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The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequence’s "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means.
The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.
The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequences "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means.
The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.
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.
The tremendous success of metal-halide perovskites, especially in the field of photovoltaics, has triggered a substantial number of studies in understanding their optoelectronic properties. However, consensus regarding the electronic properties of these perovskites is lacking due to a huge scatter in the reported key parameters, such as work function (Φ) and valence band maximum (VBM) values. Here, we demonstrate that the surface photovoltage (SPV) is a key phenomenon occurring at the perovskite surfaces that feature a non-negligible density of surface states, which is more the rule than an exception for most materials under study. With ultraviolet photoelectron spectroscopy (UPS) and Kelvin probe, we evidence that even minute UV photon fluxes (500 times lower than that used in typical UPS experiments) are sufficient to induce SPV and shift the perovskite Φ and VBM by several 100 meV compared to dark. By combining UV and visible light, we establish flat band conditions (i.e., compensate the surface-state-induced surface band bending) at the surface of four important perovskites, and find that all are p-type in the bulk, despite a pronounced n-type surface character in the dark. The present findings highlight that SPV effects must be considered in all surface studies to fully understand perovskites’ photophysical properties.
Photovoltaic cells based on halide perovskites, possessing remarkably high power conversion efficiencies have been reported. To push the development of such devices further, a comprehensive and reliable understanding of their electronic properties is essential but presently not available. To provide a solid foundation for understanding the electronic properties of polycrystalline thin films, we employ single-crystal band structure data from angle-resolved photoemission measurements. For two prototypical perovskites (CH3NH3PbBr3 and CH3NH3PbI3), we reveal the band dispersion in two high-symmetry directions and identify the global valence band maxima. With these benchmark data, we construct "standard" photoemission spectra from polycrystalline thin film samples and resolve challenges discussed in the literature for determining the valence band onset with high reliability. Within the framework laid out here, the consistency of relating the energy level alignment in perovskite-based photovoltaic and optoelectronic devices with their functional parameters is substantially enhanced.
The motion of tectonic plates is accommodated at fault zones. One of the unanswered questions about fault zones relates to the role they play in controlling shallow and local hydrology. This study focuses on the Arava/Araba Fault (AF) zone, the southern portion of the Dead Sea Transform (DST) in the Middle East. We combine seismic and electromagnetic methods (EM) to image the geometry and map the petro-physical properties and water occurrence in the top 100 m of this active fault. For three profiles, P-velocity and resistivity images were derived independently. Using a neural network cluster analysis three classes with similar P-velocity and resistivities could then be determined from these images. These classes correspond to spatial domains of specific material and wetness. The first class occurs primarily east of the fault consisting of 'wet' sand (dunes) and brecciated sediments, whereas the second class composed of similar material located west of the fault is 'dry'. The third class lies at depth below ca. 50 m and is composed of highly deformed and weathered Precambrian rocks that constitute the multi-branch fault zone of the AF at this location. The combination of two independent measurements like seismics and EM linked by a stringent mathematical approach has thus shown the potential to delineate the interplay of lithology and water near active faults.
Reduced Interface-Mediated Recombination for High Open-Circuit Voltages in CH3NH3PbI3 Solar Cells
(2017)
Perovskite solar cells with all-organic transport layers exhibit efficiencies rivaling their counterparts that employ inorganic transport layers, while avoiding high-temperature processing. Herein, it is investigated how the choice of the fullerene derivative employed in the electron-transporting layer of inverted perovskite cells affects the open-circuit voltage (V-OC). It is shown that nonradiative recombination mediated by the electron-transporting layer is the limiting factor for the V-OC in the cells. By inserting an ultrathin layer of an insulating polymer between the active CH3NH3PbI3 perovskite and the fullerene, an external radiative efficiency of up to 0.3%, a V-OC as high as 1.16 V, and a power conversion efficiency of 19.4% are realized. The results show that the reduction of nonradiative recombination due to charge-blocking at the perovskite/organic interface is more important than proper level alignment in the search for ideal selective contacts toward high V-OC and efficiency.
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.
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