'Quasi'-norm of an arithmetical convolution operator and the order of the Riemann zeta function

In this paper we study Dirichlet convolution with a given arithmetical function f as a linear mapping 'f that sends a sequence (an) to (bn) where bn = Pdjn f(d)an=d. We investigate when this is a bounded operator on l2 and ¯nd the operator norm. Of particular interest is the case f(n) = n¡® for its connection to the Riemann zeta function on the line 1, 'f is bounded with k'f k = ³(®). For the unbounded case, we show that 'f : M2 ! M2 where M2 is the subset of l2 of multiplicative sequences, for many f 2 M2. Consequently, we study the `quasi'-norm sup kak = T a 2M2 k'fak kak for large T, which measures the `size' of 'f on M2. For the f(n) = n¡® case, we show this quasi-norm has a striking resemblance to the conjectured maximal order of j³(® + iT )j for ® > 12 .

A generalized collectively compact operator theory with an application to integral equations on unbounded domains

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to establish solvability results for a class of systems of second kind integral equations on unbounded domains, this class including in particular systems of Wiener-Hopf integral equations with L1 convolutions kernels

L1 acquisition across Portuguese dialects: modular and interdisciplinary interfaces as sources of explanation

This study investigates the child (L1) acquisition of properties at the interfaces of morpho-syntax, syntax-semantics and syntax-pragmatics, by focusing on inflected infinitives in European Portuguese (EP). Three child groups were tested, 6–7-year-olds, 9–10-year-olds and 11–12-year-olds, as well as an adult control group. The data demonstrate that children as young as 6 have knowledge of the morpho-syntactic properties of inflected infinitives, although they seem at first glance to show partially insufficient knowledge of their syntax–semantic interface properties (i.e. non-obligatory control properties), differently from children aged 9 and older, who show clearer evidence of knowledge of both types of properties. However, in general, both morpho-syntactic and syntax–semantics interface properties are also accessible to 6–7-year-old children, although these children give preference to a range of interpretations partially different from the adults; in certain cases, they may not appeal to certain pragmatic inferences that permit additional interpretations to adults and older children. Crucially, our data demonstrate that EP children master the two types of properties of inflected infinitives years before Brazilian Portuguese children do (Pires and Rothman, 2009a and Pires and Rothman, 2009b), reasons for and implications of which we discuss in detail.

Individual differences and the role of the L1 in L2 processing: an ERP investigation

This study used ERP (event-related potentials) to examine both the role of the L1 and the role of individual differences in the processing of agreement violations. Theories of L2 acquisition differ with regard to whether or not native-like acquisition of L2 features is possible (Schwartz and Sprouse, 1994, 1996; Tsimpli and Mastropavlou, 2007), and the results of previous ERP studies are inconsistent when it comes to whether or not native-like processing is observed in response to L2 agreement violations (e.g., Sabourin, 2003; Tokowicz and MacWhinney, 2005). Furthermore, studies of learners in early stages of L2 acquisition have found variability in the emergence of native-like responses (e.g., McLaughlin et al., 2010; Tanner et al., 2009), but sources of variability have not been investigated. The current study examines responses to gender and number agreement violations in English-speaking learners of Spanish (n=24). Stimuli targeted agreement in three conditions: subject-verb agreement (el barco flota/*flotan), which is similar in Spanish and English; number agreement on adjectival predicates (la isla rocosa/*rocosas), a context in which agreement is not instantiated in English; and gender agreement on adjectival predicates (la isla rocosa/*rocoso), which is unique to Spanish. Grammaticality judgments and ERP responses were also tested for correlations with aptitude scores on the Modern Languages Aptitude Test (MLAT; Carroll and Sapon, 1959) and the Raven Advanced Progressive Matrices (Raven, 1965). Results are in line with theories that claim native-like processing is acquirable, since learners demonstrated similar ERP responses to a control group of native Spanish-speakers (n=8) with regard to all three agreement types. Additionally, the MLAT (but not the Raven) was significantly correlated with sensitivity to number violations, both in terms of grammaticality judgments and ERP amplitudes, indicating a role for verbal but not nonverbal aptitude in L2 processing.

Morphosyntactic processing in advanced second language (L2) learners: An event-related potential investigation of the effects of L1-L2 similarity and structural distance

Different theoretical accounts of second language (L2) acquisition differ with respect to whether or not advanced learners are predicted to show native like processing for features not instantiated in the native language (L1). We examined how native speakers of English, a language with number but not gender agreement, process number and gender agreement in Spanish. We compare agreement within a determiner phrase (órgano muy complejo “[DP organ-MASC-SG very complex-MASC-SG]”) and across a verb phrase (cuadro es auténtico “painting-MASC-SG [VP is authentic-MASC-SG]”) in order to investigate whether native like processing is limited to local domains (e.g. within the phrase), in line with Clahsen and Felser (2006). We also examine whether morphological differences in how the L1 and L2 realize a shared feature impact processing by comparing number agreement between nouns and adjectives, where only Spanish instantiates agreement, and between demonstratives and nouns, where English also instantiates agreement. Similar to Spanish natives, advanced learners showed a P600 for both number and gender violations overall, in line with the Full Transfer/Full Access Hypothesis (Schwartz and Sprouse, 1996), which predicts that learners can show native-like processing for novel features. Results also show that learners can establish syntactic dependencies outside of local domains, as suggested by the presence of a P600 for both within and across phrase violations. Moreover, similar to native speakers, learners were impacted by the structural distance (number of intervening phrases) between the agreeing elements, as suggested by the more positive waveforms for within than across-phrase agreement overall. These results are consistent with the proposal that learners are sensitive to hierarchical structure.

Exploring the source of differences and similarities in L1 attrition and heritage speaker competence: evidence from pronominal resolution

Several studies of different bilingual groups including L2 learners, child bilinguals, heritage speakers and L1 attriters reveal similar performance on syntax-discourse interface properties such as anaphora resolution (Sorace, 2011 and references therein). Specifically, bilinguals seem to allow more optionality in the interpretation of overt subject pronouns in null subject languages, such as Greek, Italian and Spanish while the interpretation of null subject pronouns is indistinguishable from monolingual natives. Nevertheless, there is some evidence pointing to bilingualism effects on the interpretation of null subject pronouns too in heritage speakers’ grammars (Montrul, 2004) due to some form of ‘arrested’ development in this group of bilinguals. The present study seeks to investigate similarities and differences between two Greek–Swedish bilingual groups, heritage speakers and L1 attriters, in anaphora resolution of null and overt subject pronouns in Greek using a self-paced listening with a sentence-picture matching decision task at the end of each sentence. The two groups differ in crucial ways: heritage speakers were simultaneous or early bilinguals while the L1 attriters were adult learners of the second language, Swedish. Our findings reveal differences from monolingual preferences in the interpretation of the overt pronoun for both heritage and attrited speakers while the differences attested between the two groups in the interpretation of null subject pronouns affect only response times with heritage being faster than attrited speakers. We argue that our results do not support an age of onset or differential input effects on bilingual performance in pronoun resolution.

L1 Acquisition of interface properties across Portuguese dialects: modular and interdisciplinary interfaces as sources of explanation

This study investigates the child (L1) acquisition of properties at the interfaces of morphosyntax, syntax-semantics and syntax-pragmatics, by focusing on inflected infinitives in European Portuguese (EP). Three child groups were tested, 6–7-year-olds, 9–10-year-olds and 11–12-year-olds, as well as an adult control group. The data demonstrate that children as young as 6 have knowledge of the morpho-syntactic properties of inflected infinitives, although they seem at first glance to show partially insufficient knowledge of their syntax–semantic interface properties (i.e. non-obligatory control properties), differently from children aged 9 and older, who show clearer evidence of knowledge of both types of properties. However, in general, both morpho-syntactic and syntax–semantics interface properties are also accessible to 6–7-year-old children, although these children give preference to a range of interpretations partially different from the adults; in certain cases, they may not appeal to certain pragmatic inferences that permit additional interpretations to adults and older children. Crucially, our data demonstrate that EP children master the two types of properties of inflected infinitives years before Brazilian Portuguese children do (Pires and Rothman, 2009a,b), reasons for and implications of which we discuss in detail.

Acquisition of inflected and uninflected infinitives in child L1 European Portuguese

This study investigates the child (L1) acquisition of inflected and uninflected infinitives in European Portuguese (EP). We test and contrast properties involving two interfaces, focusing on morpho-syntactic and syntax-semantics properties of inflected infinitives, in contrast with uninflected infinitives. We present experimental results from three monolingual EP child groups, between ages 6 and 12 (n=72), compared to EP adults (n=32). Results show that children as young as 6-7 have knowledge of the morpho-syntactic properties of inflected infinitives, although at first glance they show insufficient knowledge of their syntax-semantics interface properties (i.e. non-obligatory control properties), differently from older children, who show evidence of knowledge of both types of properties. We argue that, in general, morpho-syntactic and syntax-semantics interface distinctions are also accessible to 6-7 children, but children may not show the entire range of interpretations possible for adults.

The proportion of failures of the Hasse norm principle

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

Processing coordinate subject-verb agreement in L1 and L2 Greek

The present study examines the processing of subject-verb (SV) number agreement with coordinate subjects in pre-verbal and post-verbal positions in Greek. Greek is a language with morphological number marked on nominal and verbal elements. Coordinate SV agreement, however, is special in Greek as it is sensitive to the coordinate subject's position: when pre-verbal, the verb is marked for plural while when post-verbal the verb can be in the singular. We conducted two experiments, an acceptability judgment task with adult monolinguals as a pre-study (Experiment 1) and a self-paced reading task as the main study (Experiment 2) in order to obtain acceptance as well as processing data. Forty adult monolingual speakers of Greek participated in Experiment 1 and a hundred and forty one in Experiment 2. Seventy one children participated in Experiment 2: 30 Albanian-Greek sequential bilingual children and 41 Greek monolingual children aged 10–12 years. The adult data in Experiment 1 establish the difference in acceptability between singular VPs in SV and VS constructions reaffirming our hypothesis. Meanwhile, the adult data in Experiment 2 show that plural verbs accelerate processing regardless of subject position. The child online data show that sequential bilingual children have longer reading times (RTs) compared to the age-matched monolingual control group. However, both child groups follow a similar processing pattern in both pre-verbal and post-verbal constructions showing longer RTs immediately after a singular verb when the subject was pre-verbal indicating a grammaticality effect. In the post-verbal coordinate subject sentences, both child groups showed longer RTs on the first subject following the plural verb due to the temporary number mismatch between the verb and the first subject. This effect was resolved in monolingual children but was still present at the end of the sentence for bilingual children indicating difficulties to reanalyze and integrate information. Taken together, these findings demonstrate that (a) 10–12 year-old sequential bilingual children are sensitive to number agreement in SV coordinate constructions parsing sentences in the same way as monolingual children even though their vocabulary abilities are lower than that of age-matched monolingual peers and (b) bilinguals are slower in processing overall.