Rational Numbers Research Articles

Rational number conceptualization predicts mathematics achievement: Quantity matters

Knowledge integration is a crucial part of learning as concepts are built over time and modality. In mathematics, rational numbers are a particularly difficult concept that requires integration across notations to understand the quantity represented by fractions and decimals (e.g., ½ and 0.5). We investigated how 5- to 14-year-old children conceptualize rational numbers, whether it differs across development (Study 1), and whether the conceptualization is related to math achievement (Study 2). A novel Numerical SpAM task was created to document children’s unprompted rational number conceptualization. We found that most children organized rational numbers either by notation or quantity, with quantity becoming more common after age 10. Quantitative conceptualization predicted higher math achievement compared to notation, regardless of age. Moreover, conceptualizing rational numbers by quantity across distinct notations (e.g., 1/2 and 0.5) was related to children’s math achievement. Implications for numerical understanding as well as general cognition are discussed.

On Finite Subgroups in the General Linear Groups over an Algebraic Number Field

As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

The model method: Crown jewel in Singapore mathematics

The model method is synonymous with Singapore Mathematics. The spiral structure of the mathematics curriculum, with its focus on problem solving, and the concrete-pictorial-abstract approach to teaching, supports the use of the model method to solve arithmetic problems and enables the development of letter-symbolic algebra. The schematic representation of the model drawing encourages the decoding of information, and strengthens the development of the part-part-whole nature of numbers, beginning with whole numbers, followed by rational numbers, and finally the notion of the letter-symbolic algebra. First, the schematic representation is used to solve arithmetic-type word problems in the early years, then algebraic-type word problems. For some, the model method provides an avenue to track the ontogenetic development of algebra, from rhetorical algebra to, syncopated-Diophantine and finally letter-symbolic Vietan algebra, which provides a general solution to a set of problems, the crown of algebra.

Negligible Obstructions and Turán Exponents

We show that for every rational number $r \in (1,2)$ of the form $2 - a/b$, where $a, b \in \mathbb{N}^+$ satisfy $\lfloor b/a \rfloor^3 \le a \le b / (\lfloor b/a \rfloor +1) + 1$, there exists a graph $F_r$ such that the Tur\'an number $\operatorname{ex}(n, F_r) = \Theta(n^r)$. Our result in particular generates infinitely many new Tur\'an exponents. As a byproduct, we formulate a framework that is taking shape in recent work on the Bukh--Conlon conjecture.

Hodge theory of Kloosterman connections

We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be “classical,” we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.

NP-hardness of shortest path problems in networks with non-FIFO time-dependent travel times

We study the complexity of earliest arrival problems on time-dependent networks with non-FIFO travel time functions, i.e. when departing later might lead to an earlier arrival. In this paper, we present a simple proof of the weak NP-hardness of the problem for travel time functions defined on integers. This simplifies and reproduces an earlier result from Orda and Rom. Our proof generalizes to travel time functions defined on rational numbers and also implies that, in this case, the problem becomes harder, i.e. is strongly NP-hard. As arbitrary functions are impractical for applications, we also study a more realistic problem model where travel time functions are piecewise linear and represented by a sequence of breakpoints with integer coordinates. We show that this problem formulation is strongly NP-hard, too. As an intermediate step for this proof, we also show the strong NP-completeness of SubsetProduct on rational numbers.

Let’s Get Rolling! Exact Optimal Solitaire Yahtzee

Summary Yahtzee is a popular dice game that involves a combination of chance and decision making. We implemented a brute-force algorithm to understand optimal play for a solitaire version of Yahtzee using Mathematica. The novelty in our approach is that we were able to compute probabilities and expectations of the optimal player’s score exactly as rational numbers. Previous work involved only numerical approximations. We give an overview of our approach and analyze some of the data generated by our implementation. The figures provided in the article should give an insight as to how a game under optimal play would typically unfold.

The finitely generated Hausdorff spectra of a family of pro-p groups

Recently the first example of a family of pro-p groups, for p a prime, with full normal Hausdorff spectrum was constructed. In this paper we further investigate this family by computing their finitely generated Hausdorff spectrum with respect to each of the five standard filtration series: the p-power series, the iterated p-power series, the lower p-series, the Frattini series and the dimension subgroup series. Here the finitely generated Hausdorff spectra of these groups consist of infinitely many p-adic rational numbers, and their computation requires a rather technical approach. This result also gives further evidence to the non-existence of a finitely generated pro-p group with uncountable finitely generated Hausdorff spectrum.

The cover time of a (multiple) Markov chain with rational transition probabilities is rational

The cover time of a Markov chain on a finite state space is the expected time until all states are visited. We show that if the cover time of a discrete-time Markov chain with rational transitions probabilities is bounded, then it is a rational number. The result is proved by relating the cover time of the original chain to the hitting time of a set in another higher dimensional chain. We prove this result in a more general setting where k≥1 independent copies of a Markov chain are run simultaneously on the same state space.

Misconception on addition and subtraction of fractions in seventh-grade middle school students

As national and international assessments continue to show students struggle with rational concepts, which are seen as a major stumbling block for future career in STEM fields, we used the certainty of response index (CRI) as a method of analysis to investigate a group of 40 United Arab Emirates (UAE) seventh grade students’ understanding of basic fraction concepts. The types of errors and misconceptions displayed by students show that over 60% of all participants assumed that adding fractions procedurally is the same as adding whole numbers with many not being able to distinguish a numerator from a denominator. The results lead us to believe that the manner in which students were taught did not ensure the necessary transitional shift from learning whole numbers to learning rational numbers. Applying the theories of Dienes (1960) and Bruner (1966), we propose a method of teaching addition and subtraction of fractions that will allow students to build their own understanding of the rules using the graphing calculator as the medium for concept formation.

Exact solution for commensurate and incommensurate linear systems of fractional differential equations

In this paper, we introduce exact solutions for the initial value problems of two classes of a linear system of fractional ordinary differential equations with constant coefficients. This article concerns a linear system of fractional order, where the orders are equal or different rational numbers between zero and one. The conformable fractional derivative presented by [R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl. Math., \(\textbf{264}\) (2014), 65--70] is considered. Two different approaches are adopted to give analytic solutions for fractional order systems. The presented methods are illustrated by analyzing some numerical examples that show the effectiveness of the implemented methods.

Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains

Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL mathcal {ALC} in the presence of GCIs, quite strong restrictions, in sum called omega -admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of omega -admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate omega -admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield omega -admissible concrete domains. This allows us to show omega -admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the mathcal {EL} family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although omega -admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into mathcal {ALC} yields decidable DLs.

Algebraic Zero Error Training Method for Neural Networks Achieving Least Upper Bounds on Neurons and Layers

In the domain of artificial neural networks, it is important to know what their representation, classification and generalization capabilities are. There is also a need for time and resource-efficient training algorithms. Here, a new zero-error training method is derived for digital computers and single hidden layer networks. This method is the least upper bound on the number of hidden neurons as well. The bound states that if there are N input vectors expressed as rational numbers, a network having N − 1 neurons in the hidden layer and M neurons at the output represents a bounded function F: RD→RM for all input vectors. Such a network has massively shared weights calculated by 1 + M regular systems of linear equations. Compared to similar approaches, this new method achieves a theoretical least upper bound, is fast, robust, adapted to floating-point data, and uses few free parameters. This is documented by theoretical analyses and comparative tests. In theory, this method provides a new constructional proof of the least upper bound on the number of hidden neurons, extends the classes of supported activation functions, and relaxes conditions for mapping functions. Practically, it is a non-iterative zero-error training algorithm providing a minimum number of neurons and layers.

$SU(2)$-cyclic surgeries and the pillowcase

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, which is both a rational number and a boundary slope for $K$. We also show that such knots are prime and have infinitely many instanton L‑space surgeries. Our methods include the application of holonomy perturbation techniques to instanton knot homology, using a strengthening of recent work by the second author.

F-thresholds c(m) for projective curves

We show that if R is a two dimensional standard graded domain (with the graded maximal ideal m ) of characteristic p > 0 and I ⊂ R is a graded ideal with ℓ ( R / I ) < ∞ , then the F -threshold c I ( m ) can be expressed in terms of a strong HN (Harder-Narasimhan) slope of the canonical syzygy bundle on Proj R . Thus c I ( m ) is a rational number. This gives us a well defined notion, of the F -threshold c I ( m ) in characteristic 0, in terms of a HN slope of the syzygy bundle on Proj R . This generalizes our earlier result (in [13] ) where we have shown that if I has homogeneous generators of the same degree, then the F -threshold c I ( m ) is expressed in terms of the minimal strong HN slope (in char p ) and in terms of the minimal HN slope (in char 0), respectively, of the canonical syzygy bundle on Proj R . In the present more general setting, the relevant slope may not be the minimal one. Here we also prove that, for a given pair ( R , I ) over a field of characteristic 0, if ( m p , I p ) is a reduction mod p of ( m , I ) then c I p ( m p ) ≠ c ∞ I ( m ) implies c I p ( m p ) has p in the denominator, for almost all p .

On Some Properties of Fabulous Fraction Tree

Among several properties that real numbers possess, this paper deals with the exciting formation of positive rational numbers constructed in the form of a Tree, in which every number has two branches to the left and right from the root number. This tree possesses all positive rational numbers. Hence it consists of infinite numbers. We call this tree

Descent on elliptic surfaces and arithmetic bounds for the Mordell–Weil rank

We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.

High school student understanding of exponential and logarithmic functions

We use Action-Process-Object-Schema theory (APOS) to study high school student understanding of exponentiation and their construction of exponential and logarithmic functions. We extend didactic materials similar to those of Ferrari-Escolá et al. (2016) to include exponentials on the rational numbers and to help students construct logarithms as numbers. Qualitative data from the problem-solving activities of two groups of eight students each during a series of teaching episodes suggests that some students can use these materials successfully. The data analysis enabled us to give specific suggestions on how to help other students do some of the constructions needed to understand these functions. Research shows these constructions are difficult for students. The findings of our study led to contributing a new and detailed genetic decomposition that can be tested and improved in future research cycles.

On the Complexity of &lt;i&gt;p&lt;/i&gt;-Adic Continued Fractions of Rational Number

In this paper, we study the complexity of p-adic continued fractions of a rational number, which is the p-adic analogue of the Lame’s theorem. We calculate the length of Browkin expansion, and the length of Schneider expansion. Also, some numerical examples have been given.

Involutions of sl(2,k) and non-split, three-dimensional simple Lie algebras

We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of [Formula: see text], where [Formula: see text] is a field of characteristic not two. Up to equivalence, non-split three-dimensional simple Lie algebras obtained in this way are parametrized by a subgroup of the Brauer group of [Formula: see text] and are characterized by the fact that their Killing form represents [Formula: see text]. Over local and global fields we re-express this condition in terms of Hilbert and Legendre Symbols and give examples of three-dimensional simple Lie algebras which can and cannot be obtained by this construction over the field of rationals.