Rational Numbers Research Articles

English

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.

Minimality of topological matrix groups and Fermat primes

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group ST+(n,F) is minimal for every local field F of characteristic ≠2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group SL(n,F), where F is a subfield of a local field. This extends some known results of Remus–Stoyanov (1991) and Bader–Gelander (2017).One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime p the following conditions are equivalent:(1)p is a Fermat prime;(2)SL(p−1,Q) is minimal, where Q is the field of rationals equipped with the p-adic topology;(3)SL(p−1,Q(i)) is minimal, where Q(i)⊂C is the Gaussian rational field.

Counting rational points of a Grassmannian

We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical results of Schmidt ([12]) and Thunder ([20]) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.

Instantons and L-space surgeries

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the \operatorname{Spin}^c decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for r a positive rational number and K a nontrivial knot in the 3 -sphere, there exists an irreducible homomorphism \pi_1(S^3_r(K)) \to SU(2) unless r \geq 2g(K)-1 and K is both fibered and strongly quasipositive, broadly generalizing results of Kronheimer and Mrowka. We also answer a question of theirs from 2004, proving that there is always an irreducible homomorphism from the fundamental group of 4-surgery on a nontrivial knot to SU(2) . In another application, we show that a slight enhancement of the A -polynomial detects infinitely many torus knots, including the trefoil.

On the rational approximation to $p$-adic Thue–Morse numbers

Let p be a prime number and let \xi be an irrational p -adic number. Its multiplicative irrationality exponent \mu^{\times}(\xi) is the supremum of the real numbers \mu^{\times} for which the inequality |b \xi - a|_{p} \leq | a b |^{-\mu^{\times} / 2} has infinitely many solutions in nonzero integers a and b . We show that \mu^{\times} (\xi) can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of \xi . We establish that \mu^{\times} ({\xi_{\mathbf{t}, p}}) = 3 , where {\xi_{\mathbf{t}, p}} is the p -adic number 1 - p - p^2 + p^3 - p^4 + \ldots , whose sequence of digits is given by the Thue–Morse sequence over \{-1, 1\} .

A Lower Bound on the Number of Generators of a Defect Group

Assuming that the statement of the Alperin–McKay–Navarro conjecture holds for a p-block B with defect group D, we show that the number of generators of D is bounded from below by the number of height-zero characters in B fixed by a specific element of the absolute Galois group of the rational numbers.

Computability of sets in Euclidean space

Abstract We consider several concepts of computability (recursiveness) for sets in Euclidean space. A list of four ideal properties for such sets is proposed and it is shown in a very elementary way that no notion can satisfy all four desiderata. Most notions introduced here are essentially based on separability of ℝ n and this is natural when thinking about operations on an actual digital computer where, in fact, rational numbers are the basis of everything. We enumerate some properties of some naïve but practical notions of recursive sets and contrast these with others, including the widely used and accepted notion of computable set developed by Weihrauch, Brattka and others which is based on the “Polish school” notion of a computable real function. We also offer a conjecture about the Mandelbrot set.

Toward a unified theory of rational number arithmetic.

To explain children's difficulties learning fraction arithmetic, Braithwaite et al. (2017) proposed FARRA, a theory of fraction arithmetic implemented as a computational model. The present study tested predictions of the theory in a new domain, decimal arithmetic, and investigated children's use of conceptual knowledge in that domain. Sixth and eighth grade children (N = 92) solved decimal arithmetic problems while thinking aloud and afterward explained solutions to decimal arithmetic problems. Consistent with the hypothesis that FARRA's theoretical assumptions would generalize to decimal arithmetic, results supported 3 predictions derived from the model: (a) accuracies on different types of problems paralleled the frequencies with which the problem types appeared in textbooks; (b) most errors involved overgeneralization of strategies that would be correct for problems with different operations or types of number; and (c) individual children displayed patterns of strategy use predicted by FARRA. We also hypothesized that during routine calculation, overt reliance on conceptual knowledge is most likely among children who lack confidence in their procedural knowledge. Consistent with this hypothesis, (d) many children displayed conceptual knowledge when explaining solutions but not while solving problems; (e) during problem-solving, children who more often overtly used conceptual knowledge also displayed doubt more often; and (f) problem solving accuracy was positively associated with displaying conceptual knowledge while explaining, but not with displaying conceptual knowledge while solving problems. We discuss implications of the results for rational number instruction and for the creation of a unified theory of rational number arithmetic. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Linear Forms in Polylogarithms

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x <1$. Consider $\Phi_s(x,z) =\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch function with $s=1, 2, \cdots, r$. When $x=0$, this is a polylogarithmic function. Let $\alpha_1, \cdots, \alpha_m$ be pairwise distinct algebraic numbers of arbitrary degree over the rational number field, with $0<|\alpha_j|<1 \,\,\,(1\leq j \leq m)$. In this article, we show a criterion for the linear independence, over an algebraic number field containing $\mathbb{Q}(\alpha_1, \cdots, \alpha_m)$, of all the $rm+1$ numbers : $\Phi_1(x,\alpha_1)$, $\Phi_2(x,\alpha_1), $ $\cdots , \Phi_r(x,\alpha_1)$, $\Phi_1(x,\alpha_2)$, $\Phi_2(x,\alpha_2), $ $\cdots , \Phi_r(x,\alpha_2), \cdots, \cdots, \Phi_1(x,\alpha_m)$, $\Phi_2(x,\alpha_m)$, $\cdots , \Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the Lerch functions at several distinct algebraic points, not necessarily lying in the rational number field nor in quadratic imaginary fields. We give a complete proof with refinements and quantitative statements of the main theorem announced in [10], together with a proof in detail on the non-vanishing Wronskian of Hermite type.

The Optimal Correcting the Power Value of a Nuclear Power Plant Power Unit Reactor in the Event of Equipment Failures

The paper considers the problem of adjusting the power of a nuclear power plant power unit reactor for those cases when equipment failure occurs. In such circumstances, sometimes it is sufficient to reduce the reactor power, while maintaining the probabilistic level of safe operation of the power unit. The rational number of the reactor power is determined by solving the problem of minimizing the risk criterion in the integral root mean square context. In order to demonstrate the efficiency of the proposed approach, a numerical example is considered. The approach outlined in the paper is focused on improving the power unit control in case of equipment failures.

Interpreting literal symbols in algebra under the effects of the natural number bias

ABSTRACT In this study, we investigated how secondary students interpret algebraic expressions that contain literal symbols to stand for variables. We hypothesized that the natural number bias (i.e., the tendency to over-rely on knowledge and experiences based on natural numbers) would affect students to think that the literal symbols stand for natural numbers only rather than for any rational or real number (integrity effect); and that the arithmetical values of the algebraic expressions are of the same sign as the expressions’ phenomenal sign (phenomenal sign effect). The participants (138 8th and 9th graders) were asked to evaluate 48 statements about numbers that can or cannot be assigned to six algebraic expressions that contained literal symbols (e.g., a, -d-4). The results supported the main hypothesis of the study with respect to the integrity as well as the phenomenal sign effect and also indicated that the former was stronger than the latter. Additionally, the most salient characteristics of the form of each expression, such as its sign, appeared to affect students’ responses regarding the arithmetical values they may represent. Theoretical and educational implications are discussed.

Limit laws for rational continued fractions and value distribution of quantum modular forms

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in (0, 1] ordered by denominators. We show convergence to a stable law in a general setting, by proving an estimate with power-saving error term for the associated characteristic function. This extends results of Baladi and Vallée on Gaussian behaviour for costs of moderate growth. We apply our result to obtain the limiting distribution of values of several key examples of quantum modular forms. We obtain the Gaussian behaviour of central values of the Esterman function ∑ n ⩾ 1 τ ( n ) e 2 π i n x / n $\sum _{n\geqslant 1} \tau (n) {\rm e}^{2\pi i n x}/\sqrt {n}$ ( x ∈ Q $x\in {\mathbb {Q}}$ ), a problem for which known approaches based on Eisenstein series have been so far ineffective. We give a new proof, based on dynamical systems, that central modular symbols associated with a holomorphic cusp form for S L ( 2 , Z ) $SL(2,{\mathbb {Z}})$ have a Gaussian distribution, and give the first proof of an estimate for their probabilities of large deviations. We also recover a result of Vardi on the convergence of Dedekind sums to a Cauchy law, using dynamical methods.

Comparing the costs of Any Fit algorithms for bin packing

An online bin packing instance is a list of item sizes each of which is a rational number in (0,1], the goal is to partition the items into bins where the total size of items in a bin is at most 1, and the total number of bins is minimized. Here we analyze the ratio of two online algorithms for sequences where the total size of items is large and prove an old open conjecture of Johnson.

Stable commutator length in right‐angled Artin and Coxeter groups

We establish a spectral gap for stable commutator length (scl) of integral chains in right-angled Artin groups (RAAGs). We show that this gap is not uniform, that is, there are RAAGs and integral chains with scl arbitrarily close to zero. We determine the size of this gap up to a multiplicative constant in terms of the opposite path length of the defining graph. This result is in stark contrast with the known uniform gap 1 / 2 $1/2$ for elements in RAAGs. We prove an analogous result for right-angled Coxeter groups. In a second part of this paper, we relate certain integral chains in RAAGs to the fractional stability number of graphs. This has several consequences: First, we show that every rational number q ⩾ 1 $q \geqslant 1$ arises as the scl of an integral chain in some RAAG. Second, we show that computing scl of elements and chains in RAAGs is NP hard. Finally, we heuristically relate the distribution of scl $\mathrm{scl}$ for random elements in the free group to the distribution of fractional stability number in random graphs. We prove all of our results in the general setting of graph products. In particular, all above results hold verbatim for right-angled Coxeter groups.

A robust class of linear recurrence sequences

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers.

Integration with filters

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part, which can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations of real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.

On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set

We investigate Benford’s law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational numbers, and through statistical testing, we demonstrate that the numerators and denominators are a good fit for Benford’s law. We offer additional conjectures and observations about these coefficients. In particular, we highlight certain arithmetic subsequences related to the coefficients’ denominators, provide an estimate for their slope, and describe efficient methods to compute them.

ε-arithmetics for real vectors and linear processing of real vector-valued signals

In this paper, we introduce a new concept, namely ε-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the same dimension within ε range. For rational vectors of a fixed dimension m, they can form a field that is an mth order extension Q(α) of the rational field Q where α has its minimal polynomial of degree m over Q. Then, the arithmetics, such as addition, subtraction, multiplication, and division, of real vectors can be defined by using that of their approximated rational vectors within ε range. We also define complex conjugate of a real vector and then inner product and convolutions of two real vectors and two real vector sequences (signals) of finite length. With these newly defined concepts for real vectors, linear processing, such as linear filtering, ARMA modeling, and least squares fitting, can be implemented to real vector-valued signals with real vector-valued coefficients, which will broaden the existing linear processing to scalar-valued signals.

Periodic solutions for a class of perturbed sixth-order autonomous differential equations

PurposeThe objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equationsx(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)),wherepandqare rational numbers different from 1, 0, −1 andp≠q,εis a small enough parameter andF∈C2is a nonlinear autonomous function.Design/methodology/approachThe authors shall use the averaging theory to study the periodic solutions for a class of perturbed sixth-order autonomous differential equations (DEs). The averaging theory is a classical tool for the study of the dynamics of nonlinear differential systems with periodic forcing. The averaging theory has a long history that begins with the classical work of Lagrange and Laplace. The averaging theory is used to the study of periodic solutions for second and higher order DEs.FindingsAll the main results for the periodic solutions for a class of perturbed sixth-order autonomous DEs are presenting in the Theorem 1. The authors present some applications to illustrate the main results.Originality/valueThe authors studied Equation 1 which depends explicitly on the independent variablet. Here, the authors studied the autonomous case using a different approach.

Gromov–Witten invariants of Hilbert schemes of two points on elliptic surfaces

In this paper, we study the Gromov–Witten theory of the Hilbert scheme [Formula: see text] of two points on an elliptic surface [Formula: see text]. Assume that [Formula: see text] contains an element supported on the smooth fibers of [Formula: see text]. By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the [Formula: see text]-point genus-[Formula: see text] Gromov–Witten invariant [Formula: see text] up to some rational number [Formula: see text] depending only on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a smooth fiber of [Formula: see text], [Formula: see text] with [Formula: see text] being a fixed point, and [Formula: see text]. Moreover, we propose a conjecture regarding [Formula: see text], and prove that the conjecture is true for [Formula: see text] where [Formula: see text] is an elliptic curve and [Formula: see text] is a smooth curve.