Linear Inequalities Research Articles

Gravitationally induced unified relation among duality, coherence, steering, and maximal average fidelity

Recently, numerous measures have been proposed for quantifying the quantumness of a given system, and the existence of intrinsic connections among quantum resource measures has been proven. Here, we study the unified relationship between duality, first-order coherence, three-setting linear steering inequality, and maximum average fidelity between two masses due to gravity. Under gravitational inducement, an equivalent relationship was identified between the first-order coherence and duality. The coherence of a system can be controlled by adjusting arm lengths and the distance between the arms of an interferometer. In most cases, the first-order coherence of a system cannot be maximised. Furthermore, a trade-off relationship between gravitationally induced duality and steering violations was derived. We can adjust the arm length and distance between the arms of the interferometer such that the steering violation reaches its maximum at phase π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document}. The results show that the value of the steering violation is always greater than 1; that is, the state of the system is steerable. In addition, we explored the intrinsic relationship between duality and the maximal average fidelity due to gravity. In most cases, the maximum average fidelity of the system is greater than 2/3, indicating that the state is useful for quantum teleportation. These results are important for investigating the intrinsic relationships among various quantum resources within the framework of gravity.

Norm-Based Finite-Time Convergent Recurrent Neural Network for Dynamic Linear Inequality

Norm-Based Finite-Time Convergent Recurrent Neural Network for Dynamic Linear Inequality

Analisis Kemampuan Berpikir Kreatif Matematis Siswa Kelas VII SMP Pada Materi Persamaan Dan Pertidaksamaan Linear Satu Variabel Dengan Menggunakan Hands on Activity

Analisis Kemampuan Berpikir Kreatif Matematis Siswa Kelas VII SMP Pada Materi Persamaan Dan Pertidaksamaan Linear Satu Variabel Dengan Menggunakan Hands on Activity

Approximation of the Objective Function of Single-Machine Scheduling Problem

The problem of the approximation of the coefficients of the objective function of a scheduling problem for a single machine is considered. It is necessary to minimize the total weighted completion times of jobs with unknown weight coefficients when a set of problem instances with known optimal schedules is given. It is shown that the approximation problem can be reduced to finding a solution to a system of linear inequalities for weight coefficients. For the case of simultaneous job release times, a method for solving the corresponding system of inequalities has been developed. Based on it, a polynomial algorithm for finding values of weight coefficients that satisfy the given optimal schedules was constructed. The complexity of the algorithm is O(n2(N+n)) operations, where n is the number of jobs and N is the number of given instances with known optimal schedules. The accuracy of the algorithm is estimated by experimentally measuring the function ε(N,n)=1n∑j=1n∣wj−wj0∣wj0, which is an indicator of the average modulus of the relative deviation of the found values wj from the true values wj0. An analysis of the results shows a high correlation between the dependence ε(N,n) and a function of the form α(n)/N, where α(n) is a decreasing function of n.

Porous invariants for linear systems

AbstractWe introduce the notion of porous invariants for multipath affine loops over the integers. These are invariants definable in (fragments of) Presburger arithmetic and, as such, lack certain tame geometrical properties, such a convexity and connectedness. Nevertheless, we show that in many cases such invariants can be automatically synthesised, and moreover can be used to settle reachability questions for various non-trivial classes of affine loops and target sets. For the class of $$\mathbb {Z}$$ Z -linear invariants (those defined as conjunctions of linear equations with integer coefficients), we show that a strongest such invariant can be computed in polynomial time. For the more general class of $$\mathbb {N}$$ N -semi-linear invariants (those defined as Boolean combinations of linear inequalities with integer coefficients), such a strongest invariant need not exist. Here we show that for point targets the existence of a separating invariant is undecidable in general. However we show that such separating invariants can be computed either by restricting the number of program variables or by restricting from multipath to single-path loops. Additionally, we consider porous targets, represented as $$\mathbb {Z}$$ Z -semi-linear sets (those defined as Boolean combinations of equations with integer coefficients). We show that an invariant can be computed providing the target spans the whole space. We present our tool porous, which computes porous invariants.

Formally verifying decompositions of stochastic specifications

According to the principles of compositional verification, verifying that lower-level components satisfy their specification ensures that the whole system satisfies its top-level specification. The key step is to ensure that the lower-level specifications constitute a correct decomposition of the top-level specification. In a non-stochastic context, such decomposition can be analyzed using techniques of theorem proving. In industrial applications, especially in safety-critical systems, specifications are often of stochastic nature, for example, giving a bound on the probability that a system failure will occur before a given time. A decomposition of such a specification requires techniques beyond traditional theorem proving. The first contribution of the paper is a theoretical framework that allows the representation of, and reasoning about, stochastic and timed behavior of systems as well as specifications for such behavior. The framework is based on traces that describe the continuous-time evolution of a system, and specifications are formulated using timed automata combined with probabilistic acceptance conditions. The second contribution is a novel approach to verifying decompositions of such specifications by reducing the problem to checking emptiness of the solution space for a system of linear inequalities.

Global exponential stability of quaternion bidirectional associative memory neural networks with multiple delays

In this paper, the global exponential stability of quaternion BAM neural networks with mixed delays is investigated. The involved delays include multiple time‐varying transmission delays and unbounded distributed delays. Existence and uniqueness of equilibrium point of the considered neural networks are investigated by using the homeomorphism theory between topology spaces. To show sufficient conditions for global exponential stability of the unique equilibrium point, a method based on estimation of solutions is presented. The acquired stability conditions contain only several simple linear scalar inequalities, which can be easily solved. The availability of theoretical results is shown by a numerical example. A point worth emphasizing is that the method based on estimation of solutions is applicable for models of many time‐delay systems after a minor modification.

Some Inequalities between General Randić-Type Graph Invariants

The Randić-type graph invariants are extensively investigated vertex-degree-based topological indices and have gained much prominence in recent years. The general Randić and zeroth-order general Randić indices are Randić-type graph invariants and are defined for a graph G with vertex set V as RαG=∑υi∼υjdidjα and QαG=∑vi∈Vdiα, respectively, where α is an arbitrary real number, di denotes the degree of a vertex υi, and υi∼υj represents the adjacency of vertices υi and υj in G. Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between Rα and Qα have been derived so far. In this paper, we establish explicit inequality relationships between Rα and Qα. Also, we determine linear inequality relationships between these graph invariants. Moreover, we obtain some new inequalities for various vertex-degree-based topological indices by the appropriate choice of α.

Distributed continuous-time optimization for convex problems with coupling linear inequality constraints

Distributed continuous-time optimization for convex problems with coupling linear inequality constraints

Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

AbstractIn this note, we prove interior a priori first‐ and second‐order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal regularity of Calderón–Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range , being the homogeneous dimension of the group.

Errors and Misconceptions in Linear Inequalities Among Senior High Students in Mfantseman Municipality

This study aimed to investigate the different errors and misconceptions made by students when dealing with linear inequalities. The goal was to uncover the nature and causes of these errors and misconceptions among students in Senior High Schools within the Mfantseman Municipality in the Central Region of Ghana. The research employed an explanatory sequential mixed methods design and was conducted in two public Senior High Schools selected from the Municipality. A total of 180 Senior High School students and teachers participated in the study, including 10 teachers. The sample was selected using a random sampling technique, which yielded 170 students from the two chosen public Senior High Schools. Data collection encompassed results from students' tests on linear inequalities, interviews with students, and questionnaires given to mathematics teachers. The collected data was coded and analyzed using descriptive statistics. The study's findings revealed common errors made by students, such as multiplying/dividing by a negative number, substituting inequality symbols with "equal to" symbols, performing operations on only one side or different numbers on the two sides of a compound inequality, as well as errors in algebraic operations, simplification, and arithmetic. Misconceptions observed included confusion between equality and inequality, misconceptions when dividing or multiplying through an inequality by a negative number, and struggles with compound inequalities. Students' difficulties arose from an inadequate understanding of basic inequality concepts, overgeneralization, limited mastery of inequality rules, and insufficient exposure to compound inequalities. Translating word problems into algebraic symbols posed a significant challenge. The study also highlighted that mathematics teachers were aware of the errors made by students. Consequently, teachers made efforts to address these errors during linear inequality classes. The findings suggest that teachers not only need assistance in identifying errors but also in understanding how errors can emerge during the learning process. One of the recommendations is to enhance teacher education by emphasizing diverse teacher-student interactions that thoroughly consider students' mathematical ideas. This approach aims to support teachers in effectively utilizing students' experiences in the learning process.

Penerapan Program Linear untuk Mengoptimalkan Jumlah Produksi dalam Memperoleh Keuntungan Maksimal pada Rumah UMKM Produk Wedang Sehat

Linear programming is a mathematical method used to solve problems related to linear optimization (maximum value and minimum value). Quoted from Mathematics 3 competency by Johanes, linear programming is a part of mathematics in the form of models, which consist of linear inequalities as a method for solving various problems in everyday life. This linear program can help solve problems ranging from basic to complex. This article aims to apply the use of linear programming in determining the maximum profit that can be obtained for managers of healthy wedang products. In this case, traders of ginger wedang products can use linear programming using the simplex method to find maximum profits manually and analyze using POM-QM for Windows software. The results of the analysis using the linear program of the simplex method with calculations on POM-QM have the same results, namely the maximum profit of MSME houses wedang sehat buk Sumiati in one day reaches Rp. 180,000 by producing healthy wedang ginger powder (X1) as much as 1 piece and wedang sehat suetajarah (X2) as much as 1 piece.

Pengembangan E-Modul Berbasis Etnomatematika untuk Meningkatkan Kemampuan Pemahaman Konsep Matematis Siswa SMAN 1 Grobogan

This media development research was motivated by the problems experienced by SMAN 1 Grobogan, namely the limited learning resources or teaching materials and the absence of electronic learning media and the low ability to understand students mathematical concepts which only reached 59,33 and the results of the questionnaire showed that 79% of students need electronic learning media. The aims of this study were. (1) to develop an ethnomatematics-based e-module as a learning medium to improve students ability to understand mathematical concepts. (2) test the validity of the ethnomatematics-based e-module to improve students ability to understand mathematical concepts. (3) testing the practicality of ethnomatematics-based e-module to improve students understanding mathematical concepts. Learning media is tool that can be used to assist the learning process, so that it can make it easier for students to receive material from a teacher. Ethnomatematics-based e-module are learning media that contain elements of culture with material on linear equations and inequalities and contain indicators of understanding mathematical concept. The subjects of this study were class X students of SMAN 1 Grobogan, while the research steps used were the ADDIE development model (Analyze, Design, Development, Implementation and Evaluation). Data collection techniques include interviews and questionnaires. Analysis of the data used includes analysis of the process of developing electronic media, analysis of the validation of media experts and material experts, as well as analysis of the practicality of e-modules. The results of the ethnomatematics based e-module validation show very valid criteria with a percentage of 80%. The results of the teacher and student practicality assessment of the ethnomatematics-based e-module show practical criteria with a percentage 71,96%. The test results on evaluating the ability to understand mathematical concepts have experienced a significant increase, reaching 82,94%. Based on the above results, the ethnomatematics-based e-module is suitable for use as a medium for learning mathematics in class X linear equations and inequalities. The results of the research that has been carried out show that the ethnomatematics-based e-module that has been developed has been declared valid, practical and able to improve comprehension skills students mathematical concept. For future research, it is hoped that it will be able to add trigger questions and more types of questions that are not in the book.

Switched max-plus linear-dual inequalities: cycle time analysis and applications

AbstractP-time event graphs are discrete event systems suitable for modeling processes in which tasks must be executed in predefined time windows. Their dynamics can be represented by max-plus linear-dual inequalities (LDIs), i.e., systems of linear dynamical inequalities in the primal and dual operations of the max-plus algebra. We define a new class of models called switched LDIs (SLDIs), which allow to switch between different modes of operation, each corresponding to a set of LDIs, according to a sequence of modes called schedule. In this paper, we focus on the analysis of SLDIs when the considered schedule is fixed and either periodic or intermittently periodic. We show that SLDIs can model a wide range of applications including single-robot multi-product processing networks, in which every product has different processing requirements and corresponds to a specific mode of operation. Based on the analysis of SLDIs, we propose algorithms to compute: i. minimum and maximum cycle times for these processes, improving the time complexity of other existing approaches; ii. a complete trajectory of the robot including start-up and shut-down transients.

The Difficulty of Students’ Reflective Thinking in Problems Solving of Linear Program

The identification in this study with the aim is to describe how difficult it is for students to think reflectively when solving math problems, especially in linear programming material. Based on the purpose of this study, the type of research is qualitative with a descriptive exploratory approach. Data collection techniques used are: (1) test instruments; (2) interview instruments, and 3) documentation. Analysis of research data namely: (1) research data reduction, (2) data exposure, (3) data triangulation, and (4) drawing conclusions. The subjects in this study were 24 high school students. Then 2 students were selected as subjects for each category (high, medium, and low). The results of the study show that students with high mathematical abilities have difficulty in reflecting, namely, 1) difficulty connecting new information with previous understanding, so they are not careful when identifying stories in the form of mathematical models, 2) difficulties in aspects of finding relationships and formulating solutions, students mistake the sign of linear inequality two variables, 3) difficulty in evaluating aspects of the completion process, students find it difficult to recall the function graph material to solve problems using the graphical method. Students with moderate mathematical abilities, namely: 1) difficulties in the aspect of connecting new knowledge with previous understanding, students need to be careful in solving contextual problems, 2) difficulties in aspects of finding relationships and formulating solutions, students have difficulty recalling function graph material, difficult to shade the area of settlement, 3) difficulties when students evaluate the completion process. Students find it difficult to prove whether the answer is correct or not by using the graphical method. Students with low mathematical ability, 1) difficulties in the aspect of connecting new knowledge with previous understanding, students find it difficult to translate story problems into mathematical models, it is difficult to recall the material of a two-variable linear inequality system, 2) difficulties in the aspect of finding relationships and formulating solutions, students have difficulty finding coordinates, drawing graphs, finding intersection points, substituting corner points into the objective function. 3) difficulties in evaluating aspects of the completion process, students find it difficult to prove the correctness of the answers obtained by the graphical method. The difficulties experienced by students in reflective thinking were caused by students not remembering previous material related to linear programming, as well as students’ difficulties in the dimensions of fact, concept and procedural knowledge.

Transformations of mixed solution types of interval linear equations system with boundaries on its left-hand side to linear inequalities with binary variables

To determine the solution set of an interval linear equations system Ax=b, it is necessary to know the semantics of the system. In addition to strong and weak semantics, there are four other important semantics: tolerance, control, left-localized, and right-localized. A solution x with one of these semantics provides that particular behavior in every equation of the system. It could happen that there is a solution x with one semantic in some equations and another semantic in the remaining equations of the system. This solution generates another solution type. We provide fifteen combinations of solution types generated by tolerance, control, left-localized, and right-localized semantics and prove that the solution set containing all four semantics is the weak (united) solution set. We show that each of these fifteen nonnegative solution types with some given boundaries on Ax can be transformed to be the solution set of linear inequalities with binary variables. The transformed system is deterministic and can be used in a linear programming problem with interval linear equation constraints to find an optimistic solution satisfying the semantics of the interval linear equations. Task management and course assignment examples are provided to show the necessity of these solution types.

On polytopes with linear rank with respect to generalizations of the split closure

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to t-branch split cuts and t-dimensional lattice cuts for a fixed positive integer t. These inequalities are the same as split cuts when t=1 and generalize split cuts when t>1. For polytopes contained in the n-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most n, and this bound is tight as Cornuéjols and Li gave an example with split rank n. All known examples with high split rank – i.e., at least cn for some positive constant c<1 – are defined by exponentially many (as a function of n) linear inequalities. For any fixed integer t>0, we give a family of polytopes contained in [0,1]n for sufficiently large n such that each polytope has empty integer hull, is defined by O(n) inequalities, and has rank Ω(n) with respect to t-dimensional lattice cuts. Therefore the split rank of these polytopes is Ω(n). It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.

Persistence and extinction in the anti-symmetric Lotka-Volterra systems

We investigate the global dynamics of the Lotka-Volterra systems with anti-symmetric interactions. We prove that the dynamics of system can be determined by some associated linear inequalities, which are of independent interest in linear programming. Necessary and sufficient conditions for the existence of solutions to the linear inequalities are established. When the solutions of the linear inequalities exist, the solutions of corresponding Lotka-Volterra systems are uniformly bounded, and the persistence and extinction of any species can be classified by an index set, which is uniquely determined by the linear inequalities. When the solutions of linear inequalities do not exist, the solutions of the Lotka-Volterra systems are unbounded. As an application, we analyze the three-species case and classify the long-time behaviors of the solutions to the system.

A model-based hypothesis framework to define and estimate the diel niche via the 'Diel.Niche' R package.

How animals use the diel period (24-h light-dark cycle) is of fundamental importance to understand their niche. While ecological and evolutionary literature abound with discussion of diel phenotypes (e.g. diurnal, nocturnal, crepuscular, cathemeral), they lack clear and explicit quantitative definitions. As such, inference can be confounded when evaluating hypotheses of animal diel niche switching or plasticity across studies because researchers may be operating under different definitions of diel phenotypes. We propose quantitative definitions of diel phenotypes using four alternative hypothesis sets (maximizing, traditional, general and selection) aimed at achieving different objectives. Each hypothesis set is composed of mutually exclusive hypotheses defined based on the activity probabilities in the three fundamental periods of light availability (twilight, daytime and night-time). We develop a Bayesian modelling framework that compares diel phenotype hypotheses using Bayes factors and estimates model parameters using a multinomial model with linear inequality constraints. Model comparison, parameter estimation and visualizing results can be done in the Diel.Niche R package. A simplified R Shiny web application is also available. We provide extensive simulation results to guide researchers on the power to discriminate among hypotheses for a range of sample sizes (10-1280). We also work through several examples of using data to make inferences on diel activity, and include online vignettes on how to use the Diel.Niche package. We demonstrate how our modelling framework complements other analyses, such as circular kernel density estimators and animal movement modelling. Our aim is to encourage standardization of the language of diel activity and bridge conceptual frameworks and hypotheses in diel research with data and models. Lastly, we hope more research focuses on the ecological and conservation importance of understanding how animals use diel time.

Analysis of Students' Errors in Solving Story Problems Based on Newman's Error Analysis (NEA) on the Subject Matter of Linear Program in View of Students' Mathematical Ability

This study aims to describe the types of errors of students in class XI SMA Negeri 1 Sirenja in solving story problems of linear program material based on Newman's Error Analysis in terms of mathematical ability. This type of research is qualitative research with a descriptive approach. The subjects in this study were based on the results of prerequisite tests with two-variable linear inequality material with consideration of students who made the most mistakes, were able to communicate well and recommendations from mathematics teachers with one student with high, medium and low mathematics abilities. Data collection techniques in this study were written tests and interviews in the form of one essay question and interview guidelines. The interview technique used was semi-structured conducted with selected students to ascertain the types of errors made in solving story problems. The results of this study show that the errors made by subjects with high mathematics ability (ST) made mistakes in writing the final answer (encoding error). Errors made by subjects with moderate mathematics ability (SS) made errors in process skills (process skill errors), and errors in writing the final answer (encoding errors). Errors made by low ability subjects (SR) made errors in understanding the problem (comprehension error), errors in changing the problem into mathematical form (transformation error), errors in process skills (process skill error), and errors in writing the final answer (encoding error).