Monotone $L\sb 1$-approximation on the unit $n$-cube

On approximation of asymmetric separators of the n-cube

* Correspondence: boronskij@mytu. tuskegee.edu Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30059 Kraków, Poland Full list of author information is available at the end of the article Abstract A new combinatorial result intertwined with the Brouwer fixed point theorem for the ncube is given. This result can be used for any map (f1, ..., fn): [0, ...

Derangements on the n-cube

Chen, W.Y.C. and R.P. Stanley, Derangements on the n-cube, Discrete Mathematics 115 (1993) 65-15. Let Q. be the n-dimensional cube represented by a graph whose vertices are sequences of O’s and l’s of length n, where two vertices are adjacent if and only if they differ only at one position. A k-dimensional subcube or a k-face of Q. is a subgraph of Q. spanned by all the vertices u1 u2 u, with c...

Linear transformations of monotone functions on the discrete cube

For a function f : {0, 1} → R and an invertible linear transformation L ∈ GLn(2), we consider the function Lf : {0, 1} → R defined by Lf(x) = f(Lx). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I(Lf) ≥ I(f), where I(f) is the total influence of f . Second, we conjecture that if both f and L(f) are monotone, then f = L(f) (up to a permutation of the coord...

Negative Theorems on Monotone Approximation

Capacitance of the Unit Cube

Computing the capacitance of the unit cube analytically is considered to be “one of the major unsolved problems of electrostatic theory.” However, due to improvements in computer performance and error analysis for “Walk on Spheres” (WOS) Monte Carlo algorithms, we can now calculate the capacitance of the unit cube to many more significant digits than previously possible by using a modified Brow...