Abstract We investigate the following problem: what is the smallest possible distance between a cubic irrational $$\xi $$ ξ and a rational number p / q in terms of the height $$H(\xi )$$ H ( ξ ) and q ? More precisely, we consider the set $$D_{3,1}$$ D 3 , 1 consisting of all pairs ( u , v ) of positive real numbers such that $$|\xi - p/q| > cH^{-u}(\xi )q^{-v}$$ | ξ - p / q | > c H - u ( ξ ) q - v for all cubic irrationals $$\xi $$ ξ and rationals p / q . First, we transform this problem into one about the root separation of cubic polynomials. Second, under the assumption of the famous abc-conjecture, we give an almost complete description of $$D_{3,1}$$ D 3 , 1 . Namely, the points ( u , v ) with $$2\leqslant v\leqslant 3$$ 2 ⩽ v ⩽ 3 that lie in the interior of $$D_{3,1}$$ D 3 , 1 are characterised by the inequality $$u> 10-3v$$ u > 10 - 3 v . Assuming only the weaker Hall conjecture, we also obtain nontrivial results about the shape of $$D_{3,1}$$ D 3 , 1 , although these are not as strong as those derived from the abc-conjecture. Finally, we discuss an analogue of the set $$D_{3,1}$$ D 3 , 1 in function fields where we are able to give an almost complete description unconditionally.

Abstract Classic mass partition results are about dividing the plane into regions that are equal with respect to one or more measures (masses). We introduce a new concept in which the notion of partition is replaced by that of a cover. In this case, we require (almost) every point in the plane to be covered the same number of times. If all elements of this cover are equal with respect to the given masses, we refer to them as equicoverings. To construct equicoverings, we study a natural generalization of q -fan partitions, which we call spiral equicoverings. Like k -fans, these consist of wedges centered at a common point, but arranged in a way that allows overlapping. Our main result nearly characterizes all reduced positive rational numbers p / q for which there exists a covering by q convex wedges such that every point is covered exactly p times. The proofs use results about centerpoints and combine tools from classical mass partition results, and elementary number theory.

Let f t f_t be a one-parameter family of rational maps defined over a number field K K . We show that for all t t outside of a set of natural density zero, every K K -rational preperiodic point of f t f_t is the specialization of some K ( T ) K(T) -rational preperiodic point of f f . Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of K K -rational preperiodic points of f f , giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family f t ( z ) = z 2 + t f_t(z) = z^2 + t over the field of rational numbers.

We study multiplicative Chern insulators (MCIs) as canonical examples of multiplicative topological phases of matter. Constructing the MCI Bloch Hamiltonian as a symmetry-protected tensor product of two topologically nontrivial parent Chern insulators (CIs), we study two-dimensional (2D) MCIs and introduce three-dimensional mixed MCIs, constructed by requiring the two 2D parent Hamiltonians share only one momentum component. We study the 2D MCI response to time-reversal symmetric flux insertion, observing a 4 π Aharonov-Bohm effect, relating these topological states to fractional quantum Hall states via the microscopic field theories of the quantum skyrmion Hall effect. As part of this response, we observe evidence of quantization of a proposed topological invariant for compactified many-body states to a rational number, suggesting higher-dimensional topology may also be relevant. Finally, we study effects of bulk perturbations breaking the symmetry-protected tensor-product structure of the child Hamiltonian, finding the MCI evolves adiabatically into a topological skyrmion phase.

Abstract We obtain an asymptotic formula for all moments of Dirichlet L -functions L ⁢ ( 1 , χ ) {L(1,\chi)} modulo p when averaged over a subgroup of characters χ of size p - 1 d {\frac{p-1}{d}} with φ ⁢ ( d ) = o ⁢ ( log ⁡ p ) {\varphi(d)=o(\log p)} . Assuming the infinitude of Mersenne primes, the range of our result is optimal and improves and generalises the previous result of S. Louboutin and M. Munsch (2022) for second moments. We also use our ideas to get an asymptotic formula for the second moment of L ⁢ ( 1 2 , χ ) {L(\tfrac{1}{2},\chi)} over subgroups of characters of similar size. This leads to non-vanishing results in this family where the proportion obtained depends on the height of the smallest rational number lying in the dual group. This improves a recent result of this type due to É. Fouvry, E. Kowalski and Ph. Michel (2024). Additionally, we prove that, in both cases, we can take much smaller subgroups for almost all primes p .

Abstract We associate a family of ideal sheaves to any ℚ-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and 𝑉-filtrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, provide criteria for the nontriviality, and introduce the center of minimal exponent (generalizing the notion of minimal log canonical center). The main idea is to exploit the global structure of the 𝑉-filtration along an effective divisor using the notion of twisted Hodge modules. As applications, we prove new cases of conjectures by Debarre, Casalaina–Martin, and Grushevsky on singularities of theta divisors on principally polarized abelian varieties.

An Egyptian fraction is a sum of the form 1 / n 1 + ⋯ + 1 / n r 1/n_{1} + \cdots + 1/n_{r} where n 1 n_{1} , …, n r n_{r} are distinct positive integers. We prove explicit lower bounds for the cardinality of the set E N {E}_{N} of rational numbers that can be represented by Egyptian fractions with denominators not exceeding N N . More precisely, we show that for every integer k ≥ 4 k \geq 4 such that ln k ⁡ N ≥ 3 / 2 \ln _{k} N \geq 3/2 it holds ln ⁡ | E N | ln ⁡ 2 ≥ ( 2 − 3 ln k ⁡ N ) N ln ⁡ N ∏ j = 3 k ln j ⁡ N , \begin{equation*} \frac {\ln {|{{E}_{N}}|}}{\ln 2} \geq \Big (2 - \frac {3}{\ln _{k} N}\Big )\frac {N}{\ln N}\prod _{j=3}^{k} \ln _{j} N , \end{equation*} where ln k \ln _{k} denotes the k k -th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erdős [Illinois J. Math. 20 (1976), pp. 598–613] who established a similar bound but under the more stringent condition ln k ⁡ N ≥ k \ln _{k} N\geq k and with a leading constant of 1 1 . Furthermore, we provide some methods to compute the exact values of | E N | |{E}_{N}| for large positive integers N N , and we give a table of | E N | |{E}_{N}| for N ≤ 154 N \leq 154 .

Abstract Relative to fractions, decimal numbers are thought to be easier for students to learn because they employ the same base-10 system as whole numbers. However, unlike whole numbers, larger decimals can have fewer digits, leading to worse performance when comparing Inconsistent decimal pairs, like 0.8 vs 0.26, than Consistent pairs like 0.86 vs 0.2. Students may be applying the whole number rule: “more digits = larger number” or they could be ignoring the decimal points and comparing 8 vs 26. This study used neuroimaging and our specially designed stimulus set to distinguish between these possibilities. We focused on the intraparietal sulcus (IPS), implicated in numerical magnitude processing, and the anterior cingulate cortex (ACC) and insula, implicated in inhibitory control. We found no neural differences between Consistent and Inconsistent comparisons, suggesting that the number of digits does not drive brain responses in skilled adults (n=21). Instead, for Consistent comparisons, we found that the IPS was sensitive to the actual distance between the decimals, while the ACC showed this pattern for Inconsistent comparisons. Crucially, we also examined the effect of the distance between the decimal pairs when ignoring the decimal point. Here, we found sensitivity to this distance among Inconsistent comparisons in the IPS and insula, suggesting that whole number referents are automatically processed during decimal comparison and require engagement of cognitive control regions to counteract. More broadly, our results underscore the unique challenges of decimal notation, revealing the need for educational practices that emphasize differences to whole numbers rather than highlighting similarities.

Abstract Inquiry-based learning has been shown to foster conceptual understanding through cycles of hypothesis generation and testing, making it particularly relevant for conceptual change when prior knowledge conflicts with new concepts. While inquiry-based learning has been extensively applied in science education, its use in mathematics is still developing. For example, it is often connected to exploratory tasks or problem-solving phases prior to instruction (PS-I). The present study aims to address this gap by investigating whether digital inquiry, including explicit cycles of experimentation, can foster conceptual change in the transition from natural to rational numbers. Using the Scientific Discovery as Dual Search (SDDS) model, we designed a digital learning environment that enables students to generate hypotheses, create visual representations with dynamic fraction bars, test their hypotheses through experimentation, and revise their reasoning based on feedback. We examined the impact of prompts that required the use of digital tools for generating and manipulating dynamic fraction bars, as well as providing empirical feedback on students’ understanding of fractions in two contexts: basketball and color mixing. In an experiment with 231 fifth graders, we found a significant indirect effect of prompts to use the digital tools on the post-test, mediated by the quality of an external visual representation generated with dynamic fraction bars and verbal reasoning. Additionally, there was a positive and significant indirect effect of context on the post-test, favoring the basketball context, mediated by the quality of verbal reasoning. However, no effect of empirical feedback was found. The findings of this study suggest that using dynamic fraction bars and familiar contexts leads to more elaborated learning activities and helps students to shift from natural number concepts to understanding fractions.

A bstract We use methods of arithmetic geometry to find solutions to the abelian local anomaly cancellation equations for a four-dimensional gauge theory whose Lie algebra has a single $${\mathfrak{u}}_{1}$$ summand, assuming that a non-trivial solution exists. The resulting polynomial equations in the integer $${\mathfrak{u}}_{1}$$ charges define a projective cubic hypersurface over the field of rational numbers. Generically, such a hypersurface is (by a theorem of Kollár) unirational, making it possible to find a finitely-many-to-one parameterization of infinitely many solutions using secant and tangent constructions. As an example, for the Standard Model Lie algebra with its three generations of quarks and leptons (or even with just a single generation and two $${\mathfrak{s}\mathfrak{u}}_{3}\oplus {\mathfrak{s}\mathfrak{u}}_{2}$$ singlet right-handed neutrinos), it follows that there are infinitely many anomaly-free possibilities for the $${\mathfrak{u}}_{1}$$ hypercharges. We also discuss whether it is possible to find all solutions in this way.

The investigation of chaos is an important field of science and has got many significant achievements. In the earlier age of the field, the main focus is on the study of the systems that are smooth everywhere. Less attention has been paid to nonsmooth systems. Nonsmooth dynamical systems are broadly appeared in practices, such as impact oscillators, relaxation oscillators, switch circuits, neuron firing, epidemic and even economic models, and have become an active field of study recently. The typical characteristics of those systems is the abruptly variation of the dynamics after slowly evolving over a longer time. Piecewise smooth maps are a type of important models and often employed to describe the dynamics of those systems. Among them, much attention is paid to a class of generally one dimensional piecewise linear discontinuous maps since they are easy to hand and can display rich classes of dynamical phenomena with new characteristics.<br>Enclosed in this work is a discontinuous two-piece mapping function. The left branch is a linear function with slope $\alpha$, and the right is a power law function with exponent $z$. There is a gap confined by $[\mu,\mu+\delta]$ at $x=0$, where $\mu$ is the control parameter, and $\delta$ is the with of the gap. Even though the dynamics of nonsmooth and continuous maps under some special $z$ values have been intensively studied, while their discontinuous counterparts have not been investigated under arbitrary $z$ and discontinuous gap $\delta$. The appearance of the discontinuity may induce border collision bifurcations. The interplay between the bifurcations associated with stability analysis and the border collision bifurcations may produce complex dynamics with new characteristics. Therefore, the investigation on the dynamics of those maps are carried out in this paper, in which the periodic increments, periodic adding and coexistence of attractors are observed. The border collision bifurcation often interrupts a stable periodic orbit and make it transform to a chaotic state or another periodic orbit. In the neighborhood of critical parameters of this bifurcation, there often occurs the coexistence of a periodic orbit with a chaotical or another periodic attractor. A general approach is proposed to analytically and numerically calculate the critical control parameters at which the border collision bifurcations happen, which transform the problem into the solution of an algebraic equation of dimensionless control parameter $\mu/\mu_0$, where $\mu_0$ is the critical control parameter when $\delta=0$. The solution can be obtained analytically when $z$ is a simple rational number or small integer, and numerically for an arbitrary real number. By this way, the stability condition and critical control parameters for the periodic orbit of type $L^{n-1}R$ are analytically or numerically obtained under the arbitrary exponent $z$ and discontinuous gap $\delta$. The results are accordance with the numerical simulations very well. Based on the stability and border collision bifurcation analysis, the phase diagrams in the plane of two dimensional parameters $\mu-\delta$ are built for different ranges of $z$. Their dynamical behaviors are discussed, and three types of co-dimension-2 bifurcations are observed, and the general expressions for the coordinates at which those phenomena occur are obtained in the phase plane. Meanwhile, a specular tripe-point induced by merging of co-dimension-2 bifurcation points $\mathrm{BC-flip}$ and $\mathrm{BC-BC}$ is observed in the phase plane, and the condition for the appearance of it is analytical obtained.

Simultaneous rational number codes: Decoding beyond half the minimum distance with multiplicities and bad primes

The mathematics learning outcomes of students at SMP Negeri 3 Bontang on Rational Numbers are still relatively low. Data from the 2019 National Examination (UN) shows an average math score of 46.43, which is in the "poor" category. Summative assessment results indicate that most students have not yet achieved the Learning Objective Achievement Criteria (KKTP). This situation indicates that the learning process tends to be conventional and lacks active student engagement. Therefore, a more innovative learning model is needed, one of which is the Team Games Tournament (TGT), which combines group work, competition, and educational games. This study aims to determine the effect of the TGT learning model on the mathematics learning outcomes of seventh-grade students at SMP Negeri 3 Bontang in the topic of Rational Numbers. This study used a quantitative approach with a quasi-experimental type and a Posttest-Only Control Group Design. The study population was 203 seventh-grade students in the 2024/2025 academic year, with a sample consisting of class VII A as the experimental group (33 students) and class VII F as the control group (34 students), selected through a purposive sampling technique. The research instrument was a five-item essay test. The analysis results showed that the average posttest of the experimental group was 67.848, higher than the control group at 61.794. The Independent Sample t-Test produced a significance value of 0.031 <0.05, so H₀ was rejected. This indicates that the Team Games Tournament (TGT) learning model has a significant effect on improving students' mathematics learning outcomes in the Rational Numbers material.

Purpose The purpose of this study is to clarify how brands accumulate value through recursive knowledge loops linking firms, customers and wider stakeholders, by framing brand value as an eigenform: a pattern that is dynamically stable yet continuously reproduced across rational, emotional and spiritual knowledge fields. Design/methodology/approach This study offers a conceptual integration of second-order cybernetics with tri-field knowledge theory. A maximum–minimum fuzzy model is introduced to formalise how dispersed interpretations converge, stabilise or fragment. Findings Brand meaning consolidates when signals from the rational, emotional and spiritual fields are coherent and repeatedly reproduced; misalignment propagates fragmentation. The fuzzy formulation yields a practical diagnostic of cross-field coherence and highlights which field is most vulnerable at a given time. Research limitations/implications This conceptual study paves the way towards empirical operationalisation; avenues include longitudinal community data, agent-based simulations and cross-cultural comparisons of field weightings. Practical implications Managers can tune process/channel design, community/interface architecture and purpose/governance mechanisms to sustain a coherent eigenform creation; tracking a fuzzy-based coherence index can surface tipping points and guide timely interventions. Originality/value This study specifies the cybernetic engine that translates distributed dialogue into value-in-use and contributes a mathematically tractable tool for monitoring convergence under uncertainty.

Abstract For an integer $n\geq 7$, we investigate the matroid realization space of a specific deformation of the regular $n$-gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic modular surface $\Xi _{1}(n)$ over the modular curve $X_{1}(n)$. In this way, we obtain a model of $\Xi _{1}(n)$ defined over the rational numbers. Furthermore, a natural geometric operator acts on these matroid realizations. On the elliptic modular surface, this operator corresponds to the multiplication by $-2$ on the elliptic curves. This provides a new geometric approach to computing multiplication by $-2$ on elliptic curves.

Abstract We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to twisted diophantine approximation, and present a simple application, related to possible correlations between trace functions and dynamical sequences.

QbyR: Proposal based on the use of a ruler to acquire rational numbers

In a previous paper, we generalized the well-known Euler-Mascheroni constant c0, in both positive and negative senses. In this sense, for each (0,1), starting from two generalizations of the string cn which converges to c0, we obtained two sequences cn, and cn,-, which converge to c and cn,-, respectively. We called these limits, c - the positive generalized Euler-Mascheroni constant or the positive generalized Euler constant, respectively c- - the negative generalized Euler-Mascheroni constant or the negative generalized Euler constant. By calculating the limits of some sequences in two different ways, we obtained the integral form of these two constants c and c- and, then, we calculated these two constants for different rational values of the number . In this paper we will present other ways of determining these generalized constants, mentioned above, and we have extended the determination of these generalized constants to different values of 1 (integers or rational numbers). With all the values obtained for c, we have presented immediate applications to determining the limits of sequences of real numbers. At the end of this paper, I proposed, to the attentive reader interested in these issues, the solution of an interesting exercise. Of course, this paper is exclusively about Mathematics Didactics and can be used / recommended to all those interested in these issues: pupils, students or Mathematics teachers.

We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial f computed by a constant-depth circuit over rational numbers, and outputs a list L of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of f computable by constant-depth circuits. This list L might also include circuits that are spurious: they either do not correspond to factors of f or are not even well-defined, e.g. the input to a division gate is a sub-circuit that computes the identically zero polynomial. The key technical ingredient of our algorithm is a notion of the pseudo-resultant of f and a factor g , which serves as a proxy for the resultant of g and f / g , with the advantage that the circuit complexity of the pseudo-resultant is comparable to that of the circuit complexity of f and g . This notion, which might be of independent interest, together with the recent results of Limaye, Srinivasan and Tavenas [19] helps us derandomize one key step of multivariate polynomial factorization algorithms — that of deterministically finding a good starting point for Newton Iteration for the case when the input polynomial as well as the irreducible factor of interest have small constant-depth circuits.

Abstract A commonly documented phenomenon involves properties from an earlier encountered set being overextended to a later encountered superset (e.g., the natural number bias). This work focuses on a related but separate phenomenon, where a formula does carry over from a set to its superset, but the justification for why the formula holds does not. Specifically, justifications of the area formula for rectangles, which is typically only justified in the natural number case, are explored. Prospective teachers were asked to justify this formula in the case of rational side lengths. The data indicated three distinct types of justifications: counting with fractional units, counting with a new unit, and expanding the rectangle. The first two of these categories are otherwise viable arguments, which inadvertently took the formula working in the rational number case for granted. We discuss the varied ways prospective teachers’ approaches take advantage of or modify the known justification for the formula in the integer case.