Abstract Given finite configurations $P_1, \dots , P_n \subset \mathbb {R}^d$ let us denote by $\mathbf {m}_{\mathbb {R}^d}(P_1, P_n)$ the maximum density a set $A \subseteq can have without containing congruent copies of any $P_i$ . We will initiate study this geometrical parameter, called independence considered configurations, and give several results we believe are interesting. For instance...

We introduce the family of Theta operators $\Theta_f$ indexed by symmetric functions $f$ that allow us to conjecture a compositional refinement Delta Haglund, Remmel and Wilson for $\Delta_{e_{n-k-1}}'e_n$. show $4$-variable Catalan theorem Zabrocki is precisely Schr\"{o}der case our conjecture, we how relate this Dyck path algebra introduced Carlsson Mellit, extending one their results. Again ...

a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...

We are interested in the statistical behavior of a certain continued fraction whose associated dynamical system has infinite invariant measure. show growth rate denominator $Q_n$ $n$-th convergent negative expansion $x$ and approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{\pi^2}{3} \quad \text{in measure} for $x$. In course proof, we reprove related resu...

We present an asymptotic preserving method for the radiative transfer equations in framework of $$P_N$$ method. An implicit and explicit numerical scheme is proposed to solve system based on order analysis expansion coefficients specific intensity, where each coefficient derived by Chapman-Enskog The at higher-order are treated explicitly while those lower-order implicitly equation system. Ener...