The aim of this article is to construct some new families of generating‐type functions interpolating a certain class of higher order Bernoulli‐type, Euler‐type, Apostol‐type numbers, and polynomials. ...Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol‐type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
Non-linear neutrosophic numbers (NLNNs) are different kinds of neutrosophic numbers with at least one non-linear membership function (either of truthiness, falsity or indeterminacy part) of the ...information. Furthermore, a linear programming problem with non-linear neutrosophic numbers as coefficients/parameters is a special type of programming problem known as a non-linear linear programming problem (NLN-LPP). This paper elaborates on the concepts of non-linear neutrosophic number (NLNN) sets, different forms of non-linear neutrosophic numbers (NLNNs), alpha, beta,gamma cuts on non-linear neutrosophic numbers (NLNNs), possibility mean, possibility standard deviation, and possibility variance of non-linear neutrosophic numbers (NLNNs). In this paper, we also propose the solution technique for non-linear neutrosophic linear programming problems (NLN-LPPs) in which all coefficients/parameters are non-linear neutrosophic numbers (NLNNs). In this continuation, we suggest a new modified possibility score function for non-linear NNs in terms of possibility means and possibility standard deviations of non-linear neutrosophic numbers (NLNNs) for better use of all parts of information. This modified score function is used to convert non-linear neutrosophic number (NLNN) coefficients/parameters of non-linear neutrosophic linear programming problem (NLN-LPP) into equivalent crisp values. Thereafter, the equivalent crisp problem is solved with the usual method to obtain the optimal solution of non-linear neutrosophic linear programming problem (NLN-LPP). The proposed solution algorithm is unique and new for solving non-linear neutrosophic linear programming problems. A numerical example is solved with the proposed algorithm to legitimate the research output. A case study is also discussed to show its applicability in solving real-life problems. Keywords: Linear programming problem; Non-linear neutrosophic numbers (NLNNs), Possibility score function of Non-linear neutrosophic numbers (NLNNs), Possibility mean of Non-linear neutrosophic numbers (NLNNs), Possibility standard deviation of Non-linear neutrosophic numbers (NLNNs).
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IZUM, KILJ, NUK, PILJ, PNG, SAZU, UL, UM, UPUK
The aim of this paper is to construct a new method related to a family of operators to define generating functions for special numbers and polynomials. With the help of this method, we investigate ...various properties of these special numbers and polynomials with their generating functions, functional equations, and differential equations. We also give some algorithms to calculate the values of these numbers and polynomials. Moreover, we introduce some fundamental properties of this operator. By applying integral method to this operator, we obtain integral formulas together with combinatorial sums. Moreover, by applying this operator, the Lagrange inversion formula and convolution formula to these generating functions, we derive some novel identities and relations including combinatorial sums and combinatorial numbers including these new numbers and polynomials, ie, the Apostol‐Bernoulli numbers, the Apostol‐Euler numbers, the Stirling numbers, the central factorial numbers, and the other special numbers and polynomials. In addition, we give some examples derived from relations between composita and this operator on the set of formal power series. Finally, we give complex form of the Fourier series for these generating functions. By using these Fourier series, we not only derive some series relations and trigonometric sums including trigonometric function and hyperbolic functions but also give an example for a differential equation.
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, ...Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek’s problem 15 “Aufgabe 1088. El. Math., 49 (1994), 126–127”. Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.
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BFBNIB, IZUM, KILJ, NMLJ, NUK, PILJ, PNG, SAZU, UL, UM, UPUK
The effectiveness of an experimental middle school fraction intervention was evaluated. The intervention was centered on the number line and incorporated key principles from the science of learning. ...Sixth graders (N = 51) who struggled with fraction concepts were randomly assigned at the student level to the experimental intervention (n = 28) or to a business-as-usual control who received their school's intervention (n = 23). The experimental intervention occurred over 6 weeks (27 lessons). Fraction number line estimation, magnitude comparisons, concepts, and arithmetic were assessed at pretest, posttest, and delayed posttest. The experimental group demonstrated significantly more learning than the control group from pretest to posttest, with meaningful effect sizes on measures of fraction concepts (g = 1.09), number line estimation as measured by percent absolute error (g = −.85), and magnitude comparisons (g = .82). These improvements held at delayed posttest 7 weeks later. Exploratory analyses showed a significant interaction between classroom attentive behavior and intervention group on fraction concepts at posttest, suggesting a buffering effect of the experimental intervention on the normally negative impact of low attentive behavior on learning. A number line-centered approach to teaching fractions that also incorporates research-based learning strategies helps struggling learners to make durable gains in their conceptual understanding of fractions.
Educational Impact and Implications Statement
A mathematics intervention that used a number line-centered approach and validated learning principles to teach fraction concepts helped struggling sixth graders improve their fraction understanding. After participating in the intervention, students performed better on assessments of fraction concepts, number line estimation, and magnitude comparisons than a group of students who received their school's regular intervention, and these improvements held seven weeks later. Findings suggest that students who are struggling with fractions, even after receiving several years of formal fraction instruction in school, can still make large gains in their understanding, preparing them for more advanced mathematics and for success in STEM related fields.
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CEKLJ, FFLJ, NUK, ODKLJ, PEFLJ, UPUK
This paper introduces two processes of ranking methods on Z‐numbers that are effectively able to deal with uncertain decision‐making data. Decision making is based on recommended Z‐ numbers. For this ...purpose first, the Z‐number is transformed to a fuzzy number and then the ranking method by using the sigmoid function and the sign method is used to mention fuzzy numbers. For the next step, the method is extended to related Z‐numbers. Finally, we use it to prioritize the items and solve some examples.
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
The study of midrash-the biblical exegesis, parables, and anecdotes of the Rabbis-has enjoyed a renaissance in recent years. Most recent scholarship, however, has focused on the aggadic or narrative ...midrash, while halakhic or legal midrash-the exegesis of biblical law-has received relatively little attention. InScripture as Logos, Azzan Yadin addresses this long-standing need, examining early, tannaitic (70-200 C.E.) legal midrash, focusing on the interpretive tradition associated with the figure of Rabbi Ishmael. This is a sophisticated study of midrashic hermeneutics, growing out of the observation that the Rabbi Ishmael midrashim contain a dual personification of Scripture, which is referred to as both "torah" and "ha-katuv." It is Yadin's significant contribution to note that the two terms are not in fact synonymous but rather serve as metonymies for Sinai on the one hand and, on the other, the rabbinic house of study, the bet midrash. Yadin develops this insight, ultimately presenting the complex but highly coherent interpretive ideology that underlies these rabbinic texts, an ideology that-contrary to the dominant view today-seeks to minimize the role of the rabbinic reader by presenting Scripture as actively self-interpretive. Moving beyond textual analysis, Yadin then locates the Rabbi Ishmael hermeneutic within the religious landscape of Second Temple and post-Temple literature. The result is a series of surprising connections between these rabbinic texts and Wisdom literature, the Dead Sea Scrolls, and the Church Fathers, all of which lead to a radical rethinking of the origins of rabbinic midrash and, indeed, of the Rabbis as a whole.
Hybrid numbers, whose components are defined as real numbers, are a mixture of complex numbers, dual numbers and hyberbolic numbers. These structures are frequently used both in pure mathematics and ...in many areas of physics. In this paper, by the help of the Fibonacci divisor numbers, we introduce the Fibonacci divisor hybrid numbers that generalize the Fibonacci hybrid numbers defined by Szynal-Liana and Wloch. We obtain miscellaneous algebraic properties of the Fibonacci divisor hybrid numbers. We also give an application related to the Fibonacci divisor hybrid numbers in matrices. Finally, using the character of the Fibonacci divisor hybrid numbers, we show that these numbers are spacelike.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The main purpose of this paper is to provide a novel approach to deriving formulas for the
p
-adic
q
-Volkenborn integral including the Volkenborn integral and
p
-adic fermionic integral. By applying ...integral equations and these integral formulas to the falling factorials, the rising factorials and binomial coefficients, we derive some various identities, formulas and relations related to several combinatorial sums, well-known special numbers such as the Bernoulli and Euler numbers, the harmonic numbers, the Stirling numbers, the Lah numbers, the Harmonic numbers, the Fubini numbers, the Daehee numbers and the Changhee numbers. Applying these identities and formulas, we give some new combinatorial sums. Finally, by using integral equations, we derive generating functions for new families of special numbers and polynomials. By using generating functions, we give relations between the Lah numbers, the Bernoulli numbers, the Euler numbers and the Laguerre polynomials. We also give further comments and remarks on these functions, numbers and integral formulas related to
q
-type operators potentially used to solve problems in the areas such as physics, quantum mechanics, quantum systems and the others. In addition, we provide some tables containing some of the
p
-adic integral formulas obtained in this paper.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
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