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It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all (odd) natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number 1, as conjectured by Lothar Collatz.
It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number one, as conjectured by Lothar Collatz.
It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto unavailable. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. Furthermore, we derive and prove several formulas that can be used to traverse the graph. The analysis covers the Collatz problem both in it’s original form 3x + 1 as well as in the generalized variant kx + 1. Finally, we transform the Collatz graph into a binary tree, following the approach of Kleinnijenhuis, which could form the basis for a comprehensive proof of the conjecture.
The remarkable progress of metal halide perovskites in photovoltaics has led to the power conversion efficiency approaching 26%. However, practical applications of perovskite-based solar cells are challenged by the stability issues, of which the most critical one is photo-induced degradation. Bare CH3NH3PbI3 perovskite films are known to decompose rapidly, with methylammonium and iodine as volatile species and residual solid PbI2 and metallic Pb, under vacuum under white light illumination, on the timescale of minutes. We find, in agreement with previous work, that the degradation is non-uniform and proceeds predominantly from the surface, and that illumination under N-2 and ambient air (relative humidity 20%) does not induce substantial degradation even after several hours. Yet, in all cases the release of iodine from the perovskite surface is directly identified by X-ray photoelectron spectroscopy. This goes in hand with a loss of organic cations and the formation of metallic Pb. When CH3NH3PbI3 films are covered with a few nm thick organic capping layer, either charge selective or non-selective, the rapid photodecomposition process under ultrahigh vacuum is reduced by more than one order of magnitude, and becomes similar in timescale to that under N-2 or air. We conclude that the light-induced decomposition reaction of CH3NH3PbI3, leading to volatile methylammonium and iodine, is largely reversible as long as these products are restrained from leaving the surface. This is readily achieved by ambient atmospheric pressure, as well as a thin organic capping layer even under ultrahigh vacuum. In addition to explaining the impact of gas pressure on the stability of this perovskite, our results indicate that covalently "locking" the position of perovskite components at the surface or an interface should enhance the overall photostability.