Constant Distribution Research Articles

The unique flexibility feature of bamboo: Cantilever-beam loading form the coupling bending-shear effects

Bamboo's remarkable attribute of flexibility has earned it widespread recognition, positioning it as an exemplary material for mimicking biological structures in advanced engineering construction. While the tension, compression, bending, shear, and torsion properties of bamboo have been extensively documented, its response under combined loads remains a topic that warrants comprehensive investigation. Filling this knowledge gap, the present study employs the cantilever-beam principle as a methodological framework to scrutinize the fracture mechanism and mechanical properties of bamboo under the coupling effect of bending-shear. The obtained results reveal a linear decrease in the bending moment from the fixed support to the free end of the strip, while the shear force maintains a constant distribution along the strip's length. Within a specific segment of the strip, the bending stress exhibits a linear increase with the distance from the neutral axis, whereas the shear stress manifests a parabolic variation above or below the neutral axis. During the plastic stage, simultaneous transverse and radial fractures occur, effectively dissipating crack energy through interfacial debonding, fiber sliding, and fiber bridging mechanisms. The presence of stepwise deflection between adjacent internodes allows for substantial overall angular deflection of the bamboo culm without incurring damage. Furthermore, the mechanical properties of the strips demonstrate a positive linear correlation with both the fiber volume fraction and the density of vascular bundles. A notable achievement of this study is the establishment of a numerical structure-activity relationship, elucidating the bending-shear coupling performance. This relationship serves to clarify the fracture toughness exhibited by bamboo and provides a valuable reference for performance testing and the formulation of standardized protocols in the realm of biomimetic material production.

Polarized microwave emission from space particles in the upper atmosphere of the Earth

ABSTRACT Tons of space particles enter the Earth atmosphere every year, being detected when they produce fireballs, meteor showers, or when they impact the Earth surface. Particle detection in the showers could also be attempted from space using satellites in low Earth orbit. Measuring the polarization would provide extra crucial information on the dominant alignment mechanisms and the properties of the meteor families. In this article, we evaluate the expected signal to aid in the design of space probes for this purpose. We have used the radmc-3d code to simulate the polarized microwave emission of aligned dust particles with different compositions: silicates, carbonates, and irons. We have assumed a constant spatial particle density distribution of 0.22 cm−3, based on particle density measurements carried during meteor showers. Four different grain size distributions with power indices ranging from −3.5 to −2.0 and dust particles with radius ranging from 0.01 $\mathrm{\mu }$m to 1 cm have been considered for the simulations. Silicates and carbonates align their minor axis with the direction of the solar radiation field; during the flight time into the Earth atmosphere, iron grains get oriented with the Earth’s magnetic field depending on their size. Alignment direction is reflected in the Q-Stokes parameter and in the polarization variation along the orbit. Polarization depends on the composition and on the size distribution of the particles. The simulations show that some specific particle populations might be detectable even with a small probe equipped with high-sensitivity, photon-counting microwave detectors operating in low Earth orbit.

Smoothed Dirichlet Distribution

When the cells are ordinal in the multinomial distribution, i.e., when cells have a natural ordering, guaranteeing that the borrowing information among neighboring cells makes sense conceptually. In this paper, we introduce a novel probability distribution for borrowing information among neighboring cells in order to provide reliable estimates for cell probabilities. The proposed smoothed Dirichlet distribution forces the probabilities of neighboring cells to be closer to each other than under the standard Dirichlet distribution. Basic properties of the proposed distribution, including normalizing constant, moments, and marginal distributions, are developed. Sample generation of smoothed Dirichlet distribution is discussed using the acceptance-rejection algorithm. We demonstrate the performance of the proposed smoothed Dirichlet distribution using 2018 Major League Baseball (MLB) batters data.

Nonlinear liquid-liquid chromatography: Modeling a binary mixture separation

In liquid-liquid chromatography (LLC), mixture components are separated due to their different distribution between the phases of a biphasic liquid system composed of three or four solvents. LLC separations are typically modeled assuming that only the solutes distribute between the two liquid phases and their distribution can be described with a concentration-independent distribution constant. With increasing solute concentration, the physicochemical properties of the biphasic system change, and the distribution of the solutes becomes a function of their concentration. However, the experimental determination of liquid-liquid equilibria in multicomponent systems is time-intensive, and its prediction using thermodynamic models is often not sufficiently accurate for process design purposes. Thus, in this work, we propose a simple approach to model and simulate LLC separations in the nonlinear (concentration-dependent) range of the solutes’ distribution equilibria, namely cannabidiol (CBD) and cannabigerol (CBG). Using the inverse method, the distribution equilibrium equation parameters were estimated from pulse injection experiments of single solutes at concentrations ranging from 1 to 100 mg/mL and 1–50 mg/mL for CBD and CBG, respectively. The obtained parameters were then successfully used to predict the elution profiles of binary mixtures of different compositions at 40 mg/mL total cannabinoid concentration. The approach was demonstrated and validated for CBD and CBG as model compounds and n-hexane/methanol/water 10/7.5/2.5 (v/v/v) as the biphasic solvent system. It should be noted that the applicability of the proposed approach is system-dependent, and hence, it should be evaluated for each separation task individually.

Virtual source of a Butterfly beam

Using the virtual source method, we derive the expressions of (1+1) and (1+2)-dimensional butterfly waves. These waves can be considered to be generated by a virtual complex source at the complex position on the z-axis, which are exact solutions of the Helmholtz equation. Through numerical calculation and experiment, we study the evolution of (1+2)-dimensional butterfly wave in free space, and find that the distribution constants and beam width can affect the focusing distance and the evolution rate of the light field. Then, we carry out Taylor expansion for the exact solution of (1+1) and (1+2)-dimensional butterfly waves, and prove that the non-paraxial solution with higher order correction is valid.

Effect of Replacing Mineral Fertilizer with Manure on Soil Water Retention Capacity in a Semi-Arid Region

The long-term and excessive use of mineral fertilizers in a semi-arid region with severe water shortage will lead to soil compaction and poor water-holding capacity. The fertilization method of manure instead of mineral fertilizer has attracted wide attention. It has adverse consequences for the growth and development of crops. Hence, the objective of this study was to determine how replacing mineral fertilizer with manure affects the soil water retention curve, soil water constant, soil water availability, and soil equivalent pore size distribution, and to seek the best scheme of applying manure in semi-arid area and provide theoretical a basis for improving soil water retention capacity. Here, 0% (CK), 25% (M25), 50% (M50), 75% (M75), and 100% (M100) of 225 kg ha−1 nitrogen from mineral fertilizer were replaced with equivalent nitrogen from manure in the Loess Plateau of China under semi-arid conditions. The centrifuge method was used to determine the soil volumetric water content under different water suction levels, and the Gardner model was used to fit and draw its soil water retention curve, and then calculate the soil water constant and equivalent pore size distribution. The results showed that the Gardner model fitted well. The soil saturated water content with the M100 treatment was the highest, whereas the specific water capacity, water availability, and soil porosity with the M75 treatment were the highest. The soil saturated water content showed a downward trend with the increase in nitrogen from manure instead of nitrogen from mineral fertilizer in the partial replacement treatments. This downward trend slowed down over time. The M75 treatment increased field capacity. The M100 treatment increased soil capillary porosity, soil available water porosity, and soil water availability compared with CK from the fifth fertilization. Replacement treatments increased the specific water capacity, soil saturated water content, soil water availability, soil porosity, and reduced the wilting point over time. In the replacement treatments, specific soil water capacity, soil water availability, and soil porosity first rose and then declined with the increase in nitrogen provided by manure replacing that provided by mineral fertilizer. Therefore, the soil water holding capacity and water supply capacity with the M75 treatment were the best.

Harmonization service and global library of models to support country-driven global information on salt-affected soils

Global distribution of salt-affected soils (SAS) has remained at about 1 billion hectares in the literature over the years despite changes in climate, sea levels, and land use patterns which influence the distribution. Lack of periodic update of input soil data, data gaps, and inconsistency are part of the reasons for constant SAS distribution in the literature. This paper proposes harmonization as a suitable alternative for managing inconsistent data and minimizing data gaps. It developed a new harmonization service for supporting country-driven global SAS information update. The service contains a global library of harmonization models for harmonizing inconsistent soil data. It also contains models for identifying gaps in SAS database and for showing global distribution where harmonization of available data is needed. The service can be used by countries to develop national SAS information and update global SAS distribution. Its data availability index is useful in identifying countries without SAS data in the global database, which is a convenient way to identify countries to mobilize when updating global SAS information. Its application in 27 countries showed that the countries have more SAS data than they currently share with the global databases and that most of their data require SAS harmonization.

E72 Distribution of cells and extracellular matrix in the marginal synovium during the postnatal period

Abstract Introduction The marginal transitional zone of the joint is of fundamental importance in joint pathology. The normal marginal synovium takes part in articular cartilage repair (Adachi et al. 2012). The inflamed marginal synovium, in turn, promotes irreversible damage to the adjacent cartilage and bone (Hitchon & El-Gabalawy, 2011). Early erosive damage in rheumatoid arthritis and osteophyte formation in osteoarthritis occurs primarily here (Allard et al. 1990). Objectives The morphology of the marginal synovium from birth till the 90th day was studied using the hip joints of Wistar rats. Methods The collection of joints was carried out on the first, seventh, 14th, 30th, 45th, 60th and 90th day (six rats in each group, 42 in total). Joints were fixed, decalcified and dehydrated. Histologic paraffine sections were stained with Mallory's trichrome and Hart's elastin staining, analyzed at the acetabular labrum by light microscopy (χ100), morphometrical and statistical (P < 0.05) methods with extrapolation to humans (Sengupta, 2013). The obtained data were presented as percentages. Results In the marginal synovium, cells and components of the extracellular matrix, which are typical for fibrous connective tissue, could be seen. The synovial lining cells are embedded in the modified basement membrane which includes collagen and elastic fibers. They form a three-dimensional reticulum in it and adjacent synovial subintima. Vessels of the microcirculatory bed are also widely represented in the marginal synovium. The overall quantity of collagen fibers increased gradually, exceeding on the 90th day (approximately corresponds to 18–19 human years) that of newborns by one and a half times (48.92 ± 3.39 vs 72.17 ± 7.77; P = 0.023). The number of arranged collagen fibers increased gradually, exceeding on the 90th day that of newborns almost twofold (37.25 ± 3.21 vs 60.08 ± 5.99; P = 0.008). The area of distribution of disarranged collagen fibers in newborns was almost four and a half times less than the overall quantity of collagen fibers (11.67 ± 1.03 vs 48.92 ± 3.39; P < 0.001) and did not change substantially throughout the experiment. The number of elastic fibers increased till the 30th day (0.63 ± 0.11 vs 18.3 ± 1.25; P < 0.001) and did not undergo significant changes later. In newborns, the area of distribution of ground substance was not considerably higher than the ratio of disarranged collagen fibers (10.42 ± 1.8 vs 11.67 ± 1.03; P = 0.562) and did not undergo substantial changes later. The distribution density of vessels increased till the 30th-45th day (4.5 ± 0.67 vs 7.5 ± 0.82 and 7.7 ± 0.54; P < 0.001; approximately corresponds to the 9th and 14th human years respectively), followed by a gradual decrease from the 60th day (7.05 ± 0.72; roughly corresponds to the 16th human years). The maximum number of cells was in newborns. Subsequently, it gradually decreased and was almost three times less on the 90th day than that of newborns (38.01 ± 3.06 vs 12.58 ± 1.53; P < 0.001). Conclusion The morphogenesis of the marginal synovium during the postnatal period is accompanied by a gradual decrease in the number of cells, an increase in the area occupied by elastic and arranged collagen fibers with a relatively constant distribution density of disarranged collagen fibers and ground substance together with an undulatory pattern in the distribution of vessels. These peculiarities emphasize the morphological uniqueness of the marginal transitional zone, which forms the innate nonspecific barrier between the joint capsule and the articular cartilage, and thereby provides the necessary anatomical and physiological integrity of the joint as a whole organ. Knowledge about the patterns of morphogenesis of the marginal synovium will help in the future to analyze the modeled pathological or boundary states in it. Ethics The study was approved by the ethics committee of the Zaporizhzhia State Medical University.

E71 Distribution of cells and extracellular matrix in the normal marginal synovium of the hip joint capsule during the postnatal period

Abstract Introduction The marginal transitional zone is of fundamental importance in joint pathology. It includes the marginal cartilage and marginal synovium. The normal marginal synovium takes part in articular cartilage repair (Adachi et al. 2012). The inflamed marginal synovium, in turn, promotes irreversible damage to the adjacent cartilage and bone (Hitchon & El-Gabalawy, 2011). Objectives The morphology of the normal marginal synovium from birth till the 90th day was studied using the hip joints of Wistar rats. Methods The collection of joints was carried out on the first, seventh, 14th, 30th, 45th, 60th and 90th day (six rats in each group, 42 in total). Joints were fixed, decalcified and dehydrated. Histologic paraffine sections were stained with Mallory's trichrome and Hart's elastin staining, analyzed at the acetabular labrum by light microscopy (χ100), morphometrical and statistical (P < 0.05) methods with extrapolation to humans (Sengupta, 2013). The obtained data were presented as percentages. Results In the marginal synovium, cells and components of the extracellular matrix, which are typical for fibrous connective tissue, could be distinguished. The synovial lining cells are embedded in the modified basement membrane which is a meshwork of collagen and elastic fibers. They form a three-dimensional reticulum in it and adjacent synovial subintima. Vessels of the microcirculatory bed are also widely represented in the marginal synovium. The overall quantity of collagen fibers increased gradually, exceeding on the 90th day (∼18–19 human years) that of newborns by one and a half times (48.92 ± 3.39 vs 72.17 ± 7.77; P = 0.023). The number of arranged collagen fibers increased gradually, exceeding on the 90th day that of newborns almost twofold (37.25 ± 3.21 vs 60.08 ± 5.99; P = 0.008). The area of distribution of disarranged collagen fibers in newborns was almost four and a half times less than the overall quantity of collagen fibers (11.67 ± 1.03 vs 48.92 ± 3.39; P < 0.001) and did not change substantially throughout the experiment. The number of elastic fibers increased till the 30th day (0.63 ± 0.11 vs 18.3 ± 1.25; P < 0.001) and did not undergo significant changes later. In newborns, the area of distribution of ground substance was not considerably higher than the ratio of disarranged collagen fibers (10.42 ± 1.8 vs 11.67 ± 1.03; P = 0.562) and did not undergo substantial changes later. The distribution density of vessels increased till the 30th-45th day (4.5 ± 0.67 vs 7.5 ± 0.82 and 7.7 ± 0.54; P < 0.001; ∼9-14th human years), followed by a gradual decrease from the 60th day (7.05 ± 0.72; ∼16 human years). The maximum number of cells was in newborns. Subsequently, it gradually decreased and was almost three times less on the 90th day than that of newborns (38.01 ± 3.06 vs 12.58 ± 1.53; P < 0.001). Conclusion The morphogenesis of the marginal synovium during the postnatal period is accompanied by a gradual decrease in the number of cells, an increase in the area occupied by elastic and arranged collagen fibers with a relatively constant distribution density of disarranged collagen fibers and ground substance together with an undulatory pattern in the distribution of vessels. These peculiarities emphasize the morphological uniqueness of the marginal transitional zone, which forms the innate nonspecific barrier between the joint capsule and the articular cartilage, and thereby provides the necessary anatomical and physiological integrity of the joint as a whole organ. Knowledge about the patterns of morphogenesis of the normal marginal synovium will help to understand joint pathology. Ethics The study was approved by the ethics committee of the Zaporizhzhia State Medical University.

Topological defects law for migrating banded vegetation patterns in arid climates.

Self-organization and pattern formation are ubiquitous processes in nature. We study the properties of migrating banded vegetation patterns in arid landscapes, usually presenting dislocation topological defects. Vegetation patterns with dislocations are investigated in three different ecosystems. We show through remote sensing data analysis and theoretical modeling that the number of dislocations N(x) decreases in space according to the law N ∼ log(x/B)/x, where x is the coordinate in the opposite direction to the water flow and B is a suitable constant. A sloped topography explains the origin of banded vegetation patterns with permanent dislocations. Theoretically, we considered well-established approaches to describe vegetation patterns. All the models support the law. This contrasts with the common belief that the dynamics of dislocations are transient. In addition, regimes with a constant distribution of defects in space are predicted. We analyze the different regimes depending on the aridity level and water flow speed. The reported decay law of defects can warn of imminent ecosystem collapse.

Extended Rate Constants Distribution (RCD) Model for Sorption in Heterogeneous Systems: 4. Kinetics of Metal Ions Sorption in the Presence of Complexing Agents-Application to Cu(II) Sorption on Polyethyleneimine Cryogel from Acetate and Tartrate Solutions.

Here, we report a new version of the extended Rate Constants Distribution (RCD) model for metal ion sorption, which includes complex-formation equilibria. With the RCD-complex model, one can predict sorbent performance in the presence of complexing agents using data on metal ion sorption from ligand-free solutions and a set of coefficients for sorption rate constants of different ionic species. The RCD-complex model was applied to breakthrough curves of Cu(II) sorption from acetate and tartrate solutions on polyethyleneimine (PEI) monolith cryogel at different flow rates and ionic speciation. We have shown that, despite the lower stability of Cu(II)-acetate complex, at high flow rates, acetate has a more pronounced negative effect on sorption kinetics than tartrate. The RCD model was successfully used to predict the shape of the breakthrough curves at an arbitrary acetate concentration but failed to predict Cu(II) sorption from tartrate solutions in a broad range of ligand concentrations. Since a twofold increase in sorption capacity was observed at low tartrate concentrations, the latter fact was related to an alteration in the sorption mechanism of Cu(II)-ions, which depended on Cu(II) ionic speciation. The obtained results emphasize the importance of information about sorption kinetics of different ionic forms for the optimization of sorption filter performance in the presence of complexing agents.

Highly scalable and low cost binder free electrode of Zinc modified NiO nanofoam for dual role of supercapacitor and its free radical scavenging activity

In this study, pristine nickel oxide (NiO) and Zinc modified NiO nanofoams were prepared by green approach using camellia sinensis leaves extract. Pristine nickel oxide and Zn2+ modified NiO nanofoam were characterized by XRD, FTIR, FL, UV and FESEM. FE-SEM micrographs were clearly shows that the synthesised porous nanofoam with spherical shaped were constant distribution. The as prepared foam electrodes showed excellent supercapacitive behaviour with increase in specific capacitance with decrease in scan rate. The maximum specific capacitance 1530, 1706 and 1847Fg-1 was obtained at scan rate of 10 mVs-1 for increasing the Zn concentrations. After 3,000 cycles at 1 A g−1, the cyclic stability remains excellent at 88.1% of the initial capacitance. Moreover, the as-prepared asymmetric supercapacitor exhibits a high energy density of 30.6 W h·kg−1 at power density of 398 W kg−1. This study is expected to provide new insights into exploring the potential mechanism of catalyst action. These findings show that Zinc @ NiO nanofoam could be a potentially useful electrode material for energy storage devices.

Real-Time Monitoring and Control of Nanoparticle Formation.

Methods capable of controlling synthesis at the level of an individual nanoparticle are a key step toward improved reproducibility and scalability in engineering complex nanomaterials. To address this, we combine the spatially patterned activation of the photoreductant sodium pyruvate with interferometric scattering microscopy to achieve fast, label-free monitoring and control of hundreds of gold nanoparticles in real time. Individual particle growth kinetics are well-described by a two-step nucleation-autocatalysis model but with a distribution of individual rate constants that change with reaction conditions.

NMR Signatures and Electronic Structure of Ti Sites in Titanosilicalite-1 from Solid-State 47/49Ti NMR Spectroscopy.

Although titanosilicalite-1 (TS-1) is among the most successful oxidation catalysts used in industry, its active site structure is still debated. Recent efforts have mostly focused on understanding the role of defect sites and extraframework Ti. Here, we report the 47/49Ti signature of TS-1 and molecular analogues [Ti(OTBOS)4] and [Ti(OTBOS)3(OiPr)] using novel MAS CryoProbe to enhance the sensitivity. While the dehydrated TS-1 displays chemical shifts similar to those of molecular homologues, confirming the tetrahedral environment of Ti consistent with X-ray absorption spectroscopy, it is associated with a distribution of larger quadrupolar coupling constants, indicating an asymmetric environment. Detailed computational studies on cluster models highlights the high sensitivity of the NMR signatures (chemical shift and quadrupolar coupling constant) to small local structural changes. These calculations show that, while it will be difficult to distinguish mono- vs dinuclear sites, the sensitivity of the 47/49Ti NMR signature should enable distinguishing the Ti location among specific T site positions.

Accelerometer Mass Loading Study Based on a Damage Identification Method using Fundamental Laws in Closed Systems

Many civil engineering structures can be evaluated as closed systems in which additional mass spatial distribution can change within the system in terms of the fundamental continuity and energy equations. Considering the heavy hours of traffic, it can be assumed that the position of the vehicle mass on the bridge is variable, but the total mass remains constant. This study aims to show that the damage to the closed systems can be successfully estimated by using a damage indicator that is valid for systems with constant mass distribution. In the analytical study, a 100-element 200 degree of freedom fixed-free beam is investigated and the assessment of the damage position is verified numerically and the validity of the parameter is examined in an experimental study. Except for the piece at the free end of the beam; It has been determined that the damage indicator calculated on the damaged elements is 60 times larger than the damage indicator calculated at the undamaged elements. In the elements at the free end of the beam; it was observed that this ratio is between 6 and 40 depending on the mass of the accelerometer. Therefore, a criterion for accelerometer mass and damage indicator is proposed.

Clarithromycin as soil and environmental pollutant: Adsorption-desorption processes and influence of pH

Antibiotics pollution is a growing environmental issue, as high amounts of these compounds are found in soil, water and sediments. This work studies the adsorption/desorption of the macrolide antibiotic clarithromycin (CLA) for 17 agricultural soils with different edaphic characteristics. The research was carried out using batch-type experiments, with an additional assessment of the specific influence of pH for 6 of the soils. The results show that CLA adsorption reaches between 26 and 95%. In addition, the fit of the experimental data to adsorption models provided values between 1.9 and 19.7 Ln μmol1−n kg−1 for the KF, Freundlich affinity coefficient, and between 2.5 and 10.5 L kg−1 for Kd, distribution constant of Linear model. Regarding the linearity index, n, it varied between 0.56 and 1.34. Desorption showed lower scores than adsorption, with an average of 20%, and with values of 3.1 and 93.0 Ln μmol1−n kg−1 for KF(des) and 4.4 and 95.0 L kg−1 for Kd(des). The edaphic characteristics with the highest influence on adsorption were the silt fraction content and the exchangeable Ca content, while in the case of desorption, they were the total nitrogen, organic carbon, and exchangeable Ca and Mg contents. Regarding the pH, within the range studied (between 3 and 10), its value did not decisively affect the adsorption/desorption process. Overall, the set of these results could be of help to program appropriate measures leading to the retention/elimination of this antibiotic when it reaches the environment as a pollutant.

Burst-by-Burst Measurement of Rotational Diffusion at Nanosecond Resolution Reveals Hot-Brownian Motion and Single-Chain Binding.

We record dark-field scattering bursts of individual gold nanorods, 52 × 15 nm2 in average size, freely diffusing in water suspension. We deduce their Brownian rotational diffusion constant from autocorrelation functions on a single-event basis. Due to spectral selection by the plasmonic resonance with the excitation laser, the distribution of rotational diffusion constants is much narrower than expected from the size distribution measured by TEM. As rotational diffusion depends on particle hydrodynamic volume, viscosity, and temperature, it can sense those parameters at the single-particle level. We demonstrate measurements of hot Brownian rotational diffusion of nanorods in temperature and viscosity gradients caused by plasmonic heating. Further, we monitor hydrodynamic volumes of gold nanorods upon addition of very low concentrations of the water-soluble polymer PVA, which binds to the particles, leading to measurable changes in their diffusion constant corresponding to binding of one to a few polymer coils. We propose this analysis technique for very low concentrations of biomolecules in solution.

Modeling the Liberation of Iron Ores with Different Grain Sizes Considering Intergranular Fracture

ABSTRACT Predicting the liberation distribution of minerals in comminuted ore particles is one of the fundamental problems in mineral processing. However, there is still no widely accepted liberation model due to the complexity of the mineral fracture mechanism. In this study, the shape constants of the cumulative beta distribution were further optimized based on the investigation of the liberation distribution and fracture mechanism of iron oxide minerals in three types of iron ores with different grain sizes by locked-cycle grinding and batch grinding to model the liberation distribution of iron oxide minerals with different grinding methods. The results showed that the liberation distribution of iron oxide minerals in the three types of iron ores with locked-cycle grinding and batch grinding was related to intergranular fracture proportion and the closeness of ore particle size and mineral grain size. By relating the shape constants of cumulative beta distribution to ore particle size, mineral grain size, and the proportion of intergranular fracture, the developed liberation model well predicted the liberation distribution of iron oxide minerals with different grinding methods and its coefficient of determination (R2) was between 0.85–0.93.

Волновые пакеты в граничных задачах квантовой механики

System of two independent linear quantum equations with symbols representing polynoms of the \textit{n}-th order is considered. Boundary conditions are non-linear. They functionally connect amplitudes of the direct and inverse wave functions by mapping $\Phi :I \mapsto I$. It is demonstrated that 1) if mapping $ \Phi $ is linear, the amplitude of the falling wave at $ t\rightarrow\infty $ tends to zero or infinity; 2) if $ \Phi $ is nonlinear but single-valued, at $ t\rightarrow\infty $, the amplitude of the falling wave tends to a double-periodic – constant function with one singular point per a period; 3) if $ \Phi $ is multi-valued, asymptotically periodic --- constant distributions of square amplitude of the wavefunction with finite or infinite number of singularities per a period are possible. The limiting solutions of this type we shall call distributions of pre-turbulent or turbulent type. Applications to the study of the emergence of spatial-temporal bright and dark asymptotic solitons in a limited resonator with non-linear feedback between the amplitudes of two optical beams on the resonator surface are presented.

Plane-Parallel Laminar Flow of Viscous Fluid in the Transition Zone of the Inlet Section

A study was conducted to analyze how hydrodynamic parameters change in the entrance region of plane-parallel flow under stationary flow conditions, with an initial arbitrary distribution of velocities in the entrance section. This study was based on boundary layer equations, and a boundary problem was formed under the conditions of plane-parallel flow. The boundary conditions were chosen to reflect the pattern of arbitrary velocity distribution in the entrance section. A general solution of the approximating Navier-Stokes equations is provided, which depends on the initial conditions and the Reynolds number. The boundary conditions are established based on the nature of the motion, and the boundary value problem is described. A method for integrating the boundary value problem has been developed, and regularities for the change in velocities along the length of the inlet section have been obtained for both constant and parabolic velocity distributions in the entrance sections. Analytical solutions have been derived to obtain patterns of velocity and pressure changes in any given flow direction. Through computer analysis, velocity change patterns in various sections along the inlet transition area have been constructed, allowing for the determination of fluid velocity at any point on the section and an estimation of the length of the transition area. These proposed solutions provide a framework for accurately constructing individual units of hydromechanical equipment.