Quadratic form representations via generalized continuants

H. J. S. Smith proved Fermat's two-square theorem using the notion of palindromic continuants. In this paper we extend Smith's approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of integers and rings of polynomials over fields of odd characteristic. Also, we present new deterministic algorithms for finding the corresponding proper representations.Comment: arXiv admin note: text overlap with arXiv:1112.453

On problems related to multiple solutions of Pell's equation and continued fractions over function fields

We study old problems, connected to the theory of continued fractions, with a new twist: changing the setting from the real numbers to the field of formal Laurent series in 1/t. In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers √D is the solution to Pell's equation for D. It is well-known that, once an integer solution to Pell's equation exists, we can use it to generate all other solutions (u_n, v_n)_{n∈Z}. Our object of interest is the polynomial version of Pell's equation, where the integers are replaced by polynomials with complex coefficients. We then investigate the factors of v_n(t). In particular, we show that over the complex polynomials, there are only finitely many values of n for which v_n(t) has a repeated root. Restricting our analysis to Q[t], we give an upper bound on the number of ``new'' factors of v_n(t) of degree at most N. Furthermore, we show that all ``new'' linear rational factors of v_n(t) can be found when n≤ 3, and all ``new'' quadratic rational factors when n≤ 6. Another application of continued fractions arises from the theory of rational approximations to real irrational numbers. There, if we truncate the continued fraction expansion of \alpha\in\Ree, the resulting rational number ``best'' approximates it. This consequence remains true when we replace real numbers by formal Laurent series in $1/t$. In the framework of power series over the rational numbers, we define the Lagrange spectrum, related to Diophantine approximation of irrationals, and the Markov spectrum, related to elements represented by indefinite binary quadratic forms. We compute both spectra, by showing they equal sets whose elements are quantities attached to doubly infinite sequences of non-constant polynomials. Moreover, we prove that Lagrange and Markov spectra coincide and exhibit no gaps, contrary to what happens over the real numbers

The changing role of sound symbolism for small versus large vocabularies

Natural language contains many examples of sound-symbolism, where the form of the word carries information about its meaning. Such systematicity is more prevalent in the words children acquire first, but arbitrariness dominates during later vocabulary development. Furthermore, systematicity appears to promote learning category distinctions, which may become more important as the vocabulary grows. In this study, we tested the relative costs and benefits of sound-symbolism for word learning as vocabulary size varies. Participants learned form meaning mappings for words which were either congruent or incongruent with regard to sound-symbolic relations. For the smaller vocabulary, sound-symbolism facilitated learning individual words, whereas for larger vocabularies sound-symbolism supported learning category distinctions. The changing properties of form-meaning mappings according to vocabulary size may reflect the different ways in which language is learned at different stages of development

Non-adjacent dependency learning in infancy, and its link to language development

To acquire language, infants must learn how to identify words and linguistic structure in speech. Statistical learning has been suggested to assist both of these tasks. However, infants’ capacity to use statistics to discover words and structure together remains unclear. Further, it is not yet known how infants’ statistical learning ability relates to their language development. We trained 17-month-old infants on an artificial language comprising non-adjacent dependencies, and examined their looking times on tasks assessing sensitivity to words and structure using an eye-tracked head-turn-preference paradigm. We measured infants’ vocabulary size using a Communicative Development Inventory (CDI) concurrently and at 19, 21, 24, 25, 27, and 30 months to relate performance to language development. Infants could segment the words from speech, demonstrated by a significant difference in looking times to words versus part-words. Infants’ segmentation performance was significantly related to their vocabulary size (receptive and expressive) both currently, and over time (receptive until 24 months, expressive until 30 months), but was not related to the rate of vocabulary growth. The data also suggest infants may have developed sensitivity to generalised structure, indicating similar statistical learning mechanisms may contribute to the discovery of words and structure in speech, but this was not related to vocabulary size

Calculation of functionals of matrices arising in solid state physics and quantum chemistry

Analytic function calculation of matrices in solid state physics and quantum chemistr

Real quadratic fields with a universal form of given rank have density zero

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients.Comment: 18 pages, minor change