Geodesic Mappings of Semi-Riemannian Manifolds with a Degenerate Metric

This article introduces the concept of geodesic mappings of manifolds with idempotent pseudo-connections. The basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric are obtained. It is proved that semi-Riemannian manifolds admitting concircular fields admit completely canonical geodesic mappings and form a closed class with respect to these mappings.

Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

Sentence first, arguments after: Mechanisms of morphosyntax acquisition

Natural languages contain complex grammatical patterns. For example, in German, finite verbs occur second in main clauses while non-finite verbs occur last, as in 'dein Bruder möchte in den Zoo gehen' (“Your brother wants to go to the zoo”). Children easily acquire this type of morphosyntactic contingency (Poeppel &amp; Wexler, 1993; Deprez &amp; Pierce, 1994). There is extensive debate in the literature over the nature of children’s linguistic representations, but there are considerably fewer mechanistic ideas about how knowledge is actually acquired. Regarding German, one approach might be to learn the position of prosodically prominent open-class words (“verbs go 2nd or last”) and then fill in the morphological details. Alternatively, one could work in the opposite direction, learning the position of closed-class morphemes (“-te goes 2nd and -en goes last”) and fitting open-class items into the resulting structure. This second approach is counter-intuitive, but I will argue that it is the one learners take.Previous research suggests that learners focus distributional analysis on closed-class items because of their distinctive perceptual properties (Braine, 1963; Morgan, Meier, &amp; Newport, 1987; Shi, Werker &amp; Morgan, 1999; Valian &amp; Coulson, 1988). The Anchoring Hypothesis (Valian &amp; Coulson, 1988) posits that, because these items tend to occur at grammatically important points in the sentence (e.g., phrase edges), focusing on them helps learners acquire grammatical structure. Here I ask how learners use closed-class items to acquire complex morphosyntactic patterns such as the verb form/position contingency in German. Experiments 1-4 refute concerns that morphosyntactic contingencies like those in German are too complex to learn distributionally. Experiments 5-8 explore the mechanisms underlying learning, showing that adults and children analyze closed-class items as predictive of the presence and position of open-class items, but not the reverse. In these experiments, subtle mathematical distinctions in learners’ input had significant effects on learning, illuminating the biased computations underlying anchored distributional analysis. Taken together, results suggest that learners organize knowledge of language patterns relative to a small set of closed-class items—just as patterns are represented in modern syntactic theory (Rizzi &amp; Cinque, 2016).

Biased statistical learning of closed-class items

In natural languages, closed-class items predict open-class items but not the other way around. For example, in English, if there is a determiner there will be a noun, but nouns can occur with or without determiners. Here we asked whether statistical learning of closed-class items is also asymmetrical. In three experiments we exposed adults to a miniature language with the one-way dependency “if X then Y”: if X was present, Y was also present, but Y could occur without X. We created different versions of the language in order to ask whether learning depended on which category (X or Y) was an open or closed class. In one condition, X had the main properties of a closed class and Y had the main properties of an open class; in a contrasting condition, X had properties of an open class and Y had properties of a closed class. Learners’ exposure in these two conditions was otherwise identical. Learning was significantly better with closed-class X. Additional experiments demonstrated that it is the perceptual distinctiveness of closed-class items that drives learners to analyze them differently, and that the mathematical relationship between closed- and open-class items influences learning more strongly than their linear order. These results suggest that statistical learning is biased: learners privilege computations in which closed-class items are predictive of, rather than predicted by, open-class items. We suggest that the distributional asymmetries of closed-class items in natural languages—and perhaps the asymmetrical structure of linguistic representations—may arise in part from this learning bias.

Function words and polarity

Function words are commonly considered to be a small and closed class of words in which each element is associated with a specific and fixed logical meaning. Unfortunately, this is not always true as witnessed by negation: on the one hand, negation does reverse the truth-value conditions of a proposition, and the other hand, it does not, realizing what is called Expletive Negation. This chapter aims to investigate whether a word that is established on the basis of its function can be ambiguous by discussing the role of the syntactic derivation in some instances of so-called Expletive Negation clauses, a case in which negation seems to lose its capacity to deny the proposition associated with its sentence. Both a theoretical and an experimental approach has been adopted.

A broadened estimate of syntactic and lexical ability from the MB-CDI

A critical question in the study of language development is to understand lexical and syntactic acquisition, which play different roles in speech to the extent it would be natural to surmise they are acquired differently. As measured through the comprehension and production of closed-class words, syntactic ability emerges at roughly the 400-word mark. However, a significant proportion of the developmental work uses a coarse combination of function and content words on the MacArthur-Bates Communicative Development Inventory (MB-CDI). Using the MB-CDI Wordbank database, we implemented a factor analytic approach to distinguish between lexical and syntactic development from the Words and Sentences (WS) form that involves both function words and the explicit categorizations. Although the Words and Gestures (WG) form did not share the factor structure, common WG/WS elements recapitulate the expected age-related changes. This parsing of the MB-CDI may prove simple, yet fruitful in subsequent investigation.

On the Lattice of ESI-closed Classes of Multifunctions on Two-elements Set

The paper considers multifunctions on a two-element set with superposition and the equality predicate branching operator. The superposition operator is based on the intersection of sets. The main purpose of the work is to describe all closed classes with respect to the considered operators. The equality predicate branching operator allows the task to be reduced to a description of all closed classes generated by 2-variable multifunctions. Using this, it is shown that the lattice of classes closed with respect to the considered operators contains 237 elements. A generating set is specified for each closed class. The result obtained in the paper extends the known result for all closed classes of partial functions on a two-element set.

Be happy when your stomach is

In this paper I provide a description of the role of body-part terms in expressions of emotion and other semantic extensions in MalakMalak, a non-Pama-Nyungan language of the Daly River area. Body-based expressions denote events, emotions, personality traits, significant places and people and are used to refer to times and number. Particularly central in the language are men ‘stomach’, pundu ‘head’ and tjewurr ‘ear’ associated respectively with basic emotions, states of mind and reason. The figurative extensions of these body parts are discussed systematically, and compared with what is known for other languages of the Daly River region. The article also explores the grammatical make up of body-based emotional collocations, and in particular the role of noun incorporation. In MalakMalak, noun incorporation is a central part of forming predicates with body parts, but uncommon in any other semantic domain of the language and only lexemes denoting basic emotions may also incorporate closed-class adjectives.

On bases of closed classes of Boolean vector functions

A functional system of Boolean vector functions with a naturally defined superposition operation is considered. It is shown that each closed class of vector functions with α- or δ-functions as components has a finite basis.