In this paper, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates. We then use elementary linear subspaces to derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable (MDS) codes. Using these properties, we show that, for MRD codes with error correction capability t, the deco...

This paper is concerned with the study of the stationary solutions of the equation ut −∇ · (A(x)∇u) = f(x, u), x ∈ R , where the diffusion matrix A and the reaction term f are periodic in x. We prove existence and uniqueness results for the stationary equation and we then analyze the behaviour of the solutions of the evolution equation for large times. These results are expressed by a condition...

The parameters of a linear code C over GF(q) are given by [n, k, d], where n denotes the length, k the dimension and d the minimum distance of C . The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n−k+1. Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which hav...

We study a class of ergodic BSDEs related to PDEs with Neumann boundary conditions. The randomness of the driver is given by a forward process under weakly dissipative assumptions with an invertible and bounded diffusion matrix. Furthermore, this forward process is reflected in a convex subset of R not necessarily bounded. We study the link of such EBSDEs with PDEs and we apply our results to a...

Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions i...

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the so-called generalized Kimura diffusion operators [8, 9, 6]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scaleinvariant Harnack inequality fo...

The MacWilliams identity, which relates the weight distribution of a code to the weight distribution of its dual code, is useful in determining the weight distribution of codes. In this paper, we first provide an alternative proof to the MacWilliams identity for linear codes with the Hamming metric. An intermediate result of our approach is that the Hamming weight enumerator of the dual of any ...

The work concerns the a posteriori error estimation for the primal and the dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting, with inhomogeneous mixed Dirichlet/Neumann boundary conditions. The new, functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and...

In this paper, a recently developed fiber tracking algorithm to be used with diffusion tensor (DT) fields acquired via magnetic resonance imaging (MRI) is improved and applied to real brain DT-MR images. The method performs satisfactorily in regions where branching and crossing fibers exist and offers the capability of reporting a probability value for the computed tracts. This certainty figure...

In this thesis, we study continuous diffusions in random environment on Rd . The randomness in the environment is incorporated in the diffusion matrix and the drift term that are stationary random variables. Once the environment is chosen, it remains fixed in time, and the diffusion evolving in this frozen environment is a Markov process. To restore some stationarity, it is common to average bo...