A movement analysis of objects contained in visual scenes can be performed by means of linear multidimensional filters, which have already been analyzed in the past. While the soundness of the results was convincing, interest in those systems declined due to the limited computational power of contemporary computers. Recent advances in design and implementation of integrated circuits and hardware architectures allow realizing velocity filters if the n-D system is carefully adapted to the analyzed...

Here, we present determinants of some square matrices of field elements. First, the determinat of 2 * 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is the same as the determinant of its transpose.

The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code $C_L(D,G)$ on a curve is the differential code $C_{\Omega }(D,G)$ . We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples...

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and ${\mu }_i\in -1,1(i=1,2)$, and let $h(-2^{1-e}D)(e=0or1)$ denote the class number of the imaginary quadratic field $\mathbb{Q}\left(\surd (-2^{1-e}D)\right)$. In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then $h(-2^{1-e}D)\equiv 0\left(modn\right)$, where D, and n satisfy $kⁿ-2^{e+1}=Dx²$, x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.

Nous donnons ici deux résultats sur le déterminant $\zeta$-régularisé ${det}_{\zeta }A$ d’un opérateur de Schrödinger $A={\Delta }_g+V$ sur une variété compacte $ℳ$. Nous construisons, pour $ℳ=S^1\times S^1$, une suite $(G_n,{\rho }_n,{\Delta }_n)$ où $G_n$ est un graphe fini qui se plonge dans $ℳ$ via ${\rho }_n$ de telle manière que ${\rho }_n\left(G_n\right)$ soit une triangulation de $ℳ$ et où ${\Delta }_n$ est un laplacien discret sur $G_n$ tel que pour tout potentiel $V$ sur $ℳ$, la suite de réels $det({\Delta }_n+V)$ converge après renormalisation vers ${det}_{\zeta }({\Delta }_g+V)$. Enfin, nous donnons sur toute variété riemannienne compacte $(ℳ,g)$ de dimension inférieure ou égale à $3$...