A hybrid Laplace transform/finite difference boundary element method for diffusion problems

The solution process for diffusion problems usually involves the time development separately from the space solution. A finite difference algorithm in time requires a sequential time development in which all previous values must be determined prior to the current value. The Stehfest Laplace transform algorithm, however, allows time solutions without the knowledge of prior values. It is of interest to be able to develop a time-domain decomposition suitable for implementation in a parallel environment. One such possibility is to use the Laplace transform to develop coarse-grained solutions which act as the initial values for a set of fine-grained solutions. The independence of the Laplace transform solutions means that we do indeed have a time-domain decomposition process. Any suitable time solver can be used for the fine-grained solution. To illustrate the technique we shall use an Euler solver in time together with the dual reciprocity boundary element method for the space solution

Reuseable Silicon IP Cores for Discrete Wavelet Transform Applications

Architectures and methods for the rapid design of silicon cores for implementing discrete wavelet transforms over a wide range of specifications are described. These architectures are efficient, modular, scalable, and cover orthonormal and biorthogonal wavelet transform families. They offer efficient hardware utilization by exploiting a number of core wavelet filter properties and allow the creation of silicon designs that are highly parameterized, including in terms of wavelet type and wordlengths. Control circuitry is embedded within these systems allowing them to be cascaded for any desired level of decomposition without any interface glue logic. The time to produce chip designs for a specific wavelet application is typically less than a day and these are comparable in area and performance to handcrafted designs. They are also portable across a wide range of silicon foundries and suitable for field programmable gate array and programmable logic data implementation. The approach described has also been extended to wavelet packet transforms.

Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.