Interval Of Real Numbers Research Articles

Set-Valued $$\alpha $$-Fractal Functions

In this paper, we introduce the concept of the $$\alpha $$ -fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its $$\alpha $$ -fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an $$\alpha $$ -fractal function is an attractor of an iterated function system.

Existence and uniqueness of Nash equilibrium in discontinuous Bertrand games: a complete characterization

Since (Reny in Econometrica 67:1029–1056, 1999) a substantial body of research has considered what conditions are sufficient for the existence of a pure strategy Nash equilibrium in games with discontinuous payoffs. This work analyzes a general Bertrand game, with convex costs and an arbitrary sharing rule at price ties, in which tied payoffs may be greater than non-tied payoffs when both are positive. On this domain, necessary and sufficient conditions for (i) the existence of equilibrium (ii) the uniqueness of equilibrium are presented. The conditions are intuitively easy to understand and centre around the relationships between intervals of real numbers determined by the primitives of the model.

Hesitant Fuzzy Variable and Distribution

In recent decades, the hesitant fuzzy set theory has been used as a main tool to describe the hesitant fuzzy phenomenon, which usually exists in multiple attributes of decision making. However, in the general case concerning numerous decision-making problems, values of attributes are real numbers, and some decision makers are hesitant about these values. Consequently, the possibility of taking a number contains several possible values in the real number interval [0, 1]. As a result, the hesitant possibility of hesitant fuzzy events cannot be inferred from the given hesitant fuzzy set which only presents the hesitant membership degree with respect to an element belonging to this one. To address this problem, this paper explores the axiomatic system of the hesitant possibility measure from which the hesitant fuzzy theory is constructed. Firstly, a hesitant possibility measure from the pattern space to the power set of [0, 1] is defined, and some properties of this measure are discussed. Secondly, a hesitant fuzzy variable, which is a symmetric set-valued function on the hesitant possibility measure space, is proposed, and the distribution of this variable and one of its functions are studied. Finally, two examples are shown in order to explain the practical applications of the hesitant fuzzy variable in the hesitant fuzzy graph model and decision-making considering hesitant fuzzy attributes. The relevant research results of this paper provide an important mathematical tool for hesitant fuzzy information processing from another new angle different from the theory of hesitant fuzzy sets, and can be utilized to solve decision problems in light of the hesitant fuzzy value of multiple attributes.

Asymptotic analysis of the transient response of a thermoelastic assembly involving a thin layer

We study the transient response of a thermoelastic structure made of two tridimensional bodies connected by a thin adhesive layer. Once more we highlight the powerful flexibility of Trotter's theory of approximation of semi-groups of operators acting on variable spaces: considering the geometrical and physical characteristics of the thin layer as parameters, we are able to show in a unitary way that this situation leads to a huge variety of limit models the properties of which are detailed. In particular, according to the relative behaviors of the different parameters involved, new features are evidenced such as the apparition of an added specific heat coefficient for the interface or of additional thermomechanical state variables defined not only on the limit geometric interface but on its cartesian product by any interval of real numbers.

Optimal values range of interval polynomial programming problems

Uncertainty exists in many real-life engineering and mechanical problems. Here, we assume that uncertainties are caused by intervals of real numbers. In this paper, we consider the interval nonlinear programming (INLP) problems where the objective function and constraints include interval coefficients. So that the variables are deterministic and sign-restricted. Additionally, the constraints are considered in the form of inequalities. A basic task in INLP is calculating the optimal values range of objective function, which may be computationally very expensive. However, if the boundary functions are available, the problems become much easier to solve. By making these assumptions, an efficient method is proposed to compute the optimal values range using two classic nonlinear problems. Then, the optimal values range are obtained by direct inspection for a special kind of interval polynomial programming (IPP) problems. Two numerical examples are given to verify the effectiveness of the proposed method.

ROBUST SOLUTION IDENTIFICATION FOR UNCERTAINTY MANAGEMENT IN MOLP - AN INTERACTIVE APPROACH

This paper presents a robust deterministic method for the treatment of uncertainty in decision-making systems based on Sonar Method. It introduces uncertainty into mathematical models as interval numbers. This work proposes an interactive approach that identifies the efficient vertices of the Multiobjective Linear Programming (MOLP), where some or all the coefficients of the objective functions, constraints or limits of the constraints are interval of real numbers. The method identifies robust solutions to MOLP under uncertainty, showing how the interactive approach presented in this paper can provide the possibility of exploring in new decision support methodologies.

Scheduling with gaps: new models and algorithms

We consider scheduling problems for unit jobs with release times, where the number or size of the gaps in the schedule is taken into consideration, either in the objective function or as a constraint. Except for several papers on minimum-energy scheduling, there is no work in the scheduling literature that uses performance metrics depending on the gap structure of a schedule. One of our objectives is to initiate the study of such scheduling problems. We focus on the model with unit-length jobs. First we examine scheduling problems with deadlines, where we consider two variants of minimum-gap scheduling: maximizing throughput with a budget for the number of gaps and minimizing the number of gaps with a throughput requirement. We then turn to other objective functions. For example, in some scenarios gaps in a schedule may be actually desirable, leading to the problem of maximizing the number of gaps. A related problem involves minimizing the maximum gap size. The second part of the paper examines the model without deadlines, where we focus on the tradeoff between the number of gaps and the total or maximum flow time. For all these problems we provide polynomial time algorithms, with running times ranging from O(nlog n) for some problems to O(n^7) for other. The solutions involve a spectrum of algorithmic techniques, including different dynamic programming formulations, speed-up techniques based on searching Monge arrays, searching X+Y matrices, or implicit binary search. Throughout the paper, we also draw a connection between gap scheduling problems and their continuous analogues, namely hitting set problems for intervals of real numbers. As it turns out, for some problems the continuous variants provide insights leading to efficient algorithms for the corresponding discrete versions, while for other problems completely new techniques are needed to solve the discrete version.

Some Bounds on Bond Incident Degree Indices with Some Parameters

It is considered that there is a fascinating issue in theoretical chemistry to predict the physicochemical and structural properties of the chemical compounds in the molecular graphs. These properties of chemical compounds (boiling points, melting points, molar refraction, acentric factor, octanol-water partition coefficient, and motor octane number) are modeled by topological indices which are more applicable and well-used graph-theoretic tools for the studies of quantitative structure-property relationships (QSPRs) and quantitative structure-activity relationships (QSARs) in the subject of cheminformatics. The π -electron energy of a molecular graph was calculated by adding squares of degrees (valencies) of its vertices (nodes). This computational result, afterwards, was named the first Zagreb index, and in the field of molecular graph theory, it turned out to be a well-swotted topological index. In 2011, Vukicevic introduced the variable sum exdeg index which is famous for predicting the octanol-water partition coefficient of certain chemical compounds such as octane isomers, polyaromatic hydrocarbons (PAH), polychlorobiphenyls (PCB), and phenethylamines (Phenet). In this paper, we characterized the conjugated trees and conjugated unicyclic graphs for variable sum exdeg index in different intervals of real numbers. We also investigated the maximum value of SEIa for bicyclic graphs depending on a > 1 .

Selection by differential mortality rates

Abstract In this work, a partial differential equation model for evolutionary dynamics is presented that describes changes in densities of phenotypes in a population. We consider that the traits of individuals of a population are distributed at an interval of real numbers where a mortality rate is assigned for each value of this interval. We present some conditions for stability of stationary solutions and apply the model in theoretical scenarios of natural selection. Particularly we approach cases of stabilising, disruptive and directional selection, including the scenario of the survival of the flattest. Some computational simulations are performed to illustrate the results obtained.

On the density of ℚ in ℝ: Imaginary dialogues scripted by undergraduate students

This study aims to explore the notion of the density of the set of rational numbers in the set of real numbers, as interpreted by undergraduate mathematics students. The data comprise 95 responses to a scripting task, in which participants were asked to extend a hypothetical dialog between two student characters, who argue about the existence of one or infinitely many rational numbers in a real number interval. The analysis leans on the framework of reducing abstraction to provide explanations for the participants’ mathematical behaviour when coping with the task. The findings point to students’ informal ideas related to density that can be mapped to formal proofs, as well as to unconventional understandings of related concepts and ideas, including rational and irrational numbers, infinity, and mathematical justification. Implications are drawn.

Interval valued trapezoidal neutrosophic set: multi-attribute decision making for prioritization of non-functional requirements

This paper discusses the trapezoidal fuzzy number(TrFN); Interval-valued intuitionistic fuzzy number(IVIFN); neutrosophic set and its operational laws; and, trapezoidal neutrosophic set(TrNS) and its operational laws. Based on the combination of IVIFN and TrNS, an Interval Valued Trapezoidal Neutrosophic Set (IVTrNS) is proposed followed by its operational laws. The paper also presents the score and accuracy functions for the proposed Interval Valued Trapezoidal Neutrosophic Number (IVTrNN). Then, an interval valued trapezoidal neutrosophic weighted arithmetic averaging (IVTrNWAA) operator is introduced to combine the trapezoidal information which is neutrosophic and in the unit interval of real numbers. Finally, a method is developed to handle the problems in the multi attribute decision making(MADM) environment using IVTrNWAA operator followed by a numerical example of NFRs prioritization to illustrate the relevance of the developed method.

PRESERVATION OF LOG-CONCAVITY AND LOG-CONVEXITY UNDER OPERATORS

Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$, θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

Sampling Almost Periodic and Related Functions

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We obtain in this way very sparse sets of real numbers (and of integers) on which two different almost periodic functions cannot agree, which makes them amenable to be used in sampling theorems for these functions. These sets can be made as sparse as to have zero asymptotic density or as to be t-sets, i.e., to be sets that intersect any of their translates in a bounded set. Many of these results are proved not only for almost periodic functions but also for classes of functions generated by more general complex exponential functions, including chirps or polynomial phase functions.

An ordinal multi-criteria decision-making procedure under imprecise linguistic assessments

Many decision-making problems such as quality control analysis, market surveys or sensory analysis require ordered qualitative scales, rather than numerical ones. It is very common to assign some cardinal mathematical objects, such as numerical values, intervals of real numbers or fuzzy numbers, to the linguistic terms of ordered qualitative scales. However, when agents perceive that the psychological proximity between each pair of consecutive terms of the scale is not identical, these conversions are meaningless and an ordinal approach to deal with these non-uniform ordered qualitative scales is more appropriate. The aim of this paper is to introduce an ordinal multi-criteria decision-making procedure for ranking alternatives in the setting of ordered qualitative scales that are non-necessarily uniform. The possibility of doubt is also considered, by allowing agents to assign two consecutive terms of the scale when they hesitate. The proposed procedure is applied to a real case study in which nine experts assessed eight wines regarding different criteria.

INTERVAL VALUES AND RATIONAL CHOICE

Abstract:Decision theorists have suggested that it is sometimes appropriate to represent the value of certain options as intervals of real numbers. In this note I show that doing so is incompatible with some deeply held intuitions about rational choice.

Performance optimization of dual stage algorithm for lossless data compression and decompression

In this paper, an optimized dual stage architecture is proposed which is the combination of Lempel-Ziv-Welch (LZW) Algorithm at the first phase and Arithmetic Coding being the later part of Architecture. LZW Algorithm is a lossless compression algorithm and code here for each character is available in the dictionary which reduces 5-bits per cycle as compared to ASCII. In arithmetic coding the numbers are represented by an interval of real numbers from zero to one according to their probabilities. It is an entropy coding and is lossless in nature. The text information is allowed to pass through the proposed architecture and it gets compressed to the higher rate.

Keimel\u2019s problem on the algebraic axiomatization of convexity

Convex sets may be viewed as algebras equipped with a set of binary convex combinations that is indexed by the open unit interval of real numbers. Convex sets generate the variety of barycentric algebras, which also includes semilattices where the semilattice multiplication is repeated uncountably many times. Barycentric algebras are defined by three axioms: idempotence, skew-commutativity, and skew-associativity. Since the skew-associativity axiom is rather complicated, Klaus Keimel has asked whether it can simply be replaced by the entropic law. It turns out that the answer is negative. The counterexamples presented and studied in this paper are known as threshold barycentric algebras, depending on a threshold taken from the left-hand side of the closed unit interval. They offer an entire spectrum of algebras, ranging from barycentric algebras for threshold 0 to commutative idempotent entropic groupoids for threshold 1/2.

On generalization of midpoint type inequalities with generalized fractional integral operators

The Hermite–Hadamard inequality is the first principal result for convex functions defined on a interval of real numbers with a natural geometrical interpretation and a loose number of applications for particular inequalities. In this paper we proposed the Hermite–Hadamard and midpoint type inequalities for functions whose first and second derivatives in absolute value are s-convex through the instrument of generalized fractional integral operator and a considerable amount of results for special means which can naturally be deduced.

Vague semiprime ideals of a $$\Gamma $$ Γ -semiring

The concept of vague semiprime ideal of a $$\Gamma $$ -semiring with membership and non-membership functions taking the values in unit interval of real numbers are introduced and investigated some results.

Fully privacy-preserving determination of point-range relationship

The privacy-preserving determination of point-range relationship is widely applied to range queries. However, most existing solutions only protect one partys privacy but not the other partys; moreover, they only cover integers or rational numbers. Focusing on these problems, we design two protocols with the notion of secure multiparty computation. We first present the fully privacy-preserving protocol 1,linebreak which can determine whether an integer is in a discrete integer interval using the 0-1 coding in combination with the Goldwasser-Micali homomorphic encryption. Subsequently, we present the fully privacy-preserving protocol 2,linebreak which can determine whether a real number is in a continuous real number interval using the monotonic function in combination with Pailliers homomorphic encryption. Finally, we present a practical example based on our protocol. Theoretical and experiential analyses show that both of our protocols achieve a low communication cost, while preserving the full privacy of two parties. In addition, protocol 2 in this paper first gives a solution to privately determine whether a real number is in a continuous real number interval, while showing higher efficiency and better performance.