A viscosity solution of the Hamilton–Jacobi equation with exponential dependence of Hamiltonian on the momentum

The initial – boundary value problem is considered for the Hamilton-Jacobi of evolutionary type in the case when the state space is one-dimensional. The Hamiltonian depends on the state and momentum variables, and the dependence on the momentum variable is exponential. The problem is considered on fixed bounded time interval, and the state variable changes from a given fixed value to infinity. The initial and boundary functions are subdifferentiable. It is proved that such a problem has a continuous generalized viscosity) solution. The representative formula is given for this solution. Sufficient conditions are indicated under which the generalized solution is unique. Hamilton-Jacobi equations with an exponential dependence on the momentum variable are atypical for theory, but such equations arise in practical problems, for example, in molecular genetics.

FEATURES OF CALCULATION OF THE REQUIRED QUANTITY OF GRAIN TERMINAL TRACKS IN SEA PORTS IN MODERN CONDITIONS

Purpose. Ukraine is one of the world's five largest producers of grain crops. Much of the grain is exported. Traditionally, grain cargoes are exported through the western land crossings and seaports of the Black and Azov Seas, with the bulk of grain exports going through the port of Nikolaev and the ports of Greater Odessa. Rail transport provides delivery to exporting ports up to 2/3 of the total volume of grain transshipment in ports. Given the significant number of operations in the general technological process of processing grain wagons in ports, the required capacity of technical equipment of railway transport (number of tracks, locomotives, weighing devices, etc.) plays a significant role in the duration of the transshipment process and total operating costs. In this regard, determining the required number of ways to service the flow of grain for export is an urgent task. Methods. Comprehensive analysis of technical equipment and technology of railway transport of grain terminals in seaports, statistical analysis, graph-analytical modeling. Results. Based on the processing of statistics on the delay of wagons due to commercial marriages with grain cargo in the seaport, the conditional division of detained wagons due to and, consequently, the duration of delay of wagons into long and short. The characteristics of random values of the delay of the car are established and the dependence of the number of detained cars on the value of the total car traffic with grain cargo entering the transshipment is obtained. For short-term delays it is a polynomial dependence of the second degree, for long-term delays it is characterized by exponential dependence. Based on research, an additional track capacity has been established for the sedimentation of detained cars and recommendations have been developed for the number of such tracks and their length. Based on the rationing of the duration of shunting operations with delayed wagons, an additional working fleet of shunting locomotives for servicing delayed wagons was determined and recommendations for the location of wagon scales for commercial weighing of wagons in the process of processing grain wagons in the port. Practical significance. The obtained results can be used to determine the number of tracks and the total capacity of grain terminals in seaports. The duration of additional shunting operations to service the detained cars allows to clarify the calculation of the required locomotive fleet of grain terminals.

Dynamics and 2D temperature distribution of plasma obtained by femtosecond laser-induced breakdown

Recently, plasma produced by focusing femtosecond laser in gases has been introduced as an etching tool in materials processing. Proper control of the plasma in this application necessitates the apt understanding of the different morphological features of the plasma. In this contribution we show that, the plasma produced in air goes through several stages of morphological development – from ellipsoidal to spherical to toroidal plasma, whereas in argon, axial compression of an ellipsoidal plasma is observed. To explain this dissimilarity, we have quantified the temperature by emission spectroscopy (Planck analysis with Wien’s approximation). The evolution of temperature shows a triple exponential dependence in time which can be correlated with different stages of morphological changes of the plasma. Open Source Field Operation and Manipulation (OpenFOAM) simulations using experimentally determined temperature values show that – (i) the reverse pressure gradient propagates radially inwards and compresses the plasma in both air and argon and forms a localized high pressure zone at the center that generates a secondary pressure wave in air, but not in argon, and (ii) the baroclinic torque that is generated because of the Richtmyer-Meshkov instability, dominates the rate of vorticity in air, whereas effects of flow compressibility and velocity gradients dominate the vortices in argon. Knowledge of the initial state and the dynamics of the subsequent stages of the plasma formation can be utilized for control and optimization of laser-induced plasma applications.

Experimental and numerical investigation of Poole-Frenkel effect on dynamic R ON transients in C-doped p-GaN HEMTs

In this paper, we investigate the influence of Poole-Frenkel Effect (PFE) on the dynamic R ON transients in C-doped p-GaN HEMTs. To this aim, we perform a characterization of the dynamic R ON transients acquired during OFF-state stress (i.e., V GS,STR = 0 V < V T, V DS,STR = 25–125 V and we interpret the results with the aid of numerical simulations. We find that dynamic R ON transients at room temperature accelerate with V DS,STR 1/2, which is signature of PFE, as further confirmed by the simultaneous decrease of the activation energy (E A) extracted from the Arrhenius plot of the dynamic R ON transients at V DS,STR = 50 V and T = 30–110 °C. Results obtained by means of calibrated numerical simulations reproduce the exponential dependence of transients time constants (τ) on V DS,STR 1/2 and consequent E A reduction only when including PFE enhancement of hole emission from dominant acceptor traps in the buffer related to C doping. This result is consistent with the model that considers hole emission from acceptor traps (rather than electron capture) as the mechanism underlying dynamic R ON increase during OFF-state stress.

Influence of C=O groups on the optical extinction coefficient of graphene exfoliated in liquid phase

LiLiquid phase exfoliation of graphite is currently one of the most promising graphene production methods at large scale. For this reason, an accurate calculation of the concentration in graphene dispersions is important for standardization and commercialization. Here, graphene dispersions, at high concentrations, were produced by electrochemical exfoliation. Furthermore, a cleaner methodology to obtain graphene oxide by electrochemical exfoliation at high acid concentrations was implemented. The absorption coefficient for graphene and graphene oxide was determined in the optical range ($\alpha_{660nm}=$ 1414 ($\pm$3\%) mL mg$^{-1}$ m$^{-1}$ and $\alpha_{660nm}=$ 648 ($\pm$ 7\%) mL mg$^{-1}$ m$^{-1}$, respectively) with an exponential dependence with the wavelength. The difference in $\alpha$ for both materials is attributed to an increased presence of C=O groups as evidenced by FTIR, UV-vis and Raman spectroscopy, as well as, in the calculation of the opical extinction coefficient and optical band-gap via Tauc-plots.

Thermal oxidation stability of lubricating greases – can a rapid small-scale method replace the classical method?

The article presents the results of tests of the oxidation resistance of 26 samples of selected lubricating greases available on the market. Various types of lubricant samples (according to the type of thickener), produced with oils of different chemical nature and viscosity, were tested. The basic parameters of the greases were determined: the worked penetration and the dropping point. Two different test methods were used to determine the thermal oxidation stability: the classical oxidation method according to PN-C-04143 and the rapid small-scale test method according to ASTM D 8206. The method of determining the correlation between these methods was presented. A correlation was found between the two methods of testing the resistance to oxidation, which can be roughly described using the exponential dependence. For the quick method, better compliance with the classical method was obtained at 140°C than at 160°C, which is confirmed by the determination coefficients determined by the ranking method. Based on the results of the quick method, using the determined exponential dependency, the results of the classical method can be estimated. However, the determined correlation between the PN-C-04143 and ASTM D 8206 methods is insufficient to use these methods interchangeably.

Statistical analysis of vapor distribution in a cavitation flow based on an ensemble of instantaneous liquid velocity fields

A new method was developed for statistical analysis of ensembles of instantaneous velocity fields measured by PIV in liquid (continuous phase) to determine the distribution of the vapor phase in cavitating flow. The method is based on two main principles: the absence of tracers used for PIV measurements in vapor, and the statistical independence of individual measurements. This allowed establishing an exponential dependence of repeatability of the vapor phase at a certain point of a cavitating flow. Compliance with this theoretical law was verified using the Pearson chi-square test. All theoretical distributions were divided into several groups depending on the time-averaged local vapor content calculated over the entire ensemble of realizations and the probability of a single event. As a result, dimensions of the stationary part of an attached cavity and the place of detachments of cloud cavities from the hydrofoil surface were determined using the new method of statistical analysis for an unsteady cloud cavitation regime.

Scaling of energy gaps in phosphorene nanoflakes

Electronic structure of phosphorene nanoflakes which consist of hundreds of phosphorus atoms are studied in the framework of unrestricted Hartree-Fock approach. On the base of Pariser-Parr-Pople model for electron-electron interactions, a simplified Bethe-Salpeter formalism is established for the calculation of excitation states of the system. Taking into account the electron-hole interaction in various dielectric environments, the optical gap of a triangular phosphorene nanoflake is shown to increase as the screening effect becomes stronger while its graphene counterpart exhibits just the opposite dependence. After confirming an exponential dependence of the optical gap on the effective dielectric constant, the quasiparticle and optical gaps are also found to obey an exponential scaling rule against the total number of atoms in the nanoflakes, respectively. By extrapolating the dependence on the size of the system, one is able to estimate the exciton binding energy of a monolayer phosphorene sheet on a SiO2 substrate to be 0.894 eV. The result is found to agree well with the previous experimental result of $ 0.9 eV.

About Strong Dependence of the Complexity of Analysis of the Random 3-CNF Formulas on the Ratio of Number of Clauses to the Number of Variables

The results of a computational experiment on the assessment of the complexity of proving the unsatisfiability of random 3-CNF logical formulas are presented. The dependence of the complexity of this proving on the R-ratio of the number of clauses to the number of variables is demonstrated. The computational experiment was carried out for the range of the N-number of variables from 256 to 512. An exponential dependence of the median complexity of proving the unsatisfiability of formulas on the number of variables was revealed for each of R value: 4.3, 4.6, 5.0, 5.5, 6.0. A formula is constructed that approximates the results of the experiment. According to this formula the exponential component of the median complexity of the analysis of random 3-CNF is estimated as 2 to the power N / (8.4R-17.8).

Synthesis, Self-Assembly and In Vitro Cellular Uptake Kinetics of Nanosized Drug Carriers Based on Aggregates of Amphiphilic Oligomers of N-Vinyl-2-pyrrolidone

Development of nanocarrier-based drug delivery systems is a major breakthrough in pharmacology, promising targeted delivery and reduction in drug toxicity. On the cellular level, encapsulation of a drug substantially affects the endocytic processes due to nanocarrier–membrane interaction. In this study we synthesized and characterized nanocarriers assembled from amphiphilic oligomers of N-vinyl-2-pyrrolidone with a terminal thiooctadecyl group (PVP-OD). It was found that the dissolution free energy of PVP-OD depends linearly on the molecular mass of its hydrophilic part up to M¯n = 2 × 104, leading to an exponential dependence of critical aggregation concentration (CAC) on the molar mass. A model hydrophobic compound (DiI dye) was loaded into the nanocarriers and exhibited slow release into the aqueous phase on a scale of 18 h. Cellular uptake of the loaded nanocarriers and that of free DiI were compared in vitro using glioblastoma (U87) and fibroblast (CRL2429) cells. While the uptake of both DiI/PVP-OD nanocarriers and free DiI was inhibited by dynasore, indicating a dynamin-dependent endocytic pathway as a major mechanism, a decrease in the uptake rate of free DiI was observed in the presence of wortmannin. This suggests that while macropinocytosis plays a role in the uptake of low-molecular components, this pathway might be circumvented by incorporation of DiI into nanocarriers.