinear distributed parameter systems. Empirical Gramians can be computed by generalizing linear Gramians onto nonlinear systems, which results in calculations that only require standard matrix operations. Associated model reduction is described under the framework of Galerkin projection. In this study, two numerical examples were used to evaluate the efficacy of the proposed approach. Lower-order reduced models were achieved with the required modelling accuracy compared to linear Gramian-based combined spatial basis function- and spectral eigenfunction-based methods.     

Criterion of Bari basis property for 2 × 2 Dirac-type operators with strictly regular boundary conditions

The paper is concerned with the Bari basis property of a boundary value problem associated in $L^2([0,1];{\mathbb{C}}^2)$ with the following 2 × 2 Dirac-type equation for $y=col(y_1,y_2)$: with a potential matrix $Q\in L^2([0,1];{\mathbb{C}}^{2\times 2})$ and subject to the strictly regular boundary conditions $Uy:=\{U_1,U_2\}y=0$. If $b_2=-b_1=1$, this equation is equivalent to one-dimensional Dirac equation. We show that the normalized system of root vectors ${\{f_n\}}_{n\in \mathbb{Z}}$ of the operator $L_U(Q)$ is a Bari basis in $L^2([0,1];{\mathbb{C}}^2)$ if and only if the unperturbed operator $L_U(0)$ is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let $Q\in L^p([0,1];{\mathbb{C}}^{2\times 2})$, $p\in [1,2]$, boundary conditions be strictly regular, and let ${\{g_n\}}_{n\in \mathbb{Z}}$ be the sequence biorthogonal to the normalized system of root vectors ${\{f_n\}}_{n\in \mathbb{Z}}$ of the operator $L_U(Q)$. Then, $$\{\parallel f_n-g_n{{\parallel }_2\}}_{n\in \mathbb{Z}}\in {({\ell }^p(\mathbb{Z}))}^{\ast }\iff L_U(0)=L_U{(0)}^{\ast }.$$ These abstract results are applied to noncanonical initial-boundary value problem for a damped string equation.     

An efficient estimator for the expectation of a bounded function under the residual distribution of an autoregressive process

Consider a stationary first-order autoregressive process, with i.i.d. residuals following an unknown mean zero distribution. The customary estimator for the expectation of a bounded function under the residual distribution is the empirical estimator based on the estimated residuals. We show that this estimator is not efficient, and construct a simple efficient estimator. It is adaptive with respect to the autoregression parameter.     

Radial basis function Hermite collocation approach for the numerical simulation of the effect of precipitation inhibitor on the crystallization process of an over‐saturated solution

This work is concerned with the analysis of the effect of precipitation inhibitors on the growth of crystals from over‐saturated solutions, by the numerical simulation of the fundamental mechanisms of such crystallization process. The complete crystallization process in the presence of precipitation inhibitor is defined by a set of coupled partial differential equations that needs to be solved in a recursive manner, due to the inhibitor modification of the molar flux of the mineral at the crystal interface. This set of governing equations needs to satisfy the corresponding initial and boundary conditions of the problem where it is necessary to consider the additional unknown of a moving interface, i.e., the growing crystal surface. For the numerical solution of the proposed problem, we used a truly meshless numerical scheme based upon Hermite interpolation property of the radial basis functions. The use of a Hermitian meshless collocation numerical approach was selected in this work due to its flexibility on dealing with moving boundary problems and their high accuracy on predicting surface fluxes, which is a crucial part of the diffusion controlled crystallization process considered here. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006