On affine classification of permutations on the space GF(2)3

We give an elementary proof that by multiplication on left and right by affine permutations A, B ∈ AGL(3, 2) each permutation π : GF(2)3 → GF(2)3 may be reduced to one of the 4 permutations for which the 3 × 3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory.

Arithmetic-Analytic Representation of Peano Curve

In this work, we obtained a nonmatrix analytic expression for the generator of the Peano curve. Applying the iteration method of fractal, we established a simple arithmetic-analytic representation of the Peano curve as a function of ternary numbers. We proved that the curve passes each point in a unit square and that the coordinate functions satisfy a Hölder inequality with index α=1/2, which implies that the curve is everywhere continuous and nowhere differentiable.

Improved asymptotic estimates for the numbers of correlation-immune and k-resilient vectorial Boolean functions

We refine local limit theorems for the distribution of a part of the weight vector of subfunctions and for the distribution of a part of the vector of spectral coefficients of linear combinations of coordinate functions of a random binary mapping. These theorems are used to derive improved asymptotic estimates for the numbers of correlation-immune and k-resilient vectorial Boolean functions.

Plotting the map projection graticule involving discontinuities based on combined sampling

This article presents new algorithm for interval plotting the projection graticule on the interval $\varOmega=\varOmega_{\varphi}\times\varOmega_{\lambda}$ based on the combined sampling technique. The proposed method synthesizes uniform and adaptive sampling approaches and treats discontinuities of the coordinate functions $F,G$. A full set of the projection constant values represented by the projection pole $K=[\varphi_{k},\lambda_{k}]$, two standard parallels $\varphi_{1},\varphi_{2}$ and the central meridian shift $\lambda_{0}^{\prime}$ are supported. In accordance with the discontinuity direction it utilizes a subdivision of the given latitude/longitude intervals $\varOmega_{\varphi}=[\underline{\varphi},\overline{\varphi}]$, $\varOmega_{\lambda}=[\underline{\lambda},\overline{\lambda}]$ to the set of disjoint subintervals $\varOmega_{k,\varphi}^{g},$$\varOmega_{k,\lambda}^{g}$ forming tiles without internal singularities, containing only "good" data; their parameters can be easily adjusted. Each graticule tile borders generated over $\varOmega_{k}^{g}=\varOmega_{k,\varphi}^{g}\times\varOmega_{k,\lambda}^{g}$ run along singularities. For combined sampling with the given threshold $\overline{\alpha}$ between adjacent segments of the polygonal approximation the recursive approach has been used; meridian/parallel offsets are $\Delta\varphi,\Delta\lambda$. Finally, several tests of the proposed algorithms are involved.