A block diagonalization theorem in the energy-momentum method

We prove a geometric generalization of a block diagonalization theorem first found by the authors for rotating elastic rods. The result here is given in the general context of simple mechanical systems with a symmetry group acting by isometries on a configuration manifold. The result provides a choice of variables for linearized dynamics at a relative equilibrium which block diagonalizes the second variation of an augmented energy these variables effectively separate the rotational and internal vibrational modes. The second variation of the effective Hamiltonian is block diagonal. separating the modes completely. while the symplectic form has an off diagonal term which represents the dynamic interaction between these modes. Otherwise, the symplectic form is in a type of normal form. The result sets the stage for the development of useful criteria for bifurcation as well as the stability criteria found here. In addition, the techniques should apply to other systems as well, such as rotating fluid masses

Some New Bounds For Cover-Free Families Through Biclique Cover

An $(r,w;d)$ cover-free family $(CFF)$ is a family of subsets of a finite set such that the intersection of any $r$ members of the family contains at least $d$ elements that are not in the union of any other $w$ members. The minimum number of elements for which there exists an $(r,w;d)-CFF$ with $t$ blocks is denoted by $N((r,w;d),t)$. In this paper, we show that the value of $N((r,w;d),t)$ is equal to the $d$-biclique covering number of the bipartite graph $I_t(r,w)$ whose vertices are all $w$- and $r$-subsets of a $t$-element set, where a $w$-subset is adjacent to an $r$-subset if their intersection is empty. Next, we introduce some new bounds for $N((r,w;d),t)$. For instance, we show that for $r\geq w$ and $r\geq 2$ $$N((r,w;1),t) \geq c{{r+w\choose w+1}+{r+w-1 \choose w+1}+ 3 {r+w-4 \choose w-2} \over \log r} \log (t-w+1),$$ where $c$ is a constant satisfies the well-known bound $N((r,1;1),t)\geq c\frac{r^2}{\log r}\log t$. Also, we determine the exact value of $N((r,w;d),t)$ for some values of $d$. Finally, we show that $N((1,1;d),4d-1)=4d-1$ whenever there exists a Hadamard matrix of order 4d