Sum Of Rates Research Articles

Sum rules in multiphoton coincidence rates

We show that sums of carefully chosen coincidence rates in a multiphoton interferometry experiment can be simplified by replacing the original unitary scattering matrix with a coset matrix containing 0s. The number and placement of these 0s reduces the complexity of each term in the sum without affecting the original sum of rates. In particular, the evaluation of sums of modulus squared of permanents is shown to turn in some cases into a sum of modulus squared of determinants. The sums of rates are shown to be equivalent to the removal of some optical elements in the interferometer.

Social-Aware Collaborative Caching Based on User Preferences for D2D Content Sharing

With rapid growth of content demands, device-to-device (D2D) content sharing is exploited to effectively improve the service quality of users. Considering the limited storage space and various content demands of users, caching schemes are significant. However, most of them ignore the influence of the asynchronous content reuse and the selfishness of users. In this work, the user preferences are defined by exploiting the user-oriented content popularity and the current caching situation, and further, we propose the social-aware rate, which compre-hensively reflects the achievable contents download rate affected by the social ties, the caching indicators, and the user preferences. Guided by this, we model the collaborative caching problem by making a trade-off between the redundancy of caching contents and the cache hit ratio, with the goal of maximizing the sum of social-aware rate over the constraint of limited storage space. Due to its intractability, it is computationally reduced to the maximi-zation of a monotone submodular function, subject to a matroid constraint. Subsequently, two social-aware collaborative caching algorithms are designed by leveraging the standard and continuous greedy algorithms respectively, which are proved to achieve different approxima-tion ratios in unequal polynomial-time. We present the simulation results to illustratethe performance of our schemes.

Efficient Channel Allocation using Matching Theory for QoS Provisioning in Cognitive Radio Networks.

The focus of research efforts in cognitive radio networks (CRNs) has primarily remained confined to maximizing the utilization of the discovered resources. However, it is also important to enhance the user satisfaction in CRNs by finding a suitable match between the secondary users and the idle channels available from the primary network while taking into consideration not only the quality of service (QoS) requirements of the secondary users but the quality of the channels as well. In this work, the Gale Shapley matching theory was applied to find the best match, so that the most suitable channels from the available pool were allocated that satisfy the QoS requirements of the secondary users. Before applying matching theory, two objective functions were defined from the secondary user’s perspective as well as from the channel’s perspective. The objective function of secondary users is the weighted sum of the data rate of the secondary users and the probability of reappearance of the primary user on the channel. Whereas, the objective function of the channel is the maximum utilization of the channel. The weight factors included in the objective functions allow for diverse service classes of secondary users (SUs) or varying channel quality characteristics. The objective functions were used in developing the preference lists for the secondary users and the idle channels. The preference lists were then used by the Gale Shapely matching algorithm to determine the most suitably matched SU-channel pairs. The performance of the proposed scheme was evaluated using Monte–Carlo simulations. The results show significant improvement in the overall satisfaction of the secondary users with the proposed scheme in comparison to other contemporary techniques. Further, the impact of changing the weight factors in the objective functions on the secondary user’s satisfaction and channel utilization was also analyzed.

The water vapor foreign-continuum in the 1.6 µm window by CRDS at room temperature

Water vapour foreign-continuum absorption cross-sections, CF, are measured for the first time at room temperature in the 1.6 µm transparency window, of importance for atmospheric applications. The measurements are performed by cavity ring down spectroscopy (CRDS) at 15 selected spectral data points. These data, covering the 5700–6640 cm−1 spectral range, are derived from the variation of the absorption signal during pressure ramps of humidified air up to 1 atm with a typical 1% water vapour relative concentration. The foreign-continuum absorption was obtained as the excess of the measured loss rate compared to the sum of the loss rate measured with dry air, the local water monomer contribution and the self-continuum absorption. CF values were derived from the linear dependence of the foreign-continuum absorption with the product of the partial pressures of water vapour and air. The semi-empirical MT_CKD CF values are found significantly underestimated in the centre of the window.

RDT study of non-isentropic compressible homogeneous sheared turbulence

The objective of this work is to formulate governing equations for non-isentropic compressible homogeneous sheared turbulence using rapid distortion theory (RDT). Furthermore, we develop a code able to solve these linearized equations. It is possible for this code to address both kinematic and thermal properties i.e. mass density, temperature variances, mass and heat fluxes which are the specific properties of compressible turbulence. RDT code resolves directly evolution equations of two-point spectral correlation. The results of our RDT code are compared with the direct numerical simulation data of Simone et al (1997) and have demonstrated good agreement. The comparisons concern different contributions such as the kinetic turbulent energy, the temporal energy growth rate, the non-dimensional production term, the solenoidal and dilatational contributions of non-dimensional production and the sum of pressure-dilatation terms and rate of kinetic energy dissipation, scaled by the production term. Some results concerning the polytropic coefficient and the energy partition parameter are presented and discussed.

Temperature controls production but hydrology regulates export of dissolved organic carbon at the catchment scale

Abstract. Lateral carbon flux through river networks is an important and poorly understood component of the global carbon budget. This work investigates how temperature and hydrology control the production and export of dissolved organic carbon (DOC) in the Susquehanna Shale Hills Critical Zone Observatory in Pennsylvania, USA. Using field measurements of daily stream discharge, evapotranspiration, and stream DOC concentration, we calibrated the catchment-scale biogeochemical reactive transport model BioRT-Flux-PIHM (Biogeochemical Reactive Transport–Flux–Penn State Integrated Hydrologic Model, BFP), which met the satisfactory standard of a Nash–Sutcliffe efficiency (NSE) value greater than 0.5. We used the calibrated model to estimate and compare the daily DOC production rates (Rp; the sum of the local DOC production rates in individual grid cells) and export rate (Re; the product of the concentration and discharge at the stream outlet, or load). Results showed that daily Rp varied by less than an order of magnitude, primarily depending on seasonal temperature. In contrast, daily Re varied by more than 3 orders of magnitude and was strongly associated with variation in discharge and hydrological connectivity. In summer, high temperature and evapotranspiration dried and disconnected hillslopes from the stream, driving Rp to its maximum but Re to its minimum. During this period, the stream only exported DOC from the organic-poor groundwater and from organic-rich soil water in the swales bordering the stream. The DOC produced accumulated in hillslopes and was later flushed out during the wet and cold period (winter and spring) when Re peaked as the stream reconnected with uphill and Rp reached its minimum. The model reproduced the observed concentration–discharge (C–Q) relationship characterized by an unusual flushing–dilution pattern with maximum concentrations at intermediate discharge, indicating three end-members of source waters. A sensitivity analysis indicated that this nonlinearity was caused by shifts in the relative contribution of different source waters to the stream under different flow conditions. At low discharge, stream water reflected the chemistry of organic-poor groundwater; at intermediate discharge, stream water was dominated by the organic-rich soil water from swales; at high discharge, the stream reflected uphill soil water with an intermediate DOC concentration. This pattern persisted regardless of the DOC production rate as long as the contribution of deeper groundwater flow remained low (<18 % of the streamflow). When groundwater flow increased above 18 %, comparable amounts of groundwater and swale soil water mixed in the stream and masked the high DOC concentration from swales. In that case, the C–Q patterns switched to a flushing-only pattern with increasing DOC concentration at high discharge. These results depict a conceptual model that the catchment serves as a producer and storage reservoir for DOC under hot and dry conditions and transitions into a DOC exporter under wet and cold conditions. This study also illustrates how different controls on DOC production and export – temperature and hydrological flow paths, respectively – can create temporal asynchrony at the catchment scale. Future warming and increasing hydrological extremes could accentuate this asynchrony, with DOC production occurring primarily during dry periods and lateral export of DOC dominating in major storm events.

Work–energy analysis of granular assemblies validates and calibrates a constitutive model

This study presents a rate-independent constitutive model for granular materials which is based on the analysis of discrete element modelling (DEM) results. We examine the elastic and plastic strain rate at any state of shear, and explore how the plastic strain rate and the stress rate are related to some state variables (e.g. stress, void ratio, etc.). The elastic and plastic strain rate are evaluated by assuming that the rate of conservative energy defined in term of elastic strain rate equals the energy rate from work–energy analysis of granular assemblies. The verification of the theoretical work–energy analysis and the acquire of data regarding the mechanical rates at contacts between grains are done through DEM. It is shown that for a representative element volume of granular materials, the external work rate equals the sum of an elastic energy rate and a dissipation rate at contacts. The work rate is either the product of stress and strain rate or the sum of discrete work rate of external forces. Equations regarding the dilatancy, plastic deviatoric strain rate and stress rate are proposed. In the model, a yield surface is not explicitly defined and the deviatoric stress rate is related to not only the elastic deviatoric strain rate, but also a rate of dilatancy. Simulations show that the predictions match both virtual and Toyoura sand satisfactorily in various shear tests within a board range of densities. Additionally, the calibration of parameters is extremely simple. The treatment of cyclic loading condition is also presented.

Association between running quality and loads seen with accelerometry data obtained from female soccer players

BACKGROUND: Global positioning system (GPS) data, when obtained from athletes offers unique information on their performance. Given the information GPS data provides, it is important to identify data most pertinent to an athlete’s performance. OBJECTIVE: We evaluated a GPS-based running quality variable to predict the variance in total player load (TPL) and player load per minute (PLPM) from female soccer player (n= 26) data. METHODS: Running quality was the ratio of the displacement per minute to total distance covered. TPL was quantified as the sum of velocity change rates in all three planes of motion, while PLPM was a ratio of load generated per minute of activity. RESULTS: With a logarithmic transformation of TPL data, a Pearson Product Moment Correlation analysis revealed running quality accounted for significant (r=-0.65) amounts of our criterion’s variance, which implied higher running quality led to lower log (TPL) values. With PLPM as our dependent variable, running quality correlated with significant (r= 0.63) amounts of our criterion’s variance. CONCLUSIONS: Movement efficiency appears to be an important contributor to our correlations. We suggest running quality be examined as a correlate to performance in other sports in which running is crucial to success.

Observer-based control design of semi-Markovian jump systems with uncertain probability intensities and mode-transition-dependent sojourn-time distribution

This paper deals with the problem of H∞ observer-based control for a class of continuous-time semi-Markovian jump systems (S-MJSs) with more detailed observational information. First, to explore the impact of uncertain probability intensities, a mathematical analysis is accomplished, from which some useful inequality conditions on the sum of transition rates (TRs) are obtained. Further, to come up with more accurate bounds of TRs, the mode-transition-dependent probability distribution of sojourn time is imposed on the mechanism of forming TRs. Lastly, by devising a compatible relaxation process that can embrace all the conditions found in our derivation, the resultant observer-based stabilization conditions are formulated in terms of linear matrix inequalities.

Particle-resolved simulation and modeling of the conversion rate of coal char in chemical looping with oxygen uncoupling

The high concentration of CO2 and low concentration of O2 at relatively low temperatures of 1123-1223 K are typical reactive atmospheres in the fuel reactor of a chemical looping with oxygen uncoupling (CLOU) unit. Char within this environment can be converted via lean-O2 oxidation and/or rich-CO2 gasification. Conventional models, which either considered the oxidation only or considered the oxidation and gasification contributions independently by simply adding the two individual conversion rates together, seem arbitrary. Therefore, it is necessary to quantitatively explore the contributions of the oxidation and the gasification reactions as well as their interactions in CLOU conditions through more detailed particle-resolved simulations.In this work, a single particle model which described a spherical reacting porous particle with its reacting boundary layer was developed to investigate the conversion characteristics of pulverized coal char in lean-O2 and/or rich-CO2 conditions (analogous CLOU conditions). The heterogeneous reaction kinetics developed by Tilghman and Mitchell [35] and the simplified GRI-Mech 3.0 gas-phase reaction kinetics were adopted. It was found that the char conversion rate in CLOU conditions could be basically modeled as the sum of the full oxidation consumption rate and the partial gasification consumption rate, i.e. rmix=φ1rgasi+roxid. The interactions between the oxidation and gasification reactions were elaborated in two aspects: on the one side, the oxidation of coal char in CLOU conditions is within the weak zone II burning region, where the oxidation reaction only happens peripherally and is nearly unaffected by the high concentration of CO2; on the other side, the gasification reaction within the external layer of the coal char particle is inhibited by the oxidation reaction, consequently the gasification reaction only happens inside the coal char particle where O2 cannot penetrate into. Based on these understandings, a correlation between the coefficient, φ1, and the effectiveness factor of the oxidation reaction, η, was proposed to describe the overall char conversion rate in CLOU conditions. Through fitting the simulation results under typical CLOU conditions, the quantitative relationship between φ1 and η was finally attained.

On the dynamic allocation of assets subject to failure

Motivated by situations arising in surveillance, search and monitoring, in this paper we study dynamic allocation of assets which tend to fail, requiring replenishment before once again being available for operation on one of the available tasks. We cast the problem as a closed-system continuous-time Markov decision process with impulsive controls, maximising the long-term time-average sum of per-task reward rates. We then formulate an open-system continuous-time approximative model, whose Lagrangian relaxation yields a decomposition (innovatively extending the restless bandits approach), from which we derive the corresponding Whittle index. We propose two ways of adapting the Whittle index derived from the open-system model to the original closed-system model, a naïve one and a cleverly modified one. We carry out extensive numerical performance evaluation of the original closed-system model, which indicates that the cleverly modified Whittle index rule is nearly optimal, being within 1.6% (0.4%, 0.0%) of the optimal reward rate 75% (50%, 25%) of the time, and significantly superior to uniformly random allocation which is within 22.0% (16.2%, 10.7%) of the optimal reward rate. Our numerical results also suggest that the Whittle index must be cleverly modified when adapting it from the open-system, as the naïve Whittle index rule is not superior to a myopic greedy policy.

Evaluation of hole-specific injection rate based on momentum flux measurement in GDI systems

A novel methodology for the estimation of the mass flow rate delivered by each hole of a multi-hole GDI nozzle is presented and discussed in this paper. The proposed method is based on the measurement of the impact force of each jet and it is able to evaluate the individual hole injection rate and injected mass using the relationship between the spray momentum flux and the mass flow rate. Three different nozzles are tested, one featuring equal hole diameters and two with differentiated hole-to-hole dimensions. Firstly, results are validated in terms of total injection rate, comparing the sum of the individual hole flow rates with the signal from a Zeuch-based flow meter. Secondly, outcomes are compared with the direct measurement of the injected quantity by means of a special device able to collect and weight the fuel delivered from each hole.Results evidence an excellent agreement in terms of mass flow rate dynamic profile as the proposed method is able to detect both opening and closure transients and the static flow rate. The proposed method proved to be able to capture the dynamic mass flow rate in transient conditions, i.e. very short injections, overcoming a limitation of the methodology proposed in a previous study. Moreover, in terms of injected mass, the results show a percentage error lower than 5% for medium and long energizing times and a maximum error of 9.5% for short injections in ballistic operating conditions.

Socio-educational interventions as complementary strategies to improvement clinical parameters of Type 2 diabetes mellitus

Type 2 diabetes mellitus (T2DM) is one of the most prevalent chronic diseases amongthe elderly, which impairs their functionality and the quality of life. Since high levels of socialisolation are observed in the elderly population, we hypothesized that low social support network (SSN) could have negative effects in clinical evolution of T2DM. Therefore, the practiceof socio educational activities could contribute to a favorable disease control and improvement of T2DM clinical parameters.AimsWe evaluated, cross-sectionally and retrospectively, the relationship of SSN size withmetabolic parameters of TD2M and the influence of socio-educational activities (i.e. crafts,jewelry, fitness, literacy, dance, guitar, singing, ballroom dancing, theater, yoga, flamenco,among others) in changes of the glycemic and lipid indexes in elderly patients (> 60 yearsold).MethodsSix clinical parameters, fasting blood glucose (FBG), glycated hemoglobin (HbA1c),circulating triglycerides, HDL, LDL and total cholesterol levels were compared with socialsupport network (SSN) which was measured in 89 T2DM patients by minimal map of relationship in older people (MMRI). The same parameters were compared between the grouppracticing socio educational activities (n = 23) and non-practicing group (n = 33). The changes in values were established between two phases: 2001 to 2009 (before) and 2010 to 2011(after), in both groups. The results were analyzed in two ways: a) vertical comparison ofthe averages of each individual clinical parameter; and b) horizontal comparison of total improvement rate (TIR), resulting from the sum of the individual improvement rate (iIR). Tocalculate iIR, clinical improvement received an arbitrary value (0.25) whereas no clinicalimprovement was ascribed a 0 value.ResultsWe observed a discrete positive correlation between SSN size and glycemic control improvement of elderly T2DM patients. When socio-educational interventions are evaluated, weobserved a significant improvement in HbA1c levels of up 34.5% (odds ratio=4.25; 95% CI,1.17 - 15.45; p = 0.0535) in elderly diabetics practicing socio educational activities. However,the fasting glucose and total cholesterol values had no significant improvement. In the otherhand, the horizontal analysis showed a significant difference between IMT means from elderly diabetics practicing socio educational activities when compared to the control group (p =0.03). It was observed a greater number of cases of clinical improvement over 50% (IMT 0.75and 1.0) in the socializing group (66% vs 36%).ConclusionsOur results suggested an apparent relationship between high SSN and practices of socio educational activities with significantly improvement of glicaemic control elderly T2DMpatients. We speculated that enhancing social support affects direct or indirectly some health-related variables, although potential pathways activated by social and educational components to regulating glucose metabolism remains unclear. However, our data are suggestingthat inclusion of socio-educational elements as non-pharmacological strategies for glicaemiccontrol may contribute for an effective management of T2DM in public health.

DC-Bias and Power Allocation in Cooperative VLC Networks for Joint Information and Energy Transfer

Visible light communications (VLC) have emerged as a strong candidate for meeting the escalating demand for high data rates. In this paper, we consider a VLC network, where multiple access points (APs) serve both energy-harvesting users (EHUs), i.e., users who harvest energy from light emitted by diodes and information users (IUs), i.e., users who gather data information. In order to jointly balance the achievable sum rate at the IUs and the energy harvested by the EHUs, the paper considers maximizing a network-wide utility, which consists of a weighted sum of the IUs sum rate and the EHUs harvested energy, subject to individual IU rate constraint, individual EHU harvested-energy constraint, and AP power constraints, so as to jointly determine the direct current (DC) bias value at each AP, and the power of the alternating-current (AC) signals of the users. A difficult non-convex optimization problem is solved using an iterative approach which relies on inner convex approximations, and compensates for the used approximations using proper outer-loop updates. The paper further considers solving the special cases of the problem, i.e., maximizing the sum rate, and maximizing the total harvested-energy, both subject to the same constraints. Numerical results highlight the significant performance improvement of the proposed algorithms, and illustrate the impacts of the network parameters on the performance trade-off between the sum rate and harvested-energy.

TiO2 nanotube array modified with polypyrrole for efficient photoelectrocatalytic decolorization of methylene blue

In this work, polypyrrole (PPy) was in situ polymerized on the surface of TiO2 nanotube array (TiO2 NTA, with average pore diameter of 110 nm and length of 490–750 nm) using cyclic voltammetry method to develop the nanocomposite photoelectrode of PPy@TiO2 NTA. The morphology and the physicochemical properties of the photoelectrode were characterized by SEM, XRD, FTIR, Raman Spectra, and XPS. It was found that the PPy modification enlarged the light adsorption range of the photoelectrode, as well as increased the photocurrent density under UV light irradiation. Furthermore, the performance of PPy@TiO2 NTA photoelectrode was investigated for the decolorization of methylene blue (MB) by means of photocatalytic (PC), electrocatalytic (EC) and photoelectrocatalytic (PEC) degradation. The decolorization rate of MB using the PPy@TiO2 NTA increased by 24.94% and 10.85% under PC and EC conditions, respectively, in comparison with the decolorization rate obtained using pristine TiO2 NTA. The improved decolorization rate of MB was mainly attributed to the separation efficiency of electron-hole pairs originated from TiO2, and the enhanced utilization of visible-light due to the presence of PPy. Furthermore, the decolorization rate of MB under the PEC condition was also significantly improved, compared with the sum of the decolorization rate of PC and EC systems, indicating the synergistic effect of MB oxidation in the photoelectrocatalysis.

MnCeOx/diatomite catalyst for persulfate activation to degrade organic pollutants

Persulfate (PS)-based oxidation technologies are attracting increasing attentions in water treatment due to their high efficiency and stability. In this study, a novel diatomite supported MnCeOx composite (MnCeOx/diatomite) was prepared and characterized for activation of PS to degrade organic pollutants. Results indicated that diatomite not only dispersed MnCeOx and increased the specific surface area of catalyst, but also improved the low-valence metal site (Mn2+ and Ce3+) and reactive oxygen species site (–OH) of MnCeOx, thus enhancing the activities of MnCeOx. MnCeOx/diatomite/PS showed high efficiency for multiple dyes and pharmaceutical pollutants. Constant rate (k) of MnCeOx/diatomite (kMnCeOx/diatomite) was three times higher than the sum of constant rate of MnCeOx (kMnCeOx) and constant rate of diatomite (kdiatomite). In addition, MnCeOx/diatomite showed wide pH application (5–9). Cl− and NO32− had no effect while SO42− and humid acid had slightly negative effects on MnCeOx/diatomite/PS system. Moreover, MnCeOx/diatomite showed good reusability and stability. Mechanism analyses indicated that electron transfer of Mn and Ce attributed to the activation of PS and oxygen to produce free radicals. SO4−, OH and O2− on the surface of catalyst were the main active free radicals to attack pollutants.

BP-C1 in treatment of pretreated metastatic breast cancer: A meta-analysis of two randomized, double-blind and placebo-controlled multicenter studies.

12 Background: To compare the effect of BP-C1 versus equal looking placebo in heavily pre-treated metastatic breast cancer (MBC) patients and to evaluate the efficacy of BP-C1 according to number of negative receptors. Methods: The study sample consisted of 61 patients from two randomized, double blinded and placebo controlled multicentre studies with semi-cross-over design. By randomization, 31 patients were allocated to BP-C1 and 30 to equal looking placebo treatment for 32 days. Twenty-five of the 30 patient allocated to placebo were switched to BP-C1 after finalizing the placebo period. A joint sample of 56 patients received one daily intramuscular injection of 0.07 ml/kg BW of BP-C1 for at least one cycle of 32 days. Receptor status including Estrogen, Progesterone and Human epidermal growth factor receptor was analysed prior to entering the study and available in 54 of the 56 patients. CT scans were performed at screening and after every 32-days of treatment, and were blindly analysed by one independent radiologist. The sum of target lesion diameters (SLD), RECIST classification and disease control rate (DCR) defined as patients obtaining at least stabile disease (SD) at the end of treatment was the main variables. Results: The SLD increased non-significantly by 1.5mm (5.7%) in the BP-C1 group and significantly (p < 0.001) by 13.6 mm (26.2%) in the placebo group during the 32 days of treatment. The difference between groups with respect to SLD and DCR was significant in favour of BP-C1 (p < 0.001). The tumour growth in the joint BP-C1 group was 2.5mm (5.9%) and DCR was 89.1% [95% CI: 38.1 – 96.0]. Nine of the 56 patients reported 26 Adverse Events (AE) related to BP-C1; all of mild to moderate degree. SLD increased 16.2% (p = 0.03) in patients with three positive receptors, 7.5% (p = 0.02) in the group with one negative receptor, 2.5% in the group of two negative receptors and 0.8% in triple negative patients. The tumor growth declined with increasing number of negative receptors (r = -0.26; p = 0.05). Conclusions: BP-C1 treatment of pre-treated MBC patients controlled the cancer growth with few mainly mild AE’s. The BP-C1 efficacy increased with the number of negative receptors. Clinical trial information: NCT03603197. Clinical trial information: NCT02783794.

Partial Non-Orthogonal Multiple Access (P-NOMA)

In this letter, we propose a novel hybrid type of power-domain non-orthogonal multiple access (NOMA) in downlink environments, called partial NOMA (P-NOMA). In conventional NOMA, where the signals of user equipments (UEs) are fully overlapped, the utilization of the bandwidth can be maximized, but doing so also maximizes the interference from the other UEs. P-NOMA partially overlaps the UEs’ signals by controlling the extent of the overlap and can thus reduce the interference from other UEs. To verify whether this is a possibility with P-NOMA, we observed the performance tendency of the achievable rate versus the overlap ratio and then confirmed that there is an overlap ratio that achieves performance that is better than that of conventional NOMA with respect to the sum of the achievable rate. In other words, slightly reducing the overlap ratio improves the sum of the achievable rate compared to the full overlap ratio.

Convenient analytical solution for vibrational distribution function of molecules colliding with a wall

We study formation of the Vibrational Distribution Function (VDF) in a molecular gas at low pressure, when vibrational levels are excited by electron impact and deactivated in collisions with walls and show that this problem has a convenient analytical solution that can be used to obtain VDF and its dependence on external parameters. The VDF is determined by excitation of vibrational levels by an external source and deactivation in collisions with the wall. Deactivation in wall collisions is little known process. However, we found that the VDF is weakly dependent on the functional form of the actual form of probability for a vibrational number to transfer into a lower level at the wall. Because for a given excitation source of vibrational states, the problem is linear the solution for VDF involves solving linear matrix equation. The matrix equation can be easily solved if we approximate probability, in the form: In this case, the steady-state solution for VDF() simply involves a sum of source rates for levels above with a factor of As an example of application, we study the vibrational kinetics in a hydrogen gas and verify the analytical solution by comparing with a full model.

Singlet Fission Rate: Optimized Packing of a Molecular Pair. Ethylene as a Model.

A procedure is described for unbiased identification of all π-electron chromophore pair geometry choices that locally maximize the rate of conversion of a singlet exciton into a singlet biexciton (triplet pair), using a simplified version of the diabatic frontier orbital model of singlet fission (SF). The resulting approximate optimal geometries provide insight and are expected to represent useful starting points for searches by more advanced methods. The general procedure is illustrated on a pair of ethylenes as the simplest model of a π-electron system, but it is applicable to pairs of much larger molecules, with dozens of non-hydrogen atoms, and not necessarily planar. We first examine the value of |TA|2, the square of the electronic matrix element for SF with initial excitation fully localized on partner A, on a grid of several billion geometries within the six-dimensional space of physically realizable possibilities. Several of the optimized pair geometries are somewhat unexpected, but all are found to follow the qualitative guidance proposed earlier. In the neighborhood of each local maximum of |TA|2, consideration of mixing with charge-transfer configurations and of excitonic interaction between partners A and B determines the SF energy balance and yields squared matrix elements |T*|2 and |T**|2 for the lower and upper excitonic states S* and S**, respectively. Assuming Boltzmann populations of these states, the geometry is further optimized to maximize k, the sum of the SF rates obtained from Marcus theory, and this reorders the suitable geometries substantially. At 87 pair geometries, the |T*|2 and |T**|2 values are compared with those obtained from high-level ab initio nonorthogonal configuration interaction calculations and found to follow the same trend. Finally, the biexciton binding energy at the optimized geometries is calculated. Altogether, 13 significant local maxima of SF rate for a pair of ethylenes are identified in the physically relevant part of space that avoids molecular interpenetration in the hard-sphere approximation. The three best geometries are twist-stacked, slip-stacked, and L-shaped. The maxima occur at the (five-dimensional) surfaces of seven six-dimensional "parent" regions of space centered at physically inaccessible geometries at which the calculated SF rate is very large but the two ethylenes interpenetrate. The results are displayed in interactive graphics. The computer code ("Simple") written for these calculations is flexible in that it permits a choice of performing the search for local maxima in six dimensions on |TA|2, |T*|2, or k. It is available as freeware at https://cloud.uochb.cas.cz/simple .