System Of Inequalities Research Articles

Orbit-finite linear programming

An infinite set is orbit-finite if, up to permutations of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.

Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution

Inspired by the layered structure models in pavement mechanics research, in this study, a class of multilayer elastic contact systems with interlayer frictional contact conditions and deformable supporting frictional contact conditions on the foundation has been constructed. Based on the nonlinear elastic constitutive equations, the corresponding system of partial differential equations and variational inequalities are respectively introduced. Under the framework of variational inequalities, the existence and uniqueness of solutions for such models, along with the approximation properties of finite element numerical solutions, are proven and analyzed. The aforementioned conclusions provide fundamental and broadly applicable theoretical support for addressing mechanical problems in multilayer elastic contact systems within the framework of variational inequalities. Finally, the numerical experimental results based on the mixed finite element method also substantiate our theoretical conclusions.

COMPARATOR REGRESSION ANALYSIS IN CONDITIONS OF FUZZY INITIAL DATA

Abstract. The canonical task of regression analysis consists in studying the problem of estimation and mathematical description of the influence caused by one or more independent variables (predictors, explanatory variables) on the dependent (explained) variable. The subject of comparator identification problem is a specific modification of the standard regression analysis problem in the case when, for some reason, it is not possible to measure values of the dependent, explained variable. However, the problem of reconstructing a regression polynomial can be solved if it is possible to qualitatively compare the results of experiments and rank them, for example, in the order of increasing. Taking into account the results of ranking the values of the explained variable in different experiments, a corresponding system of inequalities is formed. The purpose of the research consists in developing a method for mathematical processing of the resulting system of inequalities for analytical assessment of the level of influence of explanatory variables on the explained variable. The method for solving the problem is based on transforming a system of inequalities into a system of linear algebraic equations. A significant drawback of the known approach to solving this problem consists in the fact that the resulting system of linear algebraic equations is an underdetermined one (the number of unknowns in this system exceeds the number of equations). At the same time, the system has an infinite number of solutions, among which there may be an uncontrollable number of unacceptable ones. The exhaustive search method for finding acceptable solutions is futile. In this regard, the problem of developing a correct method of comparator identification remains relevant. Another difficulty that accompanies the actual procedure of comparator identification when solving practical problems consists in the fact that measurements of values of the controlled parameters are not accurate. The uncertainty that arises in the conditions of a small sample of initial data can be removed using the tools of fuzzy mathematics. It is assumed that, based on preliminary research, a mathematical description of the corresponding membership function can be obtained for each of the controlled parameters. As a result, effective approaches to solving emerging problems have been proposed, based on the use of developed optimization procedures in the conditions of a small sample of fuzzy initial data.

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

Analysis of High School Students' Mathematical Literacy Skills in Solving Contextual Problems on SPtDV Material

Mathematical literacy is one of the abilities that must be possessed in today's 21st-century education. This study aims to describe the mathematical literacy skills of grade X students at one of the Yogyakarta high schools in Bantul Regency in solving contextual problems on SPtDV (System of Inequality of Two Variables) material. The method used was descriptive qualitative. Data were collected through multiple-choice tests, descriptions, and questionnaires using SPtDV material. Data analysis was done by reducing and presenting data, and drawing conclusions based on indicators of students' mathematical literacy (mathematical process). Sampling of the population was carried out using a non-purposive sampling technique. To determine the number of samples to be taken, researchers used a saturated sampling technique. The research subjects were students of class X of Yogyakarta Private Senior High School in Bantul Regency in the first semester of the 2023/2024 academic year, with a total of 20 students. The results showed that students with high ability were able to fulfill all three aspects of the indicators in mathematical literacy skills, namely formulating, using, and interpreting. While students with moderate ability were able to fulfill two aspects of the indicator, namely formulating and using. And students with low ability were only able to fulfill one aspect of the indicator, namely formulating. The percentage of results obtained is based on the Mathematical Literacy Profile Criteria, namely: 10% proficient, 60% proficient, 15% basic, and 15% need special intervention. This study implies that it is important for other researchers to explore the factors that influence the mathematical literacy skills of low-ability students, especially in high school students or equivalent.

Resolution of the sec-max-product fuzzy relation inequalities system and its lexicographic minimum solution

Resolution of the sec-max-product fuzzy relation inequalities system and its lexicographic minimum solution

A low-complexity solution for LMIs of hardly constrained nonlinear MIMO tracking control systems

This research is dedicated to the discrete-time tracking control of nonlinear multi-input multi-output (MIMO) Newtonian mechanics systems with an equal number of inputs and outputs, in the presence of input saturation and high-order dynamics of the actuation system. Firstly, the continuous-time model of the plant and actuation system has been transformed into the discrete-time domain using Adams–Bashforth and Adams–Moulton discretization techniques. In the next step, considering saturation boundaries, the modified position and velocity references are derived using the discrete-time reaching law-based control approach. These modified references are obtained through an optimization algorithm that calculates the closest achievable position and velocity trajectories with respect to the desired reference trajectory. Modifying desired trajectories in MIMO systems results in the formation of a system of linear matrix inequalities (LMIs). Solving such systems of inequalities requires the use of complex methods, and this difficulty increases with the increase in the system’s degrees of freedom (DOFs). The proposed strategy offers a more straightforward approach to solving these inequalities. Once the modified references are calculated, the corresponding position and velocity control signals can be obtained using the mentioned reaching law strategies. The final step of the proposed algorithm is to combine the resulting position and velocity control signals using a Gaussian-weighted averaging method to form the main control signal. A desirable performance for the control system can be achieved by appropriately adjusting the standard deviations of the Gaussian weighing functions. To evaluate the performance characteristics of the proposed method, it has been applied in the tracking control procedure of a nonlinear 3-DOF mechanical system. The numerical simulation results demonstrate the effectiveness of the method.

Subjective well-being of Russians through the prism of their identities: state and factors

The article shows the prevalence of identities of Russians related to subjective well-being (identity with people of the same material wealth, with those who are in constant need and with those who have achieved success in life) based on the research data of the Institute of Sociology of FCTAS RAS. The analysis of these identities effectively complements the traditional methods of studying well-being by assessing it not “in the moment”, but over a long period of time, since the identities are formed over a long period of time and are stable. It is shown that these three symbolic identities allow to “capture” different components of subjective well-being. Self-identification with those in constant need reflects the disadvantage caused not only by low income and experienced deprivations, but also by the gap between a person's aspirations and actual achievements. Identity with those who have achieved success is closely related to overall satisfaction with life and with one's position in the status hierarchy, reflecting the achievement of goals that are important to the individual (not necessarily related to material well-being). Identity with people of the same material well-being is more common among the well-off population, as many of the disadvantaged try to find more favorable grounds for positioning themselves in society. As these identities reflect different aspects of subjective well-being, the coexistence of apparently mutually exclusive identities is typical of Russians. It is also shown that the self-identifications of representatives of the mass strata of the country's population based on their wellbeing occupy an important place in their identification system, and that the prevalence of these identities is stable over time, although they are no longer among the “nuclear” identities. They are relatively more pronounced among representatives of polar groups in terms of their place in the system of social inequality. At the same time, as a whole, they allow to define the position of a person in the system of social inequalities quite clearly.

Solving problems of volume scheduling for shipbuilding enterprises in conditions of limited resources

The shipbuilding industry development strategy for the period up to 2035 and beyond is aimed at creating a new competitive shipbuilding industry based on the development of scientific, technical and personnel potential, optimization of production capacities, their modernization and technical re-equipment [1]. The paper is devoted to solving problems of volume-calendar production planning for shipbuilding enterprises when coordinating the production capacity of the enterprise with the resource needs to fulfill obligations determined by the generated order portfolio. Shipbuilding and ship repair enterprises are typical representatives of enterprises with single and small-scale production and a long production cycle of the main products [2-4]. The problem is solved at the stage of forming a portfolio of orders, which is based on the limited volumes of resources available to the production system. It is required to determine the amount of missing resources by calendar periods that are necessary for the effective functioning of the production system. Non-stored resources such as labor resources, equipment operating time, are usually taken into account when solving the problems of volumetric or volume-scheduling planning. Such resources being unused in a certain planning cycle cannot be used in subsequent cycles. The procedure for finding missing volumes of resources is formulated as the problem of reducing an incompatible system of linear two-sided algebraic inequalities to a compatible one. To do this, it is proposed to “expand” the initial parameters of the system, taking into account the costs of expansion, so that the system becomes compatible. In the general case, formally, the problem under consideration is close to the problem of finding a Chebyshev point [5] for an incompatible system of linear algebraic equations, i.e. a point that is geometrically the least deviating from all hyperplanes formed by the system constraints. The paper proposes statements and solutions of volume-scheduling problems for the case when the structure of the problem constraints is modeled by a root-oriented weighted tree. This hierarchical structure is typical for shipbuilding enterprises [6].

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function d e p t h R / I t depthR/I^t is asymptotically a constant, i.e. there is a number s s such that d e p t h R / I t = d e p t h R / I s depthR/I^t = depthR/I^s for t > s t > s . One calls the smallest number s s with this property the index of depth stability of I I and denotes it by d s t a b ( I ) dstab(I) . This invariant remains mysterious till now. The main result of this paper gives an explicit formula for d s t a b ( I ) dstab(I) when I I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize d s t a b ( I ) dstab(I) explicitly. The formula expresses d s t a b ( I ) dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

An elementary approach based on variational inequalities for modeling a friction-based locomotion problem

We propose an elementary proof based on a penalization technique to show the existence and uniqueness of the solution to a system of variational inequalities modeling the friction-based motion of a two-body crawling system. Here for each body, the static and dynamic friction coefficients are equal.

Diversifying Gender Categories and the Sex/Gender System

The growing visibility of transgender and nonbinary people raises important sociological questions about how the structure of sex and gender is shifting and underscores necessary changes to research practice. We review what is known about emerging gender identities and their implications for sociological understandings of the relationship between sex and gender and the maintenance of the sex/gender system of inequality. Transgender and nonbinary identities are increasingly common among younger cohorts and improved survey measurements of sex and gender are expanding information about these changes. In the United States, an additional gender category seems to be solidifying in public usage even as the higher status of masculinity over femininity persists. The continuing power of the normative binary contributes to both violent backlash and characteristic patterns of discrimination against gender diverse people; yet, underlying support for nondiscrimination in the workplace is stronger than commonly recognized. New, more consistent efforts to account for gender diversity in social science research are needed to fully understand these changes.

Liouville-type results for semilinear inequalities involving the Dunkl Laplacian operator

Let Δ k be the Dunkl generalized Laplacian operator associated to a root system R of R N and a nonnegative multiplicity function k defined on R and invariant by the finite reflection group W. In this paper, we establish Liouville-type theorems for the semilinear inequality − Δ k u ≥ | u | p in R N and the system of inequalities − Δ k u ≥ | v | p , − Δ k v ≥ | u | q in R N , where N ≥ 1 and p,q>1. To the best of our knowledge, this contribution is the first work dealing with Liouville-type results for nonlinear problems involving the Dunkl Laplacian.

Модификация модели прогнозирования финансово-экономического состояния для задачи управления развитием медицинских учреждений

The article analyzes the model of forecasting the financial and economic condition, considers the possibility of its transformation for managing the development of medical institutions, and provides an appropriate modification using methods of discriminant analysis and mathematical programming. The formulated modification of the model has a clear mathematical justification, which allows, on the one hand, to prove its validity, and on the other hand constructively shows the possibility of its use, taking into account the specifics of the medical industry. The purpose of the work. The purpose of this work is to study changes in the forecasting model of the financial and economic condition, taking into account the specifics of the industry, as well as to evaluate the application of methods of discriminant analysis, reducing them to a mathematical programming problem, the limitations of which are reduced to systems of linear inequalities obtained by building maximally joint subsystems. Materials and methods. It is proposed to use the method of discriminant analysis with extensible precedent sets. It is assumed that in order to improve the quality of solving this problem, expert judgments about objects that were not initially included in the training sample are involved. At the same time, the goal is to determine the best objects to involve them in the learning material in order to maximize the solution of the problem. To formalize the model based on the predicted values, we will encrypt all available data into a vector of states. The considered methods show their effectiveness. The results of the study. During the modification of the economic and financial condition model, a conclusion is made about the linearization of the model, which is also associated with the differentiation operation. The possibility of using neural networks is also demonstrated for the case of differentiation of complex functions, and in the case of convex programming, for marginal estimates obtained using the directional differentiation operation. Conclusion. The results obtained allow us to assert that the use of strict mathematical tools, a modified model of the economic and financial condition for the medical industry is easy to use for development management, since it can be implemented as some component of a computer program (neural network) into which the administrator enters data, and at the output receives a processed result.

Student Errors in Solving System of Linear Inequalities Story Questions Based on Newman’s Error Analysis (NEA)

This paper aims to describe the types of error and the causes of student error in solve math story questions on the subject matter of systems of linear inequalities based on Newman's Error Analysis (NEA). The linear inequalities system material was chosen for this study because the linear inequalities system has components that students must understand so they can easily determine the error made by student. This study is a qualitative descriptive. The subjects of this study were taken using a purposive sampling technique and obtained 10 students in the 5th semester of math education study program, Muhammadiyah University of Tangerang. This research was divided into three stage, namely the preparation of research instrument, implementation and colection of data study and data analysis. The instrument used in this research is a description of 3 questions. Based on data analysis, the factor that led to student error in solve math story question on the subject matter of the system of linear inequalities based on NEA were inaccurate, unable to read questions, unable to understand the question, and unable to carry out procedure or steps that would used to problem solving. This study contributes to providing information on errors that are commonly made by student in solve math story questions on the subject matter system linear inequalities based on NEA.

On the geometry and refined rate of primal–dual hybrid gradient for linear programming

AbstractWe study the convergence behaviors of primal–dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does.

Development of Electronic Teaching Materials Based on Contextual Teaching and Learning (CTL) Approach to Improve High School Students' Mathematical Problem Solving Skills

The aim of this research is to develop approach-based electronic teaching materials Contextual Teaching And Learning (CTL) on systems of equations and linear inequalities to improve high school students' valid and practical mathematical problem solving abilities. This research method uses research and development with a 4D model. Data was collected using questionnaires, observations and interviews. The data analysis technique used is quantitative descriptive analysis. Teaching materials are said to be valid and practical if the score obtained is more than 70%. The results of this research show that electronic teaching materials are based The CTL approach was developed to increase mathematical problem solving abilities for high school students get score mean 89% with very valid category. The results of student responses in the limited trial obtained a score of 90.28% in the very practical category. The student response results in the field trial were 90.35% with very practical criteria. The teacher response results were 94.14% with the very practical category and the observation sheet 90.6% with the very practical criteria. From the results of validity and practicality, it can be seen that the teaching materials developed are suitable for use as learning resources.

A note on the generalized Hessian of the least squares associated with systems of linear inequalities

The goal of this note is to point out an erroneous formula for the generalized Hessian of the least squares associated with a system of linear inequalities, that was given in the paper ‘A finite Newton method for classification’ by Mangasarian (Optim. Methods Softw. 17 (2002), pp. 913–929) and reproduced multiple times in other publications. We also provide sufficient contiditions for the validity of Mangasarian's formula and show that Slater's condition guarantees that some particular elements from the set defined by Mangasarian belong to the generalized Hessian of the corresponding function.

Analisis Hambatan Belajar Pada Materi Sistem Persamaan Dan Pertidaksamaan Linear Satu Variabel Di SMP Negeri 2 Tatapaan

This research was conducted to find out what factors hinder the learning of class VII students of SMP Negeri 2 Tatapaan for the 2023/2024 academic year on the material Systems of Linear Equations and Inequalities in One Variable. The population and sample of this research is 40. This research is a quantitative descriptive research that uses survey methods and data collection using research instruments in the form of questionnaires and interviews. The questionnaire created had 24 statement items. The results of the research show that the internal aspects of learning barriers in the One Variable System of Linear Equations and Inequalities material fall into the Moderately Obstructed category (65.74%), with the lowest indicator being student readiness which falls into the Quite Impeded category (48%). Furthermore, the external aspect of barriers to student learning in the material Systems of Linear Equations and Inequalities with One Variable was included in the Moderately Obstructed category (68.6%), with the lowest indicator being Family, which was included in the Moderately Obstructed category (55.5%).

Using Constraint Propagation to Bound Linear Programs

We present an approach to compute bounds on the optimal value of linear programs based on constraint propagation. Given a feasible dual solution, we apply constraint propagation to the complementary slackness conditions and, if propagation succeeds to prove these conditions infeasible, the infeasibility certificate (in the sense of Farkas’ lemma) is reconstructed from the propagation history. This certificate is a dual-improving direction, which allows us to improve the bound. As constraint propagation need not always detect infeasibility of a linear inequality system, the method is not guaranteed to converge to a global solution of the linear program but only to an upper bound, whose tightness depends on the structure of the program and the used propagation method. The approach is suited for large sparse linear programs (such as LP relaxations of combinatorial optimization problems), for which the classical LP algorithms may be infeasible, if only for their super-linear space complexity. The approach can be seen as a generalization of the Virtual Arc Consistency (VAC) algorithm to bound the LP relaxation of the Weighted CSP (WCSP). We newly apply it to the LP relaxation of the Weighted Max-SAT problem, experimentally showing that the obtained bounds are often not far from optima of the relaxation and proving that they are exact for known tractable subclasses of Weighted Max-SAT.