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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.
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 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.
Steuern und Abgaben auf Produkte oder Verbrauch mit gesellschaftlichen Folgekosten (externe Kosten) – sogenannte Pigou- oder Lenkungssteuern – sind ein gesellschaftliches „Win-Win-Instrument“. Sie verbessern die Wohlfahrt und schützen gleichzeitig die Umwelt und das Klima. Dies wird erreicht, indem umweltschädigende Aktivitäten einen Preis bekommen, der möglichst exakt der Höhe des Schadens entspricht. Eine konsequente Bepreisung der externen Kosten nach diesem Prinzip könnte in Deutschland erhebliche zusätzliche Einnahmen erbringen: Basierend auf bisherigen Studien zu externen Kosten wären zusätzliche Einnahmen in der Größenordnung von 348 bis 564 Milliarden Euro pro Jahr (44 bis 71 Prozent der gesamten Steuereinnahmen) möglich. Die Autoren warnen allerdings, dass die Bezifferung der externen Kosten mit erheblichen Unsicherheiten verbunden ist. Damit Lenkungssteuern und -abgaben ihre positiven Lenkungs- und Wohlstandseffekte voll entfalten können, seien zudem institutionelle Reformen notwendig.
Fault zones are the locations where motion of tectonic plates, often associated with earthquakes, is accommodated. Despite a rapid increase in the understanding of faults in the last decades, our knowledge of their geometry, petrophysical properties, and controlling processes remains incomplete. The central questions addressed here in our study of the Dead Sea Transform (DST) in the Middle East are as follows: (1) What are the structure and kinematics of a large fault zone? (2) What controls its structure and kinematics? (3) How does the DST compare to other plate boundary fault zones? The DST has accommodated a total of 105 km of left-lateral transform motion between the African and Arabian plates since early Miocene (similar to 20 Ma). The DST segment between the Dead Sea and the Red Sea, called the Arava/Araba Fault (AF), is studied here using a multidisciplinary and multiscale approach from the mu m to the plate tectonic scale. We observe that under the DST a narrow, subvertical zone cuts through crust and lithosphere. First, from west to east the crustal thickness increases smoothly from 26 to 39 km, and a subhorizontal lower crustal reflector is detected east of the AF. Second, several faults exist in the upper crust in a 40 km wide zone centered on the AF, but none have kilometer-size zones of decreased seismic velocities or zones of high electrical conductivities in the upper crust expected for large damage zones. Third, the AF is the main branch of the DST system, even though it has accommodated only a part (up to 60 km) of the overall 105 km of sinistral plate motion. Fourth, the AF acts as a barrier to fluids to a depth of 4 km, and the lithology changes abruptly across it. Fifth, in the top few hundred meters of the AF a locally transpressional regime is observed in a 100-300 m wide zone of deformed and displaced material, bordered by subparallel faults forming a positive flower structure. Other segments of the AF have a transtensional character with small pull-aparts along them. The damage zones of the individual faults are only 5-20 m wide at this depth range. Sixth, two areas on the AF show mesoscale to microscale faulting and veining in limestone sequences with faulting depths between 2 and 5 km. Seventh, fluids in the AF are carried downward into the fault zone. Only a minor fraction of fluids is derived from ascending hydrothermal fluids. However, we found that on the kilometer scale the AF does not act as an important fluid conduit. Most of these findings are corroborated using thermomechanical modeling where shear deformation in the upper crust is localized in one or two major faults; at larger depth, shear deformation occurs in a 20-40 km wide zone with a mechanically weak decoupling zone extending subvertically through the entire lithosphere.
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.
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.