On the finite factorial numbers

We introduce two families of transcendental numbers which we call finite factorial (FF) and partially finite factorial (PFF) numbers respectively, with the former one being subfamily of the latter one. These numbers arise naturally from some transcendental criterion for real numbers via their $b$-ary expansions. We show that rational numbers (eventually periodic words) can not be finite factorial. Then we consider the geometric (topological) properties of the collection of all the FF numbers, including its countability, density and Hausdorff dimension. Some numerical examples are given to illustrate certain results in the work

Quantum cluster characters of Hall algebras revisited

Let $Q$ be a finite acyclic valued quiver. We give a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective valued representations of $Q$. As an application, we recover the surjective homomorphism defined in \cite{DXZ}, which realizes the principal coefficient quantum cluster algebra \A_q(Q) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero--Chapoton formula, as well as their cluster multiplication formulas.Comment: 26 page

The use of settling velocity to predict the potential fate of aggregated sediment and associated SOC

The impacts of lateral movement of soil organic carbon (SOC) by soil erosion on global carbon (C) cycling and climate change have been the subject of a controversial debate for decades. Because of the limited availability of data on SOC erosion history, the effects of erosion on CO2 emissions have mostly been calculated by determining SOC inventories at sites of erosion and comparing against depositional sites. The use of SOC inventories to calculate C fluxes relies on the assumption that sediment properties are temporally and spatially stable during erosion events. However, on eroded lands, it always involves a temporal-dynamic pattern of SOC content, as well as a spatial enrichment and/or depletion of SOC in sediment that differs from original soils. Therefore, the approach of using SOC inventories at the slope-scale to back-calculate C fluxes caused by erosion would result in a biased assessment. Improving our assessment of soil erosion and its impact on C cycling thus requires a better understanding of the behavior of eroded SOC during transport and deposition across agricultural landscapes. In a given water layer, the transport distance of eroded sediment is mainly determined by particle settling velocity. Settling velocity distribution, calculated based on the diameter of dispersed mineral grains, has been used in some erosion models to predict the redistribution of sediment and associated SOC across landscapes. However, most eroded particles are transported in the form of aggregates rather than individual mineral grains. Aggregation dramatically increases the settling velocity of individual mineral grains that are incorporated into aggregates, as well as the transport distance of associated SOC. Consequently, the uncertainties of calculating the lateral redistribution of sediment and associated SOC may further lead to a biased estimate of the vertical C released from eroded SOC during redistribution. Therefore, identifying the settling velocity of natural aggregated-sediment represents an essential step if the redistribution of eroded SOC, as well as further assessing the potential CO2 mineralization, is to be more accurately modeled. Several laboratory-based studies conducted on dry-sieved aggregates have examined the transport fate of aggregated sediment and associated SOC based on settling velocity. However, the erodibility of the soil in the field is more temporally and spatially variable due to the impacts of tillage, rainfall, wetting-drying cycles, freezing, and biological effects. For example, rainfall kinetic energy will affect the breakdown of aggregates and the development of crust. It is not entirely clear whether changes in natural surface conditions could impact on the characteristics of sediment and thereby diminish the effect of aggregation on the fate of eroded SOC. Moreover, from the perspective of parametrizing erosion models, it would require a large amount of accurate settling velocity data from a wide range of soils in order to cover the broad spatial heterogeneity that is inherent during soil development. Identifying the settling velocity distribution based on laboratory or field tests with flumes, even if well-designed, are both far too much work to test all soils over a sufficiently wide range of rainfall conditions. Therefore, it is vital to develop a simple proxy to generate quasi-natural sediment in a time and labor-saving manner, and to further identify accurate settling information of sediment that could be incorporated into erosion models. To address the above knowledge gaps, four objectives were identified in this study. They are: 1) to quantitatively identify the potential fate of SOC eroded from a natural crusted soil surface and further compare the observations with that based on dry-sieved aggregates in the laboratory; 2) to investigate the sensitivity of the sediment settling behavior to increased kinetic energy during a series of rainfall events and thereafter examine the effect of aggregation on the quality of eroded SOC; 3) to develop a simple but efficient proxy method to generate natural or quasi-natural sediment; and 4) to evaluate the feasibility and sensitivity of such a proxy method. In this study, a series of experiments were conducted to attain those four objectives: Field Experiments 1&2 involved investigating the effect of a natural crusted soil surface on SOC transport and mineralization; Laboratory Experiments 1&2 involved developing an approach to identify the settling velocity of quasi-natural sediment. In Field Experiments 1&2, short term wind driven storms simulated with a modified portable wind and rainfall simulator (PWRS) were conducted on a natural crusted soil surface after harvesting in the village of Witterswil, in northwest Switzerland. The collected sediment was fractionated with a settling tube according to their respective settling velocities. The sediment mass, SOC concentration and cumulative CO2 emission of each fraction were measured. The results show: 1) 53% of eroded sediment and 62% of eroded SOC would potentially deposit across landscapes. This is six times and three times higher compared to that implied by mineral grains, respectively; 2) the underestimation of eroded SOC deposited across landscapes can mainly be attributed to underestimating mineral-associated organic carbon (MOC); 3) the preferential deposition of SOC-rich fast-settling sediment leads to a higher SOC stock than that at a comparable depth of non-eroded original soil. This would potentially release approximately 50% more CO2 than the same layer of the non-eroded original soil; 4) about 15% of SOC could be mineralized during the redistribution process of sediment, especially from the silt and clay fractions; 5) the settling velocity distributions of eroded sediment, as well as the SOC concentration and cumulative mineralization of each fraction, did not change during a series of rainfall events, suggesting settling velocity distribution of eroded SOC could be regarded as a stable parameter during redistribution. The results obtained from Field Experiments 1&2 confirm in general the conclusions drawn from the laboratory-based work and thus demonstrate that aggregation can affect the redistribution of sediment associated SOC under field conditions, including an increase in CO2 emissions compared to bulk soil. This illustrates the need to integrate the effect of aggregation on SOC redistribution into soil erosion models, which could help precisely distinguish SOC potentially re-deposited across landscapes from that possibly transported to aquatic systems, and further assess the impacts on global C cycling. In order to capture the effect of aggregation on settling behavior and thus the redistribution of eroded sediment, in Laboratory Experiments 1&2, a combined Raindrop Aggregate Destruction Device-Settling Tube (RADD-ST) proxy was developed to effectively simulate aggregate breakdown under raindrop impact, and further identify the settling velocity of aggregated sediment and associated SOC. The results show: 1) for an aggregated soil, applying dispersed mineral grain size distribution, rather than actual aggregate distribution, to soil erosion models would lead to an underestimate of deposition of eroded sediment and SOC across landscapes; 2) the RADD-ST designed in this study effectively captures the effects of raindrop impact on aggregate destruction and is thus able to simulate the quasi-natural sediment spatial redistribution; 3) the combined RADD-ST approach is adequately sensitive to measure actual settling velocities of differently aggregated soils; 4) this combined RADD-ST approach provides an effective tool to improve the parameterization of settling velocity input for erosion models. Overall, the results observed from this study confirm that aggregation effects, even on crusted soil surfaces, considerably reduce the likely transport distance of eroded SOC. It thus potentially skews the re-deposition of SOC-rich coarse sediment fractions towards terrestrial systems and contributes additional CO2 to the atmosphere. Therefore, current erosion models urgently need to be optimized by the development of a computable parameter integrating aggregated sediment settling velocity and the associated SOC distribution. The RADD-ST approach developed in this study has the potential to provide actual settling information generated under relatively simple simulated rainfall conditions to optimize the parametrization of sediment behavior and quality in erosion models. If further extrapolated appropriately to a specific erosion scenario, the RADD-ST derived sediment quality parameters can also help improve our understanding of sediment movement through watersheds and thus contribute to reaching consensus on the role of erosion on C cycling

Employing a Narrow-Band-Gap Mediator in Ternary Solar Cells for Enhanced Photovoltaic Performance.

Ternary organic solar cells (OSCs) provide a convenient and effective means to further improve the power conversion efficiency (PCE) of binary ones via composition control. However, the role of the third component remains to be explored in specific binary systems. Herein, we report ternary blend solar cells by adding the narrow-band-gap donor PCE10 as the mediator into the PBDB-T:IDTT-T binary blend system. The extended absorption, efficient fluorescence resonance energy transfer, enhanced charge dissociation, and induced tighter molecular packing of the ternary blend films enhance the photovoltaic properties of devices and deliver a champion PCE of 10.73% with an impressively high open-circuit voltage (VOC) of 1.03 V. Good miscibility and similar molecular packing behavior of the components guarantee the desired morphology in the ternary blend films, leading to solar cell devices with over 10% PCEs at a range of compositions. Our results suggest that ternary systems with properly aligned energy levels and overlapping absorption among the components hold great promises to further enhance the performance of corresponding binary ones

The use of a raindrop aggregate destruction device to evaluate sediment and soil organic carbon transport

Raindrop impact and subsequent aggregate breakdown can potentially change the movement behaviour of soil fractions and thus alter their transport distances when compared against non-impacted aggregates. In a given water layer, the transport distances of eroded soil fractions, and thus that of the associated substances across landscapes, such as soil organic carbon (SOC) and phosphorous, are determined by the settling velocities of the eroded soil fractions. However, using mineral size distribution to represent the settling velocities of soil fractions, as often applied in current erosion models, would ignore the potential influence of aggregation on the settling behaviour of soil fractions. The destructive effects of raindrops impacting onto aggregates are also often neglected in current soil erosion models. Therefore, the objective of this study is to develop a proxy method to effectively simulate aggregate breakdown under raindrop impact, and further identify the settling velocity of eroded sediment and the associated SOC. Two agricultural soils with different sandy and silty loam textures were subjected to rainfall using a raindrop aggregate destruction device (RADD). The aggregates sustained after raindrop impact were fractionated by a settling tube into six different classes according to their respective settling velocities. The same mass amount of bulk soil of each soil type was also dispersed and sieved into the same six classes, to form a comparison in size distribution. The SOC content was measured for each settled and dispersed class. Our results show the following: (1) for an aggregated soil, applying dispersed mineral grain size distribution, rather than its actual aggregate distribution, to soil erosion models would lead to a biased estimation on the redistribution of eroded sediment and SOC; (2) the RADD designed in this study effectively captures the effects of raindrop impact on aggregate destruction and is thus able to simulate the quasi-natural sediment spatial redistribution; (3) further RADD tests with more soils under standard rainfall combined with local rainfalls are required to optimize the method