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The ability to differentiate between two languages sets the stage for bilingual learning. Infants can discriminate languages when hearing long passages, but language switches often occur on short time scales with few cues to language identity. As bilingual infants begin learning sequences of sounds and words, how do they detect the dynamics of two languages? In two studies using the head-turn preference procedure, we investigated whether infants (n = 44) can discriminate languages at the level of individual words. In Study 1, bilingual and monolingual 8- to 12-month-olds were tested on their detection of single-word language switching in lists of words (e.g., “dog… lait [fr. milk]”). In Study 2, they were tested on language switching within sentences (e.g., “Do you like the lait?”). We found that infants were unable to detect language switching in lists of words, but the results were inconclusive about infants’ ability to detect language switching within sentences. No differences were observed between bilinguals and monolinguals. Given that bilingual proficiency eventually requires detection of sound sequences across two languages, more research will be needed to conclusively understand when and how this skill emerges. Materials, data, and analysis scripts are available at https://osf.io/9dtwn/.

When communicating with other people, adults reduce or lengthen words based on their predictability, frequency, and discourse status. But younger listeners have less experience than older listeners in processing speech variation across time. In 2 experiments, we tested whether English-speaking parents reduce word durations differently across utterances in child-directed speech (CDS) versus adult-directed speech (ADS). In a child-friendly game with an array of objects and destinations, adult participants (N = 48) read instructions to an experimenter (adult-directed) and then to their own 2- to 3-year-old children (child-directed). In Experiment 1, speakers produced sentences containing high-frequency target nouns, and in Experiment 2, they produced sentences containing low-frequency target nouns. In both CDS and ADS in both experiments, speakers reduced repeated mentions of target nouns across successive utterances. However, speakers reduced less in CDS than in ADS, and low-frequency nouns in CDS were overall longer than low-frequency nouns in ADS. Together, the results suggest that repetition reduction may be beyond speaker control, but that speakers still engage in audience design when producing words for relatively inexperienced listeners. We conclude that language production involves nested audience-driven and speaker-driven processes. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

The ability to differentiate between two languages sets the stage for bilingual learning. Infants can discriminate languages when hearing long passages, but language switches often occur on short time scales with few cues to language identity. As bilingual infants begin learning sequences of sounds and words, how do they detect the dynamics of two languages? In two studies using the head-turn preference procedure, we investigated whether infants (n = 44) can discriminate languages at the level of individual words. In Study 1, monolingual and bilingual 8- to 12-month-olds were tested on their detection of single-word language switching in lists of words (e.g., “dog… lait [fr. milk]”). In Study 2, they were tested on language switching within sentences (e.g., “Do you like the lait?”). Infants detected language switching within sentences, but not in lists of words. Moreover, there was no difference between bilingual and monolingual infants’ performance. Based on these contrasting effects for natural sentences versus lists of words, we conclude that infants may detect language switches more successfully if preceded by sequences of sounds and words in a single language. The ability to detect disruptions in such sequences is likely important in supporting the beginnings of bilingual proficiency.

Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gajw between summands for m € [Fn,Fn+1) converge to geometric decay for almost all m as n→ ∞. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions, which are easily checked, that yield geometric decay in the individual gap measures of generalized Zerkendorf decompositions attached to many linear recurrence relations.

Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over F q x , first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of F q x which avoid 3-term geometric progressions. This new setting gives us a parameter q to vary and study how our bounds converge to 1 as it changes, and positive characteristic introduces some extra combinatorial structure that increases the tractability of common questions in this area.

Let $\mathcal E: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic curves over $\mathbb{Q}(T)$, with $A(T), B(T) \in \mathbb Z(T)$, and consider the $k$\textsuperscript{th} moments $A_{k,\mathcal{E}}(p) := \sum_{t (p)} a_{\mathcal{E}_t}(p)^k$ of the Dirichlet coefficients $a_{\mathcal{E}_t}(p) := p + 1 - |\mathcal{E}_t (\mathbb{F}_p)|$. Rosen and Silverman proved a conjecture of Nagao relating the first moment $A_{1,\mathcal{E}}(p)$ to the rank of the family over $\mathbb{Q}(T)$, and Michel proved that if $j(T)$ is not constant then the second moment is equal to $A_{2,\mathcal{E}}(p) = p^2 + O(p^{3/2})$. Cohomological arguments show that the lower order terms are of sizes $p^{3/2}, p, p^{1/2}$, and $1$. In every case we are able to analyze in closed form, the largest lower order term in the second moment expansion that does not average to zero is on average negative, though numerics suggest this may fail for families of moderate rank. We prove this Bias Conjecture for several large classes of families, including families with rank, complex multiplication, and constant $j(T)$-invariant. We also study the analogous Bias Conjecture for families of Dirichlet characters, holomorphic forms on GL$(2)/\mathbb{Q}$, and their symmetric powers and Rankin-Selberg convolutions. We identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point in families of $L$-functions., Comment: Version 1.0, 40 pages, 2 appendices

In bilingual language environments, infants and toddlers listen to two separate languages during the same key years that monolingual children listen to just one, and bilinguals rarely learn each of their two languages at the same rate. Learning to understand language requires them to cope with challenges not found in monolingual input, notably the use of two languages within the same utterance (e.g., Do you like the perro? or ¿Te gusta el doggy?). For bilinguals of all ages, switching between two languages can reduce efficiency in real-time language processing. But language switching is a dynamic phenomenon in bilingual environments, presenting the young learner with many junctures where comprehension can be derailed or even supported. In the current study, we tested 20 Spanish-English bilingual toddlers (18- to 30-months) who varied substantially in language dominance. Toddlers’ eye movements were monitored as they looked at familiar objects and listened to single-language and mixed-language sentences in both of their languages. We found asymmetrical switch costs when toddlers were tested in their dominant vs. non-dominant language, and critically, they benefited from hearing nouns produced in their dominant language, independent of switching. While bilingualism does present unique challenges, our results suggest a united picture of early monolingual and bilingual learning. Just like monolinguals, experience shapes bilingual toddlers’ word knowledge, and with more robust representations, toddlers are better able to recognize words in diverse sentences.

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The distribution of the number of summands in such decomposition converges to a Gaussian, the gaps between summands converges to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work needed to assume the coefficients in the recurrence relation are non-negative and the first term is positive. We extend these results by creating an infinite family of integer sequences called the $m$-gonal sequences arising from a geometric construction using circumscribed $m$-gons. They satisfy a recurrence where the first $m+1$ leading terms vanish, and thus cannot be handled by existing techniques. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the decompositions and prove that it displays Gaussian behavior. There is geometric decay in the distribution of gaps, both for gaps taken from all integers in an interval and almost surely in distribution for the individual gap measures associated to each integer in the interval. We end by proving that the distribution of the longest gap between summands is strongly concentrated about its mean, behaving similarly as in the longest run of heads in tosses of a coin., Version 1.1, 22 pages