Matrices with Polynomial Entries

A positive integer $n$ is the area of a Heron triangle if and only if there is a non-zero rational number $\\tau$ such that the elliptic curve\n\\begin{equation*}\nE_{τ}^{(n)}: Y^{2} = X(X-nτ)(X+nτ^{-1})\n\\end{equation*}\nhas a rational point of order different than two. Such integers $n$ are called $\\tau$-congruent numbers. In this paper, we show that for a given positive integer $p$, and a given non-zero rational number $\\tau$, there exist infinitely many $\\tau$-congruent numbers in every residue class modulo $p$ whose corresponding elliptic curves have rank at least two.

On a problem of Diophantus for rationals

8 - THE SYSTEM OF RATIONAL NUMBERS

Rational numbers are of critical importance both in mathematics and in other fields of science. However, they form a stumbling block for learners. One widely known source of the difficulty learners have with rational numbers is the natural number bias, that is the tendency to (inappropriately) apply natural number properties in rational number tasks. Still, it has been shown that a good understanding of natural numbers is highly predictive for mathematics achievement in general, and for performance on rational number tasks in particular. In this study, we further investigated the relation between learners' natural and rational number knowledge, specifically in cases where a natural number bias may lead to errors. Participants were 140 sixth graders from six different primary schools. Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. Learners' natural number knowledge was found to be a good predictor of their rational number knowledge. However, after first controlling for learners' general mathematics achievement, their natural number knowledge only predicted the subaspect of operations with rational numbers. The results of this study suggest that the relation between learners' natural and rational number knowledge can largely be explained by their relation with learners' general mathematics achievement.

It is known that a positive integer $n$ is the area of a right triangle with rational sides if and only if the elliptic curve $E^{(n)}: y^{2} = x(x^{2}-n^{2})$ has a rational point of order different than 2. A generalization of this result states that a positive integer $n$ is the area of a triangle with rational sides if and only if there is a nonzero rational number $\tau$ such that the elliptic curve $E^{(n)}_{\tau}: y^{2} = x(x-n\tau)(n+n\tau^{-1})$ has a rational point of order different than 2. Such $n$ are called $\tau$-congruent numbers. It is shown that for a given integer $m>1$, each congruence class modulo $m$ contains infinitely many distinct $\tau$-congruent numbers.

Simultaneous rational periodic points of degree-2 rational maps

Jordan triple maps

A Thue equation is a Diophantine equation of the form f(x; y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D13-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D13-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D13-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x3 - y3 = 1, we show that the only cubic-triangular number is 1.

The k-Secure t-Conference key distribution scheme (kStC-KDS) provides dynamic conferences in wireless sensor networks, which any group of t sensor nodes can derive a conference key using only its pre-distributed piece in sensor nod concept is based on a t-variate symmetric polynomial of degree k. However, the storage size of each sensor node is exponentially proportional to the size of group. Recently, Harn and Hsu used the multiplication of t onevariate polynomials of degree k instead to reduce the storage size, but the dealer should prepare various (t-1) polynomials of degree k for each sensor node. It is observed that we only need to store the polynomial terms with various coefficients for this t-variate k-degree symmetric polynomial in Blundo and De Santis's kStCKDS. Although the number of polynomial terms with various coefficients is known from the multiset concept. In this work, we adopt partition problem to precisely determine all polynomial terms, such that we can figure out which polynomial terms should be stored in each sensor node to reduce storage size.

IntroductionRational Number KnowledgeThere is a broad agreement in the literature a good understanding of rational numbers is of critical importance for mathematics achievement in general and for performance in specific domains of the mathematics curriculum in particular (Siegler et al., 2012). For example, Siegler, Thompson, and Schneider (2011) found high correlations (all between .54 and .86) between three measures of fraction magnitude knowledge (0-1 fraction number line estimation, 0-5 fraction number line estimation, and 0-1 fraction magnitude comparison) and general mathematics achievement in upper elementary school learners. This finding was replicated by Torbeyns, Schneider, Xin, and Siegler (2015) in three countries from different continents. Similar findings emerged from a recent study of Siegler et al. (2012), who concluded fifth graders' rational number understanding predicted their overall mathematics and algebra scores in high school, even after controlling for reading achievement, IQ, working memory, number knowledge, family income, and family education.Despite the critical importance of a good rational number knowledge, a large body of literature reported children and even adults have a lot of difficulties dealing with various aspects of rational numbers (Bailey, Siegler, & Geary, 2014; Cramer, Post, & delMas, 2002; Li, Chen, & An, 2009; Mazzocco & Devlin, 2008; Merenluoto & Lehtinen, 2004; Vamvakoussi, Van Dooren, & Verschaffel, 2012; Vamvakoussi & Vosniadou, 2010; Van Hoof, Lijnen, Verschaffel, & Van Dooren, 2013). To give one example, more than one third of a representative sample of Flemish sixth graders did not reach the educational standards for rational numbers (Janssen, Verschaffel, Tuerlinckx, Van den Noortgate, & De Fraine, 2010).The difficulties learners have with rational number tasks are often - at least in part - attributed to the number (Vamvakoussi et al., 2012; see Ni & Zhou, 2005, for the closely related idea of whole number bias), which is the tendency to inappropriately use natural number properties in rational numbers tasks (Van Hoof, Vandewalle, Verschaffel, & Van Dooren, 2015). Before learners are introduced to rational numbers in the classroom, they have already formed an idea of what a number is. This idea is based on their experiences (both in daily life and in school) with natural numbers. Once the learners are then instructed about rational numbers, the properties of natural numbers are not always applicable anymore, leading to problems and misconceptions with rational numbers (Vamvakoussi & Vosniadou, 2010). This becomes apparent in learners' systematic mistakes, specifically in rational number tasks where reasoning purely in terms of natural numbers results in an incorrect solution - these tasks are called incongruent. At the same time, much higher accuracy levels are found in rational number tasks where reasoning in terms of natural numbers leads to a correct answer - these tasks are called congruent. The vast literature on this natural number bias reports three main aspects elicit such systematic errors. The first aspect relates to the density of the set of rational numbers. While natural numbers are characterized by a discrete structure (one can always indicate which number follows a given number; for example after 13 comes 14), rational numbers are characterized by a dense structure (you cannot say which number comes next, because between any two given rational numbers are always infinitely many other rational numbers) (e.g., Merenluoto & Lehtinen, 2004). The second aspect relates to the size of rational numbers. Research indicates errors in size comparison tasks are repeatedly made because students incorrectly assume that, as is the case with natural numbers, longer decimals are larger, shorter decimals are smaller, or that a fraction's numerical value always increases when its denominator, numerator, or both increase (Mamede, Nunes, & Bryant, 2005; Meert, Gregoire, & Noel, 2010; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, 2013; Resnick et al. …

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

Polynomial decomposition, also referred to as polynomial factorization, is the process of splitting a given polynomial of degree n into its constituent factors—that is, polynomials of lower degree. A reducible polynomial over a given field—such as the real (ℝ), complex (ℂ), or rational numbers (ℚ)—is one that can be factored into polynomials of lower degree with coefficients in that field; otherwise, it is irreducible over the field (Thangadurai 2007). From the fundamental theorem of algebra, we know that a polynomial with integer coefficients is completely reducible into linear factors over the complex field, whereas it is reducible to linear and quadratic factors over ℝ; however, it may be irreducible over ℚ. If a polynomial (with rational coefficients) can be factored over ℚ, Gauss's lemma states, it can be factored over the integers as well. The fundamental theorem of algebra is only an existence proof and does not provide any procedure for factoring a polynomial.

In this paper, we address the problem of ellipse recovery from blurred shape images. A shape image is a binary-valued (0/1) image in continuous-domain that represents one or multiple shapes. In general, the shapes can also be overlapping. We assume to observe the shape image through finitely many blurred samples, where the 2D blurring kernel is assumed to be known. The samples might also be noisy. Our goal is to detect and locate ellipses within the shape image. Our approach is based on representing an ellipse as the zero-level-set of a bivariate polynomial of degree 2. Indeed, similar to the theory of finite rate of innovation (FRI), we establish a set of linear equations (annihilation filter) between the image moments and the coefficients of the bivariate polynomial. For a single ellipse, we show that the image can be perfectly recovered from only 6 image moments (improving the bound in [Fatemi et al., 2016]). For multiple ellipses, instead of searching for a polynomial of higher degree, we locally search for single ellipses and apply a pooling technique to detect the ellipse. As we always search for a polynomial of degree 2, this approach is more robust against additive noise compared to the strategy of searching for a polynomial of higher degree (detecting multiple ellipses at the same time). Besides, this approach has the advantage of detecting ellipses even when they intersect and some parts of the boundaries are lost. Simulation results using both synthetic and real world images (red blood cells) confirm superiority of the performance of the proposed method against the existing techniques.

We use entropy rates and Schur concavity to prove that, for every integer k ≥2, every nonzero rational number q, and every real number α, the base-k expansions of α, q + α, and qα all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall’s 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.KeywordsShannon EntropyNonzero EntryNormal BaseNormal SequenceKolmogorov ComplexityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A nonzero rational integer has absolute value at least 1. A nonzero rational number has absolute value at least the inverse of any denominator. Liouville’s inequality (§ 3.5) is an extension of these estimates and provides a lower bound for the absolute value of any nonzero algebraic number. More specifically, if we are given finitely many (fixed) algebraic numbers γ l,...,γ t , and a polynomial P ∈ ℤ[X1,...,X t ] which does not vanish at the point (γ l,...,γ t ) then we can estimate from below |P(γ l,...,γ t )|. The lower bound will depend upon the degrees of P with respect to each of the X i ’s, the absolute values of its coefficients as well as some measure of the γ i ’s.KeywordsNumber FieldAlgebraic NumberMinimal PolynomialProduct FormulaAlgebraic IntegerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.