We present a scheme that leverage orthonormal or biorthogonal wavelets to a new system of biorthogonal wavelets. The leveraged biorthogonal wavelets will have some nice properties. If we start with orthonormal wavelets, the leveraged... more
We present a scheme that leverage orthonormal or biorthogonal wavelets to a new system of biorthogonal wavelets. The leveraged biorthogonal wavelets will have some nice properties. If we start with orthonormal wavelets, the leveraged scaling functions and wavelets are compactly supported and are diierentiable. The derivatives of the leveraged wavelets are orthogonal to their translations; the derivatives of the leveraged scaling functions are nearly or-thogonal to their translations; and the derivatives of the leveraged scaling functions and wavelets are orthogonal to each other. This feature may be valuable for the numerical solution of diier-ential equations. If we start with B-splines and cooperating with the lifting scheme of Sweldens, our leverage scheme can reproduce all of those biorthogonal wavelets by Cohen, Daubechies and Feauveau. There is a simple algorithm to calculate new lter coeecients from the old lter coeecients. In the end of this article we test the newly constru...
We develop a stable direct method with linear complexity for solving large banded sparse linear systems which are typical for the nite diierence discretization of one-dimensional diierential equations. This method was derived on the basic... more
We develop a stable direct method with linear complexity for solving large banded sparse linear systems which are typical for the nite diierence discretization of one-dimensional diierential equations. This method was derived on the basic idea of multiresolutional decompositions that are fundamental to the theory of wavelets.
We develop a stable direct method with linear complexity for solving large banded sparse linear systems which are typical for the numerical solution of one-dimensional diierential equations. This method was derived on the basic idea of... more
We develop a stable direct method with linear complexity for solving large banded sparse linear systems which are typical for the numerical solution of one-dimensional diierential equations. This method was derived on the basic idea of multiresolutional decompositions that are carried out by the theory of wavelets.
We present a scheme that will lever orthonormal or biorthogonal wavelets to a new system of biorthogonal wavelets. If we start with orthonormal wavelets, the raised scaling functions and wavelets are compactly supported and are... more
We present a scheme that will lever orthonormal or biorthogonal wavelets to a new system of biorthogonal wavelets. If we start with orthonormal wavelets, the raised scaling functions and wavelets are compactly supported and are differentiable. The derivatives of the raised biorthogonal scaling/wavelets forms an almost orthonormal system. If we start with B-splines and cooperating with the lifting scheme of Sweldens, our levering scheme can reproduce all of those biorthogonal wavelets of compact support by Cohen, Daubechies and Feauveau. There is a simple algorithm to construct from the old filter coefficients to the new filter coefficients. 1. MOTIVATION The motivation of this work is the following formula Phi(x) = Z x Gamma1 OE(t) Gamma OE(t Gamma 1) dt (1) that was suggested by Xu and Shann in their paper on Galerkin-wavelets methods [7]. Here OE(x) is an orthonormal scaling function. In that paper, they were studying the Galerkin basis functions derived from orthonormal wa...
Besides probability and data analysis, there is a trend in education all across the world to include risk assessment into the teaching of uncertainty. Taiwan’s “12 Year National Mathematics Education Outline” will be officially... more
Besides probability and data analysis, there is a trend in education all across the world to include risk assessment into the teaching of uncertainty. Taiwan’s “12 Year National Mathematics Education Outline” will be officially implemented in August this year (2019); however, the education system still lacks materials for teaching risk concepts. A probability board game called “Rise of a Singular Cloud” was designed by this study which is targeted at eighth grade students, and explores ways to teach students to understand and solve probability problems via games, while paying special attention to the development of the concept of risk assessment. This experiment also investigates students’ ability to express probabilities in words, and their decision-making thought process. It is anticipated that the eighth grade students will be able to improve their probability knowledge and ability to express probabilities in words and will be able to cultivate an understanding of risk. From data...
Abstract. Let be a group of Heisenberg type with homogeneous dimension Q. For every 0 <ϵ<Q we construct a non-divergence form operator Lϵ and a non-trivial solution uϵ ∈ L2,Q−ϵ(Ω) ∩ C(Ω) to the Dirichlet problem: Lu = 0 in Ω, u = 0... more
Abstract. Let be a group of Heisenberg type with homogeneous dimension Q. For every 0 <ϵ<Q we construct a non-divergence form operator Lϵ and a non-trivial solution uϵ ∈ L2,Q−ϵ(Ω) ∩ C(Ω) to the Dirichlet problem: Lu = 0 in Ω, u = 0 on ∂Ω. This ...
