An algorithm to prescribe the configuration of a finite graph

We provide algorithms involving edge slides, for a connected simple graph to evolve in a finite number of steps to another connected simple graph in a prescribed configuration, and for the regularization of such a graph by the minimization of an appropriate energy functional

Strong Ramsey Games in Unbounded Time

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ wins. If none of the players win, the game is declared a draw. A notorious open question of Beck asks whether the first player has a winning strategy in $\mathcal{R}(K_n,K_k)$ in bounded time as $n\rightarrow\infty$. Surprisingly, in a recent paper Hefetz et al. constructed a $5$-uniform hypergraph $\mathcal{H}$ for which they proved that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(5)},\mathcal{H})$ in bounded time. They naturally ask whether the same result holds for graphs. In this paper we make further progress in decreasing the rank. In our first result, we construct a graph $G$ (in fact $G=K_6\setminus K_4$) and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n \sqcup K_n,G)$ in bounded time. As an application of this result we deduce our second result in which we construct a $4$-uniform hypergraph $G'$ and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(4)},G')$ in bounded time. This improves the result in the paper above. An equivalent formulation of our first result is that the game $\mathcal{R}(K_\omega\sqcup K_\omega,G)$ is a draw. Another reason for interest on the board $K_\omega\sqcup K_\omega$ is a folklore result that the disjoint union of two finite positional games both of which are first player wins is also a first player win. An amusing corollary of our first result is that at least one of the following two natural statements is false: (1) for every graph $H$, $\mathcal{R}(K_\omega,H)$ is a first player win; (2) for every graph $H$ if $\mathcal{R}(K_\omega,H)$ is a first player win, then $\mathcal{R}(K_\omega\sqcup K_\omega,H)$ is also a first player win.Comment: 17 pages, 48 figures; improved presentation, particularly in section

Topology optimization and boundary observation for clamped plates

We indicate a new approach to the optimization of the clamped plates with holes. It is based on the use of Hamiltonian systems and the penalization of the performance index. The alternative technique employing the penalization of the state system, cannot be applied in this case due to the (two) Dirichlet boundary conditions. We also include numerical tests exhibiting both shape and topological modifications, both creating and closing holes

Penalization of stationary Navier-Stokes equations and applications

We consider the steady Navier-Stokes system with mixed boundary conditions, in subdomains of a holdall domain. We study, via the penalization method, its approximation properties. Error estimates and the uniqueness of the solution, obtained in a non standard manner, are also discussed. Numerical tests, including topological optimization applications, are presented

The structure and number of Erd\H{o}s covering systems

Introduced by Erd\H{o}s in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erd\H{o}s was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$. In this paper we answer another question of Erd\H{o}s, asked in 1952, on the number of minimal covering systems. More precisely, we show that the number of minimal covering systems with exactly $n$ elements is $$\exp\left( \left(\frac{4\sqrt{\tau}}{3} + o(1)\right) \frac{n^{3/2}}{(\log n)^{1/2}} \right)$$ as $n \to \infty$, where $$\tau = \sum_{t = 1}^\infty \left( \log \frac{t+1}{t} \right)^2.$$ En route to this counting result, we obtain a structural description of all covering systems that are close to optimal in an appropriate sense.Comment: 31 page