The opaque square

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length $\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots$. The current best lower bound for the length of a (not necessarily connected) barrier is $2$, as established by Jones about 50 years ago. No better lower bound is known even if the barrier is restricted to lie in the square or in its close vicinity. Under a suitable locality assumption, we replace this lower bound by $2+10^{-12}$, which represents the first, albeit small, step in a long time toward finding the length of the shortest barrier. A sharper bound is obtained for interior barriers: the length of any interior barrier for the unit square is at least $2 + 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established by Sylvester for the measure of all lines that meet two disjoint planar convex bodies, and (ii) a procedure for detecting lines that are witness to the invalidity of a short bogus barrier for the square.Comment: 23 pages, 8 figure

Distribution of discontinuous mudstone beds within wave-dominated shallow-marine deposits : Star Point Sandstone and Blackhawk Formation, Eastern Utah

Acknowledgements Funding for this study was provided from the Research Council of Norway through the Petromaks project 193059 and the FORCE Safari Project. The lidar data was collected by Julien Vallet and Samuel Pitiot of Helimap Systems SA. Riegl LMS GmbH is acknowledged for software support. The first author would like to thank Oliver Severin Tynes for assistance in the field. Tore Grane Klausen and Gijs Allard Henstra are thanked for invaluable discussions. The authors would also like to thank Janok Bhattacharya, Cornel Olariu and one anonymous revier for their insightful comments which improved this paper, and Frances Witehurst for his editorial comments.Peer reviewedPostprin

Finite-Temperature Transition into a Power-Law Spin Phase with an Extensive Zero-Point Entropy

We introduce an $xy$ generalization of the frustrated Ising model on a triangular lattice. The presence of continuous degrees of freedom stabilizes a {\em finite-temperature} spin state with {\em power-law} discrete spin correlations and an extensive zero-point entropy. In this phase, the unquenched degrees of freedom can be described by a fluctuating surface with logarithmic height correlations. Finite-size Monte Carlo simulations have been used to characterize the exponents of the transition and the dynamics of the low-temperature phase

Unstable Slope Management Program

INE/AUTC 11.1

Celeste is PSPACE-hard

We investigate the complexity of the platform video game Celeste. We prove that navigating Celeste is PSPACE-hard in five different ways, corresponding to different subsets of the game mechanics. In particular, we prove the game PSPACE-hard even without player input.Comment: 15 pages, 13 figures. Presented at 23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Game