The Navier-Stokes Equations with Time Delay

In the present paper we use a time delay epsilon > 0 for an energy conserving approximation of the nonlinear term of the non-stationary Navier-Stokes equations. We prove that the corresponding initial value problem (N_epsilon)in smoothly bounded domains G \subseteq R^3 is well-posed. Passing to the limit epsilon \rightarrow 0 we show that the sequence of stabilized solutions has an accumulation point such that it solves the Navier-Stokes problem (N_0) in a weak sense (Hopf).

Computing Generators of Free Modules over Orders in Group Algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ.

Computations in Relative Algebraic K-Groups

Let G be finite group and K a number field or a p-adic field with ring of integers O_K. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K_0(O_K[G],K) as an abstract abelian group. We solve the discrete logarithm problem, both in K_0(O_K[G],K) and the locally free class group cl(O_K[G]). All algorithms have been implemented in MAGMA for the case K = \IQ. In the second part of the manuscript we prove formulae for the torsion subgroup of K_0(\IZ[G],\IQ) for large classes of dihedral and quaternion groups.

Construction of fractal interpolation surfaces on rectangular grids

We present a general method of generating continuous fractal interpolation surfaces by iterated function systems on an arbitrary data set over rectangular grids and estimate their Box-counting dimension.

Numerical Simulation of Flows at Low Mach Numbers with Heat Sources

This work is concerned with finite volume methods for flows at low mach numbers which are under buoyancy and heat sources. As a particular application, fires in car tunnels will be considered. To extend the scheme for compressible flow into the low Mach number regime, a preconditioning technique is used and a stability result on this is proven. The source terms for gravity and heat are incorporated using operator splitting and the resulting method is analyzed.

Construction of recurrent fractal interpolation surfaces(RFISs) on rectangular grids

A recurrent iterated function system (RIFS) is a genaralization of an IFS and provides nonself-affine fractal sets which are closer to natural objects. In general, it's attractor is not a continuous surface in R3. A recurrent fractal interpolation surface (RFIS) is an attractor of RIFS which is a graph of bivariate continuous interpolation function. We introduce a general method of generating recurrent interpolation surface which are at- tractors of RIFSs about any data set on a grid.

Numerical Methods for Non-Stationary Stokes Flow

We consider a first order implicit time stepping procedure (Euler scheme) for the non-stationary Stokes equations in smoothly bounded domains of R3. Using energy estimates we can prove optimal convergence properties in the Sobolev spaces Hm(G) (m = 0;1;2) uniformly in time, provided that the solution of the Stokes equations has a certain degree of regularity. For the solution of the resulting Stokes resolvent boundary value problems we use a representation in form of hydrodynamical volume and boundary layer potentials, where the unknown source densities of the latter can be determined from uniquely solvable boundary integral equations’ systems. For the numerical computation of the potentials and the solution of the boundary integral equations a boundary element method of collocation type is used. Some simulations of a model problem are carried out and illustrate the efficiency of the method.

Image compression predicated on recurrent iterated function systems

Recurrent iterated function systems (RIFSs) are improvements of iterated function systems (IFSs) using elements of the theory of Marcovian stochastic processes which can produce more natural looking images. We construct new RIFSs consisting substantially of a vertical contraction factor function and nonlinear transformations. These RIFSs are applied to image compression.

The Parity of the Number of Irreducible Factors for Some Pentanomials

It is well known that Stickelberger-Swan theorem is very important for determining reducibility of polynomials over a binary field. Using this theorem it was determined the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on. We discuss this problem for type II pentanomials namely x^m +x^{n+2} +x^{n+1} +x^n +1 \in\ IF_2 [x]. Such pentanomials can be used for efficient implementing multiplication in finite fields of characteristic two. Based on the computation of discriminant of these pentanomials with integer coefficients, it will be characterized the parity of the number of irreducible factors over IF_2 and be established the necessary conditions for the existence of this kind of irreducible pentanomials.

Parity of the Number of Irreducible Factors for Composite Polynomials

Various results on parity of the number of irreducible factors of given polynomials over finite fields have been obtained in the recent literature. Those are mainly based on Swan’s theorem in which discriminants of polynomials over a finite field or the integral ring Z play an important role. In this paper we consider discriminants of the composition of some polynomials over finite fields. The relation between the discriminants of composed polynomial and the original ones will be established. We apply this to obtain some results concerning the parity of the number of irreducible factors for several special polynomials over finite fields.