Properties Of Operators Research Articles

On generalized distributions associated with singular partial differential operators

In this paper, we discuss various properties of the Riemann-Liouville operator over the generalized distributions D ′ w([0,+∞[×R) and other spaces. Next, we examine some properties of the convolution of the generalized distributions on the space D ′ w([0,+∞[×R).

Pseudo-differential operators associated with Gyrator transform

In this present work, a brief introduction to the Gyrator transform and its fundamental properties are given. Gyrator transform of tempered distribution is studied. This article made further discussion on the boundedness properties of pseudo-differential operators associated with Gyrator transform on Schwartz space as well as on Sobolev space.

Influence of tin on crystallization processes in iron-carbon alloys (review)

Currently, iron foundries and production facilities are experiencing great difficulties in providing such an integral charge material for smelting cast iron as high-quality unalloyed steel scrap. Due to the fact that press and forging industries and enterprises are increasingly using high-strength steels to produce their products, which significantly reduce the weight of the products obtained, microalloyed with such elements-carbidizing agents as, for example, Mo, W, Nb, V, Ti, B, the charge yards of iron foundries are supplied waste “infected” with these elements. As a rule, in such industries, products are made from several grades of steels, both microalloyed with carbide-forming elements and without them, and due to the objective complexity, the waste sorting process is not organized organizationally, therefore, waste from all available grades of steel falls uncontrollably into steel bags, mixing in arbitrary proportions. In addition to carbide-forming elements, surfactants such as Cu, Sn, As and Sb are also the most common residual elements in steel and cast iron scrap. If the accumulation of carbide-forming elements to a certain extent can be considered a positive factor, then the accumulation of surfactants, which are practically impossible to remove from steel and cast iron, has a sharply negative effect on the mechanical and operational properties of iron-carbon alloys. Tin is used as an alloying element to produce both steels and cast irons with a fully pearlitic matrix. The addition of tin in an amount of no more than 0.10–0.15% seems to be the only practical way to avoid the growth of ferrite in a stable eutectoid reaction and completely transform the metal base of steel and cast iron into pearlite in a metastable eutectoid Fe–C system. At the same time, the question of how tin affects eutectoid transformations in steels and cast irons and, in general, the entire cycle of structure formation, is still a matter of debate. In this paper, an overview of the Fe–Sn system is performed, as well as experimental and industrial data available in the literature on the Fe–C–Sn system to assess the effect of tin on crystallization processes in iron-carbon alloys.

Simulation Analysis of the Semi-Trailer Steered Wheels Control Algorithm

As part of improving road safety around trucks, a solution was proposed to reduce the swept path width of a moving tractor–semi-trailer. This article presents a mathematical analysis of the movement of a tractor unit with a traditional semi-trailer with fixed axles and steered wheels. A simulation analysis of both presented vehicles was carried out. The core of the algorithm controlling the steering angle of the semi-trailer wheels is presented. The influence of controlling the semi-trailer’s swivel wheels on the swept path width of a tractor–trailer with a semi-trailer equipped with swivel wheels is discussed. The assumptions for building the control algorithm are presented. The article presents the advantages of the solution used along with the control algorithm. Measurable benefits resulting from the use of the presented solution are presented, such as increasing cargo space, reducing cargo transport costs, and reducing aerodynamic resistance and fuel consumption. It is worth emphasizing that reducing fuel consumption is very important because it reduces the emission of harmful exhaust gases into the atmosphere. The swept path width is important especially in the case of vehicles moving in a limited area, for example in the parking lots of transhipment and logistics centers, between urban buildings. Vehicles admitted to traffic meet the minimum conditions imposed by homologation regulations, but reducing the swept path width allows for improving the operational properties of the tractor–semi-trailer. The use of the proposed control algorithm to control the turn of the semi-trailer’s steered wheels brings tangible benefits both in improving road safety and in reducing the emission of harmful substances into the environment.

P0390 Development, internal, and external validation of a novel endoscopic activity score for evaluation of Crohn’s disease (EASE-CD)

Abstract Background Endoscopy is essential for measuring luminal disease activity in Crohn’s disease (CD). Current endoscopic indices for evaluating CD activity are limited: they are only modestly correlated with disease activity, contain items that are ambiguously defined and have poor inter-observer reliability, and lack validated definitions for clinically relevant change. We aimed to develop and internally and externally validate a reliable and responsive endoscopic CD activity index. Methods In this multi-part study, candidate endoscopic items for index development with face validity derived from a prior RAND/UCLA appropriateness exercise were assessed by blinded central readers using ileocolonoscopy videos from the phase 3b adalimumab EXTEND (NCT00348283) and phase 2 risankizumab (NCT02031276) trials. Acceptable item-level operating properties for further consideration as covariables in regression for model building included at least moderate inter-rater reliability (intraclass correlation coefficient [ICC]≥0.41), and medium responsiveness defined as the probability of a patient receiving active treatment having a better endoscopy score than a patient receiving placebo or win probability (WinP) ≥0.56. Model development and internal validation with bootstrapping utilised data from 112 paired baseline and post-treatment videos from the adalimumab trial. External model validation utilised data from 108 video pairs from the risankizumab trial. Results The Endoscopic Activity Score for Evaluating CD (EASE-CD) is comprised of ulcer size, presence of deep ulcer(s), and percentage of ulcerated surface with each item evaluated and averaged across visualised bowel segments (Table 1). Total scores range from 0-100 (higher scores indicate greater endoscopic activity). EASE-CD demonstrated almost perfect inter-rater reliability (ICC 0.820 [95% CI: 0.662-0.906]) and a large degree of responsiveness (WinP 0.769 [95% CI: 0.658-0.851]). A 10-point reduction in EASE-CD had 87.8% specificity and 60.9% sensitivity for endoscopic response defined by the Simple Endoscopic Score for CD (SES-CD)/CD Endoscopic Index of Severity (CDEIS). Ulcer-free endoscopic remission results in EASE-CD=0. Conclusion EASE-CD is a novel, validated, continuous measure of endoscopic CD activity constructed using modern principles of index development, that is reliable, responsive to treatment, and linearly correlated with disease activity. EASE-CD does not include “narrowing” (which cannot be reliably evaluated between observers) or “affected area” (which is ambiguous in meaning, poorly reliable, and not independently associated with disease activity). EASE-CD is based only on ulcer-related items, the hallmark lesion in CD.

Mapping Properties of Certain Nonlinear Integral Operators Involving Hornich Operations

Mapping Properties of Certain Nonlinear Integral Operators Involving Hornich Operations

ОБОСНОВАНИЕ ПРИМЕНЕНИЯ НОВОЙ ТЕХНОЛОГИИ ПРОИЗВОДСТВА ШПИНТОНОВ

This study objective is to analyze the effectiveness of using the technology of tail suspension of cars as a way to increase the reliability and comfort of movement in railway transport. To achieve this goal, the following task was defined: identification and analysis of internal defects that are the causes of product rejection, and the subsequent development of recommendations for their elimination. The main research method is the analysis of current operating procedure of manufacturing parts, the study of materials and methods of their processing used to strengthen tails. The priority is to identify the key factors contributing to increasing the durability of the tail assembly through the use of reinforcing technology. Both theoretical analysis of existing scientific papers on this topic and practical experiments on stress tests are used as research methods. The novelty of the work is in offering a more environmentally friendly product processing technology, as well as confirming the possibility of replacing the tail material with a more cost-effective one. It is worth noting that the issue of ensuring the quality of truck tails of subway cars, the ecological purity of the heat treatment process, and cost reduction is being solved through a new heat treatment technology – quenching with a fast-moving stream of water and replacing 40X alloy steel with 35 carbon steel without reducing operational properties. As a result, the proposed measures contribute to improving the manufacturing quality of tail assemblies and reducing the percentage of their rejection at the production testing stage.

Issues in the Expansion and Contraction of Operators

This paper explores fundamental issues in the correct contraction and expansion of operators, with a primary focus on the concept of symmetry within operator theory. Special attention is given to how symmetry influences the behavior of operators, particularly regarding their approximation and convergence properties. In the domains of quantum mechanics and condensed matter physics, such operators are essential for modeling phenomena like superconductivity, excitons, and surface states. The symmetric properties of operators have a profound impact on the physical interpretations and predictions these models generate. A rigorous analysis is provided regarding the existence of correct contractions and expansions for a specific class of nonlinear operators, demonstrating how symmetry affects the structural integrity of operators under natural conditions. The study presents a comprehensive description of the set of all correct contractions, expansions, and regular expansions, with an application to a third-order nonlinear differential expression. Additionally, a condition for the unique solvability of a Bitsadze–Samarskii-type problem is derived, showcasing how symmetry plays a crucial role in guiding the solution of complex physical models. Furthermore, the paper emphasizes the importance of preserving symmetry in the construction of operators, ensuring the consistency and accuracy of mathematical models. This has significant implications for both theoretical research and practical applications in various fields, including nuclear physics and quantum theory.

Quantum–Fractal–Fractional Operator in a Complex Domain

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions.

Operational properties of cement-free concrete with porous aggregate

The article presents the results of technology development and research on the operational properties of porous aggregate and cement-free concretes based on it. The purpose of the work is to study lightweight concretes containing a liquid–glass porous aggregate for resistance to various aggressive influences. The porous granular aggregate was obtained by firing a mixture of liquid glass with the ash of thermal power plants and an ash aluminosilicate microsphere. Binders based on caustic magnesite and liquid glass with the addition of thermal energy waste were used to produce coarse-pored concretes. The choice of cement-free binders is due to the high adhesion to the filler. The behavior of the developed concretes in various aggressive environments, under the influence of low and elevated temperatures, has been studied. The resistance of magnesia concrete to the effects of water and salt solutions has been revealed. The technological and operational advantages of liquid-glazed concrete are shown, featuring increased thermal insulation ability, satisfactory resistance to aggressive media and resistance to low and high-temperature fluctuations. The developed concretes can be used in the enclosing structures of objects for various purposes.

Exploring Principal Component Analysis for Enhanced Insights into Physical and Operational Characteristics of Palladium-Based Membrane Composites: Advancing Hydrogen (H2) Energy Potential to Revolutionize the Energy Sector

In this study, we used Principal Component Analysis (PCA) to evaluate the physical and operational properties of palladium (Pd)-based membrane composites, focusing on variables like temperature, differential pressure (ΔP), thickness, hydrogen (H2) permeability, and H2 flux. The analysis revealed that the first two principal components explained 53.16% of the total variance, indicating moderate explanatory power. Interdependencies were observed among temperature, thickness, H2 flux, and H2 permeability, while ΔP functioned independently. This study found similarities among membranes, such as eco-friendly chitosan-based membranes, which performed comparably to conventional options like Pd–PSS and Pd–Cu/αAl2O3. Overall, PCA proved to be an invaluable tool for uncovering hidden patterns, optimizing experimental processes, and deepening the understanding of Pd-based membranes. The findings underscore PCA’s potential to enhance material performance and promote sustainable alternatives, with practical benefits for advancing hydrogen separation technologies. This study illustrates how data-driven approaches can refine material analysis and drive innovation in membrane design.

The Properties of Two-Fold Algebra Based on the n-standard Fuzzy Number Theoretical System

In this paper, we study the two-fold algebra based on the n-standard fuzzy number theoretical system as a special type of two-fold fuzzy algebras, where we study the elementary properties of the algebraic operations defined over this system. Also, we prove many results that describe the relations between two-fold substructures and sub-algebras defined by fuzzy number theoretical systems. On the other hand, we provide many different examples to explain our results.

A constrained optimisation framework for parameter identification of the SIRD model.

We consider a numerical framework tailored to identifying optimal parameters in the context of modelling disease propagation. Our focus is on understanding the behaviour of optimisation algorithms for such problems, where the dynamics are described by a system of ordinary differential equations associated with the epidemiological SIRD model. Applying an optimise-then-discretise approach, we examine properties of the solution operator and determine existence of optimal parameters for the problem considered. Further, first-order optimality conditions are derived, the solution of which provides a certificate of goodness of fit, which is not always guaranteed with parameter tuning techniques. We then propose strategies for the numerical solution of such problems, based on projected gradient descent, Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), nonmonotone Accelerated Proximal Gradient (nmAPG), and limited memory BFGS trust region approaches. We carry out a thorough computational study for a range of problems of interest, determining the relative performance of these numerical methods. Our results provide insights into the effectiveness of these strategies, contributing to ongoing research into optimising parameters for accurate and reliable disease spread modelling. Moreover, our approach paves the way for calibration of more intricate compartmental models.

Nonclassical dynamical behavior of solutions of partial differential-difference equations

<p>For partial differential-difference equations, a review of results regarding the relation between the type of the equation and dynamical properties of its solutions is provided. This includes the case of elliptic equations with timelike independent variables: Their solutions acquire dynamical properties (more exactly, behave as solutions of parabolic equations). The following approach to classify differential-difference equations into types, based on the property of differential-difference operators to be Fourier multipliers is applied in the following manner: An operator is treated to be elliptic if the real part of its symbol is positive, while the parabolic and hyperbolic types are defined correspondingly. It is shown that the proposed approach (being a natural extension of the classical notion of the ellipticity) is reasonable. On the other hand, fundamental novelties (compared with the classical theory of partial differential equations) occur as well. We provide conditions which guarantee the following results. For the half-space (half-plane) Dirichlet problem for elliptic equations, integral representations (of the Poisson type) of solutions are constructed, which are infinitely differentiable outside the boundary hyperplane (plane), and the asymptotic closeness of solutions (as the timelike independent variable unboundedly increases) takes place. For the Cauchy problem for parabolic equations, the same is valid, but we deal with the classical time instead of the timelike independent variable. For hyperbolic equations, multiparameter families of infinitely smooth global solutions are constructed. The said (sufficient) conditions restrict the sign of the real part of the symbol for the differential-difference operator with respect to spatial or spacelike independent variables. In a number of special cases, they might be weakened such that symbols with sign-changing real parts are admitted. The objective of the study is to observe the current stage of the classification issues for partial differential-difference equations in terms of the aforementioned approach: The ellipticity of a Fourier multiplier is defined by means of the sign (of there real part) of its symbol. Since both differential and translation operators are Fourier multipliers, methods of Fourier analysis are applicable in this study: We apply the Fourier transformation to the original partial differential-difference equation, solve the obtained ordinary differential equation, and apply the inverse Fourier transformation to the obtained solution. The main contribution obtained within this study is an efficient (workable) type concept for the fundamentally new extension of the class of partial differential equations, which is the class of partial differential-difference equations.</p>

On positive vectors in indefinite inner product spaces

Let [Formula: see text] be a linear space equipped with an indefinite inner product [Formula: see text]. Denote by [Formula: see text] the nonlinear set of positive vectors in [Formula: see text]. We demonstrate that the properties of a linear operator [Formula: see text] in [Formula: see text] can be uniquely determined by its restriction to [Formula: see text]. In particular, we prove that the bijectivity of [Formula: see text] on [Formula: see text] is equivalent to [Formula: see text] being close to a unitary operator with respect to [Formula: see text]. Furthermore, we consider a one-parameter semi-group of operators [Formula: see text], where each [Formula: see text] maps [Formula: see text] onto itself in a one-to-one manner. We show that, under this natural restriction, the semi-group [Formula: see text] can be transformed into a one-parameter group [Formula: see text] of operators that are unitary with respect to [Formula: see text]. By imposing additional conditions, we show how to construct a suitable definite inner product [Formula: see text], based on [Formula: see text], which guarantees the unitarity of the operators [Formula: see text] in the Hilbert space obtained by completing [Formula: see text] with respect to [Formula: see text].

Variational Properties of the Abstract Subdifferential Operator

Variational Properties of the Abstract Subdifferential Operator

Unveiling New Dimensions in Soft Disemigraph Theory: Lexicographic and Restricted Lexicographic Products

Soft set theory provides a systematic approach for handling imprecision and uncertainty by categorizing elements of a set based on specific parameters. In semigraph theory, soft semigraphs utilize this approach, offering a parameterized perspective that has significantly advanced the field through effective parameter management. Building on this foundation, disemigraphs extend semigraphs by incorporating directional relationships among vertices, making them ideal for modeling scenarios where the sequence and direction of connections are crucial. In this paper, we introduce and explore the lexicographic product, and restricted lexicographic product of soft disemigraphs. We provide formal definitions, illustrative examples, and detailed proofs of several theorems to investigate the properties of these product operations.

Study of the Influence of Modification and Heat Treatment of Cast Iron Grinding Balls on Their Operational Properties

The study addresses the enhancement of wear resistance and impact strength of cast iron grinding balls used under intensive abrasive wear and impact conditions at ore processing plants. The primary aim is to improve the operating properties of cast iron by optimizing its chemical composition and introducing advanced thermal and chemical-thermal treatment methods, including controlled cooling. To achieve this goal, two alloyed cast iron compositions with added chromium, molybdenum, and nickel were used, and the base composition was further modified with titanium and titanium carbonitride to increase the microhardness of the matrix phases. Experimental samples underwent various heat treatment regimes, including quenching and normalization, to evaluate the effect of these processes on the microstructure, hardness, and impact resistance of the grinding balls. Microstructural analysis revealed that additional alloying elements and modifiers contributed positively to forming a more homogeneous and fine-grained structure with an increased proportion of carbide phases. The key findings indicated that the optimized structure of the cast iron grinding balls achieved 10 units higher hardness and 2-4 units greater impact resistance than the base composition after heat treatment. Furthermore, titanium and titanium carbonitride modifications enhanced the performance characteristics of the balls by several additional units. Thus, the developed treatment regimes and modified cast iron composition significantly increase the wear resistance of grinding balls, contributing to reduced grinding costs in industrial applications.

Evaluation of Wear Resistance of a Modified Radial Bearing Design Taking into Account Compressibility and Viscosity of the Lubricant

Introduction. The challenge problem of the quality of lubricants has led to both large-scale and narrow-focused theoretical and applied studies that relate to the operational properties of lubricants. In particular, the authors of the publications are interested in the interaction of bearing and lubrication, since numerous types of machinery and equipment contain these elements. In the literature, the composites used to strengthen the bearing surface are compared, the most effective compositions are determined, and the advantages and disadvantages of the components are analyzed. Mathematical models have been proposed and tested for some of the processes of the interaction under consideration, and acceptable adequacy has been proved for some of them. However, the improvement of such solutions requires taking into account the specifics of the tribosystem. This issue has been poorly worked out, and the presented article is intended to fill this gap. When evaluating the wear resistance of a radial bearing, the compressibility of a high viscosity lubricant is taken into account.Materials and Methods. The study was based on the tribocontact scheme, which included the radius of the polymer-coated shaft, the radius of the bearing sleeve, the height of the lubrication groove, and the thickness of the lubricating layer. To create new mathematical models that took into account the compressibility of the lubricant, the authors used three equations: motion of the liquid lubricant, continuity, and state. To verify the model, the results of calculations and laboratory tests were compared. In the experiments, a bearing with a groove to preserve lubrication was used. Its rotation speed, loads and temperature conditions were changed. Friction was measured using traditional methods and modern instruments.Results. The bearing design was modified to take into account an additional factor — the compressibility of the lubricant. The new model predicted the bearing capacity of the part by 8–10% more accurately, and the coefficient of friction — by 7–9%. Fluctuations in the coefficient of friction up to 45 MPa (equivalent to a five-fold increase in load) were detected and explained. This was due to dynamic changes in the surface contact conditions and the effects of external parameters. Optimal applications of antifriction coatings based on hybrid composite materials were determined. The possibilities of practical use of calculation models of a journal bearing were expanded. Its critically important operational characteristics were evaluated in practice.Discussion and Conclusion. The scientific research results described in this article make it possible to establish the performance characteristics of the bearing at the design stage. The significant potential of this approach has been identified in terms of increasing the reliability and durability of the studied part, and this seems to be an important step in the development of bearing and lubricant technologies. In the future, the authors intend to study such factors as temperature conditions, dynamic loads, and interaction with various lubricants. This will allow us to improve bearing designs and expand their application areas.

DESIGN AND EXPERIMENTAL STUDIES OF STRENGTHENING OF BACKWATER TYPE HYDRAULIC STRUCTURES WITH COMPOSITE MATERIALS

During long-term operation of hydraulic structures of sluices, retaining walls, abutments, there are non-design settlements, tilts, displacements, opening of inter-block construction joints, corrosion of working reinforcement, change in strength characteristics of concrete, which leads to a decrease in operational properties and bearing capacity. The analysis of scientific and technical documentation is presented, the results of surveys are analysed, a spatial mathematical model based on the finite element method is developed. The computational research of the stress-strain state of sluices is carried out, including taking into account the reinforcement with prestressed basalt-composite reinforcement. The scheme of reinforcement of sluice walls has been justified by calculation. In experimental researches the complex problem on definition of strength, deflections, width of opening of inter-block construction joints, deformations of steel and prestressed basalt-composite reinforcement of reinforced concrete structures including those exposed to water environment was solved. The conducted design and experimental studies substantiate the use of prestressed basalt-composite reinforcement as an effective method of reinforcement of retaining structures, increasing their load-bearing capacity and maintaining deflections, width of opening of interlock construction joints, deformations of steel and prestressed basalt-composite reinforcement within the limits allowed by normative documents.