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Journal articles on the topic 'Alternative extension of the real numbers set'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Sukhotin, A., and M. Zvyagin. "Alternative analysis: the prime numbers theory and an extension of the real numbers set." Bulletin of Science and Practice 398, no. 10(11) (2016): 10–14. https://doi.org/10.5281/zenodo.160909.

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Here we consider the theory of prime numbers at a new methodology. The theory of prime numbers is one of the most ancient mathematical branches. We found an estimate of the all prime numbers sum using the notions of infinite lager numbers and infinitely small numbers, farther we estimated the value of the maximal prime number. We proved that Hardy–Littlewood Hypothesis has the positive decision too. The infinite small numbers define a new methodology of the well–known function o(x) application. We use the sets of the theory of prime numbers and infinitely small numbers with a linear function t

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2

Sukhotin, A. "Alternative analysis: some new methodology and extension of the real numbers set." Bulletin of Science and Practice, no. 11 (November 13, 2017): 16–20. https://doi.org/10.5281/zenodo.1048277.

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Positive definitions made it possible to prove new theorems on well-known objects of analysis: numerical sequences and series. The estimates of the quantity of all primes and the largest of them are obtained. With linear function at, we define an Alternative extension of the real numbers set.

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Panev, A., and A. Sukhotin. "The limiting values of smoothly–convex monotonic functions and properties of infinity." Bulletin of Science and Practice, no. 3 (March 15, 2017): 8–13. https://doi.org/10.5281/zenodo.399057.

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An introduction contains the short methodological agreement on symbols and on terms, on concepts and on mathematical texts. Then we have find the border between a finite and an infinite subset of the linearly ordered set. By definition each subset of the final set (except trivial) has two boundary points. The set is called as an infinite set if there [{exits}] its subset, which has less two boundary points. Further we have established some facts of the theory of smoothly convex monotonous functions. In particular, smoothly convex monotonous functions have the nonnegative first derivative and t

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Sen, Dilip Kumar, Saurav Datta, and Siba Sankar Mahapatra. "Extension of TODIM combined with grey numbers: an integrated decision making module." Grey Systems: Theory and Application 5, no. 3 (2015): 367–91. http://dx.doi.org/10.1108/gs-05-2015-0029.

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Purpose – Decision making is the task of selecting the most appropriate alternative among a finite set of possible alternatives with respect to some attributes. The attributes may be subjective or objective (or combination of both), depending upon the situation; requirements may also be conflicting. In practice, most of the real-world decision-making problems are based on subjective evaluation criteria which are basically ill-defined and vague. Since subjective human judgment bears ambiguity and vagueness in the decision making; application of grey numbers set theory may be proved fruitful in

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Remad, Fedia Daami, and Hela Moalla Frikha. "The multicriteria group decision making FlowSort method under uncertainty." Multiple Criteria Decision Making 16 (2021): 122–39. http://dx.doi.org/10.22367/mcdm.2021.16.07.

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Crisp values are insufficient to model real-life situations and imprecise ideas are frequently represented in multicriteria decision aid analysis. In fact, it is difficult to treat the evaluation criteria precisely and to fix exact preferences rating. The triangular intuitionistic fuzzy numbers succeeded to treat this kind of ambiguity in a great deal of research than other forms of fuzzy representation functions. The field of sorting issues is an active research topic in the multiple criteria decision aid (MCDA). This study extended one of the sorting methods, FLOWSORT, for solving multiple c

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Lev, Raskin, and Sira Oksana. "Development of methods for extension of the conceptual and analytical framework of the fuzzy set theory." Eastern-European Journal of Enterprise Technologies 6, no. 4(108) (2020): 14–21. https://doi.org/10.15587/1729-4061.2020.217630.

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Fuzzy set theory is an effective alternative to probability theory in solving many problems of studying processes and systems under conditions of uncertainty. The application of this theory is especially in demand in situations where the system under study operates under conditions of rapidly changing influencing parameters or characteristics of the environment. In these cases, the use of solutions obtained by standard methods of the probability theory is not quite correct. At the same time, the conceptual, methodological and hardware base of the alternative fuzzy set theory is not sufficientl

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Riaz, Muhammad, Wojciech Sałabun, Hafiz Muhammad Athar Farid, Nawazish Ali, and Jarosław Wątróbski. "A Robust q-Rung Orthopair Fuzzy Information Aggregation Using Einstein Operations with Application to Sustainable Energy Planning Decision Management." Energies 13, no. 9 (2020): 2155. http://dx.doi.org/10.3390/en13092155.

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A q-rung orthopair fuzzy set (q-ROFS), an extension of the Pythagorean fuzzy set (PFS) and intuitionistic fuzzy set (IFS), is very helpful in representing vague information that occurs in real-world circumstances. The intention of this article is to introduce several aggregation operators in the framework of q-rung orthopair fuzzy numbers (q-ROFNs). The key feature of q-ROFNs is to deal with the situation when the sum of the qth powers of membership and non-membership grades of each alternative in the universe is less than one. The Einstein operators with their operational laws have excellent

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Fedia, Remadi Daami, and Frikha Moalla Hela. "The Intuitionistic Fuzzy FlowSort Method for Multicriteria Group Decision Making." International Journal of Fuzzy System Applications 11, no. 1 (2022): 1–21. http://dx.doi.org/10.4018/ijfsa.285554.

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The real life problems are multidimensional in nature and may involve some ambiguity when it comes to decision making. It is, therefore, difficult to design the evaluation criteria precisely and determine the exact value of the attributes in the multicriteria analysis. The intuitionistic fuzzy set (IFS) achieved great success in treating this kind of ambiguity in a great deal of research. The study of sorting problems is an active research issue in the multiple criteria decision aid (MCDA) area. This paper investigated one of the sorting methods, FLOWSORT, and extended it to the multicriteria

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Solatikia, Farnaz, Erdem Kiliç, and Gerhard Wilhelm Weber. "Fuzzy optimization for portfolio selection based on Embedding Theorem in Fuzzy Normed Linear Spaces." Organizacija 47, no. 2 (2014): 90–97. http://dx.doi.org/10.2478/orga-2014-0010.

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Abstract Background: This paper generalizes the results of Embedding problem of Fuzzy Number Space and its extension into a Fuzzy Banach Space C(Ω) × C(Ω), where C(Ω) is the set of all real-valued continuous functions on an open set Ω. Objectives: The main idea behind our approach consists of taking advantage of interplays between fuzzy normed spaces and normed spaces in a way to get an equivalent stochastic program. This helps avoiding pitfalls due to severe oversimplification of the reality. Method: The embedding theorem shows that the set of all fuzzy numbers can be embedded into a Fuzzy Ba

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Hashim, Hazwani, Noor Azzah Awang, and Lazim Abdullah. "A Normalized Weighted Bonferroni Mean Aggregation Operator in Neutrosophic Vague Multi-Criteria Decision- Making." International Journal of Neutrosophic Science 22, no. 1 (2023): 86–103. http://dx.doi.org/10.54216/ijns.220107.

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Decision-making problems involve uncertain and incomplete information, which can be well represented by the Neutrosophic set (NS). Various extensions of NS are available in the literature for solving such problems. However, the published extensions of NS have some restrictions such as single based membership degree. Neutrosophic vague set (NVS) is a newly developed theory to address the shortcomings of previous set theory. NVS is structured based on interval membership in the context of dependent membership functions. Beside uncertainty information, aggregation operators (AOs) are critical com

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Sunthrayuth, Pongsakorn, Fahd Jarad, Jihen Majdoubi, Rana Muhammad Zulqarnain, Aiyared Iampan, and Imran Siddique. "A Novel Multicriteria Decision-Making Approach for Einstein Weighted Average Operator under Pythagorean Fuzzy Hypersoft Environment." Journal of Mathematics 2022 (May 9, 2022): 1–24. http://dx.doi.org/10.1155/2022/1951389.

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The experts used the Pythagorean fuzzy hypersoft set (PFHSS) in their research to discourse ambiguous and vague information in decision-making processes. The aggregation operator (AO) plays a prominent part in the sensitivity of the two forefront loops and eliminates anxiety from that perception. The PFHSS is the most influential and operative extension of the Pythagorean fuzzy soft set (PFSS), which handles the subparameterized values of alternatives. It is also a generalized form of Intuitionistic fuzzy hypersoft set (IFHSS) that provides better and more accurate assessments in the decision-

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Zulqarnain, Rana Muhammad, Imran Siddique, Shahzad Ahmad, et al. "Pythagorean Fuzzy Soft Einstein Ordered Weighted Average Operator in Sustainable Supplier Selection Problem." Mathematical Problems in Engineering 2021 (November 30, 2021): 1–16. http://dx.doi.org/10.1155/2021/2559979.

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Pythagorean fuzzy soft set (PFSS) is the most influential and operative extension of the Pythagorean fuzzy set (PFS), which contracts with the parametrized standards of the substitutes. It is also a generalized form of the intuitionistic fuzzy soft set (IFSS) and delivers a well and accurate estimation in the decision-making (DM) procedure. The primary purpose is to prolong and propose ideas related to Einstein’s ordered weighted aggregation operator from fuzzy to PFSS, comforting the condition that the sum of the degrees of membership function and nonmembership function is less than one and t

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Akram, Muhammad, and Shumaiza. "Multi-criteria decision-making methods based on q-rung picture fuzzy information." Journal of Intelligent & Fuzzy Systems 40, no. 5 (2021): 10017–42. http://dx.doi.org/10.3233/jifs-202646.

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The q-rung picture fuzzy sets serve the fuzzy set theory as a competent, broader and accomplished extension of q-rung orthopair fuzzy sets and picture fuzzy sets which exhibit excellent performance in modeling the obscure data beyond the limits of existing approaches owing to the parameter q and three real valued membership functions. The accomplished strategy of VIKOR method is established on the major concepts of regret measure and group utility measure to specify the compromise solution. Further, TOPSIS method is another well established multi-criteria decision-making strategy that finds ou

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Banai, Miklos. "Computing in terms of Takeuti's Quantum Set Theory." Journal of Artificial Intelligence & Cloud Computing 3, no. 5 (2024): 1–2. http://dx.doi.org/10.47363/jaicc/2024(3)386.

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In this letter we discuss computing in terms of Gaisi Takeuti's quantum set theory using an explicit example of a rigid body of cuboid form. The universe V(L) of Takeuti will be determined. A set of real numbers in this Universe are also explicitly described including a set of binary numbers. Thus we arrive at the foundations of von Neumann' theory of computing in terms of ordinary binary numbers. Then we see that this extension of computing to the Universe V(L) provides a sound, mathematically well defined theory of quantum computing.

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Carrillo Flores, Wilber, and Alberto Mariano Rivero Zapata. "Classification of Real Division Algebras." Pesquimat 26, no. 2 (2023): 1–10. http://dx.doi.org/10.15381/pesquimat.v26i2.25686.

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This article aims to oﬀer a unifying approach to the basic theory of division algebras by presenting the research of the German-American mathematician Max August Zorn, who classiﬁed alternative division algebras. In section 1 the basic theory of real division algebras is developed. Section 2 presents the Cayley-Dickson Process, which consists of constructing an extension algebra from an algebra provided with a conjugation, similar to the construction of complex numbers from real numbers. In Section 3 presents the classical division algebras R (real), C (complex), H (quaternions) and O (octonio

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K., Kalaivani, and Palanivel Kaliyaperumal. "A neutrosophic approach to the transportation problem using single-valued trapezoidal neutrosophic numbers." Proyecciones (Antofagasta) 42, no. 2 (2023): 533–47. http://dx.doi.org/10.22199/issn.0717-6279-5374.

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In general, a fuzzy set can’t handle situations of inconsistencies and inexact data, however, the Neutrosophic Set (NS) has been used to address such types of issues in all real world problems. The neutrosophic set is an extension of the fuzzy set and the intuitionistic fuzzy set, that can deal with imperfect, inconsistent, and indeterminate data in all the related problems. This article proposed a conventional neutrosophic approach using a ranking function for the transportation problem. This approach has considered a single-valued neutrosophic set for the entire transportation problem with t

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Shakhatreh, M. A., and A. M. Al-Shorman. "Some properties of integration of real-valued function over a fuzzy interval." International Journal of ADVANCED AND APPLIED SCIENCES 8, no. 12 (2021): 9–13. http://dx.doi.org/10.21833/ijaas.2021.12.002.

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One of the most fundamental concepts in fuzzy set theory is the extension principle. It gives a generic way of dealing with fuzzy quantities by extending non-fuzzy mathematical concepts. There are a few examples, including the concept of fuzzy distance between fuzzy sets. The extension approach is then methodically applied to real algebra, with considerable development of fuzzy number operations. These operations are computationally appealing and generalized interval analysis. Although the set of real fuzzy numbers with extended addition or multiplication is no longer a group, it retains many

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HERENCIA, JOSÉ A., and M. TERESA LAMATA. "A TOTAL ORDER FOR THE GRADED NUMBERS USED IN DECISION PROBLEMS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 07, no. 03 (1999): 267–76. http://dx.doi.org/10.1142/s0218488599000209.

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Let us consider a classical problem of decision-making with fuzzy rewards. In order to choose the alternative which corresponds to the best fuzzy reward, we need to use a total order for the fuzzy numbers involved in the problem. In this paper, we consider a definition of such a total order, which is based upon two subjective aspects: the degree of optimism/pessimism and the liking for risk/safety. We also use the graded numbers which are analogous to the fuzzy numbers. However, the operations with graded numbers are carried out as a simple extension of operations with real intervals.

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ZHU, YUANGUO. "ON PARA-NORMED SPACE WITH FUZZY VARIABLES BASED ON EXPECTED VALUED OPERATOR." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 16, no. 01 (2008): 95–106. http://dx.doi.org/10.1142/s0218488508005066.

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Fuzzy variables are functions from credibility spaces to the set of real numbers. The set of fuzzy variables is a linear space with the classic operations of addition and multiplication by numbers. Its subspace formed by fuzzy variables with finite pth absolute moments is showed to be a complete para-normed space. The concept of para-normed space is novel, and is an extension of normed space. It is seen that most properties of normed spaces hold in para-normed spaces. Also some useful inequalities in para-normed spaces are obtained.

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Baldwin, Stewart. "An extension of Šarkovskiư's Theorem to the n-od." Ergodic Theory and Dynamical Systems 11, no. 2 (1991): 249–71. http://dx.doi.org/10.1017/s0143385700006131.

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AbstractThe n-od is defined to be the set of all complex numbers z such that zn is a real number in the interval [0,1], i.e., a central point with n copies of the unit interval attached at their endpoints. Given a space X and a function f:X → X, Per (f) is defined to be the set {k: f has for a point of (least) period k, k a positive integer}. The main result of this paper is to give, for each n, a complete characterization of all possible sets Per (f), where f ranges over all continuous functions on the n-od.

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EXNER, GEORGE R., IL BONG JUNG, MI RYEONG LEE, and SUN HYUN PARK. "BACKWARD 3-STEP EXTENSIONS OF RECURSIVELY GENERATED WEIGHTED SHIFTS: A RANGE OF QUADRATIC HYPONORMALITY." Bulletin of the Australian Mathematical Society 89, no. 3 (2013): 488–93. http://dx.doi.org/10.1017/s0004972713000920.

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AbstractLet $\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with $1\leq x\leq u\leq v\leq w$ and let ${W}_{\alpha } $ be the associated weighted shift with weight sequence $\alpha $. The set of positive real numbers $x$ such that ${W}_{\alpha } $ is quadratically hyponormal for some $u, v$ and $w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 3

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Vladareanu, Victor, Florentin Smarandache, and Luige Vladareanu. "Extension Hybrid Force-Position Robot Control in Higher Dimensions." Applied Mechanics and Materials 332 (July 2013): 260–69. http://dx.doi.org/10.4028/www.scientific.net/amm.332.260.

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The paper presents an advanced method for solving contradictory problems of hybrid position-force control of the movement of walking robots by applying a 2D Extension Set. Using the linear and non-linear attraction point principle and the network of attraction curves, there is determined the 2D space Dependent Function generated by position and force in order to solve the robot real time control. The generalization of the extension distance and dependent function uses Extenics in Higher Dimensions theory eliminates the crisp logic matrix of Cantor logic which describes the position-force seque

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Mehta, G. "On a theorem of Fleischer." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40, no. 2 (1986): 261–66. http://dx.doi.org/10.1017/s1446788700027233.

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AbstractFleischer proved that a linearly ordered set that is separable in its order topology and has countably many jumps is order-isomorphic to a subset of the real numbers. The object of this paper is to extend Fleischer's result and to prove it in a different way. The proof of the theorem is based on Nachbin's extension to ordered topological spaces of Urysohn's separation theorem in normal topological spaces.

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Qi, Jiangang, and Shaozhu Chen. "On an open problem of Weidmann: essential spectra and square-integrable solutions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 2 (2011): 417–30. http://dx.doi.org/10.1017/s0308210509001681.

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In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the

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Naeem, Muhammad, Muhammad Qiyas, Thongchi Botmart, Saleem Abdullah, and Neelam Khan. "Complex Spherical Fuzzy Decision Support System Based on Entropy Measure and Power Operator." Journal of Function Spaces 2022 (March 8, 2022): 1–25. http://dx.doi.org/10.1155/2022/8315733.

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The objectives of this paper are to define novel aggregation operators (AOs) for aggregating different complex spherical fuzzy numbers (CSFNs) under the influence of their membership grades. The uncertainties included in the information are dealt with in contemporary studies of the fuzzy set and its extensions by membership grades, which are a subset of real numbers that lose some relevant information and hence alter the decision results. The conversion to these complex spherical fuzzy sets addresses the classes’ uncertainty, whose ranges differ from the specific subset of the complex subset o

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Gaber, Mohamed, Majed G. Alharbi, Abd Alwahed Dagestani, and El-Saeed Ammar. "Optimal Solutions for Constrained Bimatrix Games with Payoffs Represented by Single-Valued Trapezoidal Neutrosophic Numbers." Journal of Mathematics 2021 (June 18, 2021): 1–13. http://dx.doi.org/10.1155/2021/5594623.

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Single-valued neutrosophic set (SVNS) is considered as generalization and extension of fuzzy set, intuitionistic fuzzy set (IFS), and crisp set for expressing the imprecise, incomplete, and indeterminate information about real-life decision-oriented models. The theme of this research is to develop a solution approach to solve constrained bimatrix games with payoffs of single-valued trapezoidal neutrosophic numbers (SVTNNs). In this approach, the concepts and suitable ranking function of SVTNNs are defined. Hereby, the equilibrium optimal strategies and equilibrium values for both players can b

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Zhu, Huafeng, Xingfa Zhang, Xin Liang, and Yuan Li. "Moving Average Model with an Alternative GARCH-Type Error." Journal of Systems Science and Information 6, no. 2 (2018): 165–77. http://dx.doi.org/10.21078/jssi-2018-165-13.

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Abstract Motivated by the double autoregressive model with order p (DAR(p) model), in this paper, we study the moving average model with an alternative GARCH error. The model is an extension from DAR(p) model by letting the order p goes to infinity. The quasi maximum likelihood estimator of the parameters in the model is shown to be asymptotically normal, without any strong moment conditions. Simulation results confirm that our estimators perform well. We also apply our model to study a real data set and it has better fitting performance compared to DAR model for the considered data.

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Hend, Dawood, and Megahed Nefertiti. "A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives." Punjab University Journal of Mathematics 51, no. 11 (2019): 77–100. https://doi.org/10.5281/zenodo.3479546.

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Differentiation arithmetic is a principal and accurate technique for the computational evaluation of derivatives of first and higher order. This article aims at recasting real differentiation arithmetic in a formalized theory of dyadic real differentiation numbers that provides a foundation for first and higher order automatic derivatives. After we set the stage by putting on a systematic basis certain fundamental notions of the algebra of differentiation numbers, we begin by setting up an axiomatic theory of real differentiation arithmetic, as a many-sorted extension of the theory of a contin

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Kayid, Mohamed, Abdulrahman Abouammoh, and Ghadah Alomani. "Scaled-Invariant Extended Quasi-Lindley Model: Properties, Estimation, and Application." Symmetry 15, no. 9 (2023): 1780. http://dx.doi.org/10.3390/sym15091780.

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In many research fields, statistical probability models are often used to analyze real-world data. However, data from many fields, such as the environment, economics, and health care, do not necessarily fit traditional models. New empirical models need to be developed to improve the fit. In this study, we investigated a further extension of the quasi-Lindley model. This extension was asymmetrically distributed on the positive real number line. Maximum likelihood, least square error, Anderson–Darling, and expectation maximization algorithms were used to estimate the parameters studied. All tech

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Nowak, Georg. "On the distribution of M-tuples of B-numbers." Publications de l'Institut Math?matique (Belgrade) 77, no. 91 (2005): 71–78. http://dx.doi.org/10.2298/pim0591071n.

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In the classical sense, the set B consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field Q(i). G. J. Rieger (1965) and T. Cochrane, R. E. Dressler (1987) established bounds for the number of pairs (n; n + h), resp., triples (n; n + 1; n + 2) of B-numbers up to a large real parameter x. The present article generalizes these investigations into two directions: The result obtained deals with arbitrary M-tuples of arithmetic progressions of positive integers excluding the trivi

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Isakov, Alex, Rodion Latypov, Andrey Repin, Egor Postolit, Alexey Evseev, and Elena Sinelnikova-Muryleva. "Hard Numbers: Open Consumer Price Database." Russian Journal of Money and Finance 80, no. 1 (2021): 104–19. http://dx.doi.org/10.31477/rjmf.202101.104.

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We document a new source of consumer price microdata. The new database allows researchers studying consumer price behaviour to access current and granular raw statistical observations. The range of observed prices fully covers goods and services of the Rosstat’s CPI sample and extends beyond it. In this paper, we pursue two objectives. First, we describe the data collection mechanism, data structure, and their access protocols, as well provide four complete illustrations of their application using open API: i) training machine models of product classification based on text labels, ii) real-tim

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Yuan, Yuan, and Li Yang He. "Extension of TOPSIS for Multiple Attribute Decision Making Using Intuitionistic Fuzzy Sets." Advanced Materials Research 433-440 (January 2012): 4053–58. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.4053.

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This electronic document is a “live” template. The various components of your paper [title, text, heads, etc.] are already defined on the style sheet, as illustrated by the portions given in this document. Due to the nature of vagueness inherent to real-life situations, some fuzzy data are deemed to suitable enough to describe the qualitative and/or quantitative estimation for decision making problems. Therefore, a new method for multiple attribute decision making under fuzzy environment is discussed, in which the attribute values take the form of intuitionistic fuzzy numbers. To overcome some

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Siswanto, Siswanto, and Ade Safira Septiany. "CRAMER’S RULE IN INTERVAL MIN-PLUS ALGEBRA." BAREKENG: Jurnal Ilmu Matematika dan Terapan 19, no. 1 (2025): 571–80. https://doi.org/10.30598/barekengvol19iss1pp571-580.

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A min-plus algebra is a set , where is the set of all real numbers, equipped with the minimum and addition operations. The system of linear equations in min-plus algebra can be solved using Cramer's rule. Interval min-plus algebra is an extension of min-plus algebra, with the elements in it being closed intervals. The set is denoted by equipped with two binary operations, namely minimum and addition . The matrix with notation is a matrix over interval min-plus algebra with size . Since the structure of min-plus algebra and interval min-plus algebra are analogous, the system of linear equations

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Fivel, Oren, Moshe Klein, and Oded Maimon. "Decision Trees with Soft Numbers." International Journal of Circuits, Systems and Signal Processing 15 (January 6, 2022): 1803–16. http://dx.doi.org/10.46300/9106.2021.15.194.

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In this paper we develop the foundation of a new theory for decision trees based on new modeling of phenomena with soft numbers. Soft numbers represent the theory of soft logic that addresses the need to combine real processes and cognitive ones in the same framework. At the same time soft logic develops a new concept of modeling and dealing with uncertainty: the uncertainty of time and space. It is a language that can talk in two reference frames, and also suggest a way to combine them. In the classical probability, in continuous random variables there is no distinguishing between the probabi

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Xu, Jixiang, Yanhua Tan, Jinggui Gao, and Enmin Feng. "Pricing Currency Option Based on the Extension Principle and Defuzzification via Weighting Parameter Identification." Journal of Applied Mathematics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/623945.

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We present a fuzzy version of the Garman-Kohlhagen (FG-K) formula for pricing European currency option based on the extension principle. In order to keep consistent with the real market, we assume that the interest rate, the spot exchange rate, and the volatility are fuzzy numbers in the FG-K formula. The conditions of a basic proposition about the fuzzy-valued functions of fuzzy subsets are modified. Based on the modified conditions and the extension principle, we prove that the fuzzy price obtained from the FG-K formula for European currency option is a fuzzy number. To simplify the trade, t

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Kadyrzhan, Aruzhan, Akhat Bakirov, Dina Shaltykova, and Ibragim Suleimenov. "Application of the Algebraic Extension Method to the Construction of Orthogonal Bases for Partial Digital Convolutions." Algorithms 17, no. 11 (2024): 496. http://dx.doi.org/10.3390/a17110496.

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Mathematical tools have been developed that are analogous to the tool that allows one to reduce the description of linear systems in terms of convolution operations to a description in terms of amplitude-frequency characteristics. These tools are intended for use in cases where the system under consideration is described by partial digital convolutions. The basis of the proposed approach is the Fourier–Galois transform using orthogonal bases in corresponding fields. As applied to partial convolutions, the Fourier–Galois transform is decomposed into a set of such transforms, each of which corre

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Landver, Avner. "Baire numbers, uncountable Cohen sets and perfect-set forcing." Journal of Symbolic Logic 57, no. 3 (1992): 1086–107. http://dx.doi.org/10.2307/2275450.

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The Baire number of the real line (see 1.1(c)) has uncountable cofinality [Mi]. This number is equal to the Baire number of the space 2ω and also to the Baire number (1.1(d)) of every countable partial order. Our research is motivated by the following open question (1.13) (see also [BPS] and [Mi]): can the cofinality of nκ, the Baire number of the space (2κ)κ (1.1(b)), be less than or equal to κ? This question is nontrivial when κ is regular and 2κ = κ (1.3). A. Miller proved that the answer is “no” if κ is strongly inaccessible (see [Mi] and 1.11). Assuming CH, is equal to the Baire number of

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Burgin, M. S. "Topology in nonlinear extensions of hypernumbers." Discrete Dynamics in Nature and Society 2005, no. 2 (2005): 145–70. http://dx.doi.org/10.1155/ddns.2005.145.

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Modern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in spaces of real and complex hypernumbers by extending the concept of a hypernumber. In such a way, linear algebras of extended hypernumbers are built. A special topology of conical neighborhoods in these algebras is introduced and studied. It is proved that the space of all extended real hypernumbers

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Yan, Guiping, Richard W. Smiley, and Patricia A. Okubara. "Detection and Quantification of Pratylenchus thornei in DNA Extracted from Soil Using Real-Time PCR." Phytopathology® 102, no. 1 (2012): 14–22. http://dx.doi.org/10.1094/phyto-03-11-0093.

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The root-lesion nematode Pratylenchus thornei is one of the most important pests restricting productivity of wheat in the Pacific Northwest (PNW). It is laborious and difficult to use microscopy to count and identify the nematodes in soils. A SYBR Green I-based real-time polymerase chain reaction (PCR) assay was developed to detect and quantify this species from DNA extracts of soil. A primer set, designed from the internal transcribed spacer region (ITS1) of rDNA, was highly specific to P. thornei and did not amplify DNA from 27 isolates of other Pratylenchus spp., other nematodes, and six fu

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Ma, Jinshan, and Changsheng Ji. "Generalized Grey Target Decision Method for Mixed Attributes Based on Connection Number." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/763543.

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Grey target decision model for mixed attributes including real numbers, interval numbers, triangular fuzzy numbers, and trapezoidal fuzzy numbers is complex for its data processing in different ways and information distortion in handling fuzzy numbers. To solve these problems, the binary connection number proposed in set pair analysis is applied to unify different types of index values with their parameters’ average values and standard deviations as determinacy-uncertainty vectors. Then the target center index vectors are determined by the modules of index vectors of all alternatives under dif

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Małolepsza, Olga, Dariusz Mikołajewski, and Piotr Prokopowicz. "Adaptation of Fuzzy Systems Based on Ordered Fuzzy Numbers: A Review of Applications and Development Prospects." Electronics 14, no. 12 (2025): 2341. https://doi.org/10.3390/electronics14122341.

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This paper presents a comprehensive overview of the adaptation of fuzzy systems based on Ordered Fuzzy Numbers (OFNs), an extension of classical fuzzy set theory that allows for more accurate modeling of uncertainty and variability across diverse domains. Key adaptation techniques—including genetic algorithms, evolutionary programming, learning algorithms, reinforcement learning, and online adaptation—are systematically analyzed and compared in terms of their strengths, limitations, and application areas. The analysis reveals that, despite the considerable potential of OFN-based systems in fie

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WEI, Guiwu. "PICTURE FUZZY CROSS-ENTROPY FOR MULTIPLE ATTRIBUTE DECISION MAKING PROBLEMS." Journal of Business Economics and Management 17, no. 4 (2016): 491–502. http://dx.doi.org/10.3846/16111699.2016.1197147.

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In this paper, we investigate the multiple attribute decision making problems with picture fuzzy information. The advantage of picture fuzzy set is easily reflecting the ambiguous nature of subjective judgments because the picture fuzzy sets are suitable for capturing imprecise, uncertain, and inconsistent information in the multiple attribute decision making analysis. Thus, the cross entropy of picture fuzzy sets, called picture fuzzy cross entropy, is proposed as an extension of the cross entropy of fuzzy sets. Then, a multiple attribute decision making method based on the proposed picture f

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Jiang, Wenqi. "Limited public resources allocation model based on social fairness using an extended VIKOR method." Kybernetes 45, no. 7 (2016): 998–1012. http://dx.doi.org/10.1108/k-05-2014-0108.

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Purpose Different from manufacturing resources allocation problems, the prices and amounts of limited public service resources could not be changed with the consumers’ requirements and social fairness is the most important objective for improving allocation efficiency. To measure social fairness reasonably, the purpose of this paper is fourfold: first, divide social fairness into longitudinal comparative fairness and crosswise comparative fairness, therefore providing their calculation formula and describing the comprehensive fair degree by using the interval numbers. Second, the comparison re

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Lovyagin, Yuri N., and Nikita Yu Lovyagin. "Finite Arithmetic Axiomatization for the Basis of Hyperrational Non-Standard Analysis." Axioms 10, no. 4 (2021): 263. http://dx.doi.org/10.3390/axioms10040263.

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The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrationa

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Malafosse, Bruno de. "Extension of Some Results on the (SSIE) and the (SSE) of the Form F ⊂ ℇ + F′ x and ℇ + Fx = F". Fasciculi Mathematici 59, № 1 (2017): 107–23. http://dx.doi.org/10.1515/fascmath-2017-0020.

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Abstract Given any sequence a = (an)n≥1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n≥1 such that y/a = (yn/an)n≥1 ∈ E. In this paper we deal with the solvability of the (SSIE) of the form ℓ∞ ⊂ ℇ+F′x where ℇ is a linear space of sequences and F′ is either c0, or ℓ∞ and we solve the (SSIE) c0 ⊂ ℇ + sx for ℇ ⊂ (sα)∆ and α ∈ c0. Then we study the (SSIE) c ⊂ ℇ + s(c)x and the (SSE) ℇ + s(c)x = c. Then we apply the previous results to the solvability of the (SSE) of the form (ℓrp)∆ + Fx = F for p ≥ 1 and F is any of the sets c0, c, o

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Wang, Lin, David A. Ratkowsky, Johan Gielis, Paolo Emilio Ricci, and Peijian Shi. "Effects of the Numerical Values of the Parameters in the Gielis Equation on Its Geometries." Symmetry 14, no. 12 (2022): 2475. http://dx.doi.org/10.3390/sym14122475.

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The Lamé curve is an extension of an ellipse, the latter being a special case. Dr. Johan Gielis further extended the Lamé curve in the polar coordinate system by introducing additional parameters (n1, n2, n3; m): rφ=1Acosm4φn2+1Bsinm4φn3−1/n1, which can be applied to model natural geometries. Here, r is the polar radius corresponding to the polar angle φ; A, B, n1, n2 and n3 are parameters to be estimated; m is the positive real number that determines the number of angles of the Gielis curve. Most prior studies on the Gielis equation focused mainly on its applications. However, the Gielis equa

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Ali, Arshad, Muhammad Haris Mateen, Qin Xin, Turki Alsuraiheed, and Ghaliah Alhamzi. "$ (\epsilon, \delta) $-complex anti fuzzy subgroups and their applications." AIMS Mathematics 9, no. 5 (2024): 11580–95. http://dx.doi.org/10.3934/math.2024568.

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<abstract><p>The complex anti-fuzzy set (CAFS) is an extension of the traditional anti-fuzzy set with a wider range for membership function beyond real numbers to complex numbers with unit disc aims to address the uncertainty of data. The complex anti-fuzzy set is more significant because it provides two dimensional information and versatile representation of vagueness and ambiguity of data. In terms of the characteristics of complex anti-fuzzy sets, we proposed the concept of $ (\epsilon, \delta) $-CAFSs that offer a more comprehensive representation of the uncertainty of data tha

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Pradhan, Avik, and Biswal Prasad. "Linear programming problems with some multi-choice fuzzy parameters." Yugoslav Journal of Operations Research 28, no. 2 (2018): 249–64. http://dx.doi.org/10.2298/yjor160520004p.

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In this paper, we consider some Multi-choice linear programming (MCLP) problems where the alternative values of the multi-choice parameters are fuzzy numbers. There are some real-life situations where we need to choose a value for a parameter from a set of different choices to optimize our objective, and those values of the parameters can be imprecise or fuzzy. We formulate these situations as a mathematical model by using some fuzzy numbers for the alternatives. A defuzzification method based on incentre point of a triangle has been used to find the defuzzified values of the fuzzy numbers. We

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Nataraj, P. S. V., and Sachin Tharewal. "An Interval Analysis Algorithm for Automated Controller Synthesis in QFT Designs." Journal of Dynamic Systems, Measurement, and Control 129, no. 3 (2006): 311–21. http://dx.doi.org/10.1115/1.2397147.

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In this paper, an interval analysis algorithm is proposed for the automatic synthesis of fixed structure controllers in quantitative feedback theory (QFT). The proposed algorithm is tested on several examples and compared with the controller designs given in the QFT literature. Compared to the existing methods for QFT controller synthesis, the proposed algorithm yields considerable improvement in the high frequency gain of the controller in all examples, and improvements in the cutoff frequency of the controller in all but one examples. Notation: R denotes the field of real numbers, Rn is the

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Edeme, Chukwuma Percy, and F. Z. Okwonu. "THE ODD LOMAX TOPP LEONE DISTRIBUTION: PROPERTIES AND APPLICATION." FUDMA JOURNAL OF SCIENCES 8, no. 5 (2024): 286–94. http://dx.doi.org/10.33003/fjs-2024-0805-2849.

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Lifetime distributions are parametric models used in statistical analyses of time-to-event data. Several probability distributions have been very used in literature to model lifetime data sets which have been useful in the analysis of lifetime data, but in most cases, they are not flexible enough to analyze some complex lifetime data in practice. Due to the importance of these lifetime distributions in modeling real lifetime data, there has several modifications and generalization of lifetime distributions, particularly the Lomax distribution to develop more flexible distributions to address t

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