Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings.
more »
« less
Faithfulness and underspecification
This work is about two ‘generation problems’ for classic Optimality Theory, chain shifts and saltations. The issues for OT posed by traditional analyses of chain shifts and saltations have led to various embellishments of the classic theory, typically in the form of novel constraint types. Reiss (2021a,b) proposes a general solution to the problem of chain shifts and saltations that relies more directly on different assumptions about representations than about constraints. Specifically, Reiss assumes that underlying representations may be underspecified, and that a map ‘counts’ as a chain shift or as a saltation so long as the surface alternants from a uniform underlying representation match the respective observed alternants. We report here on three results from our ongoing formal assessment of Reiss’s proposed solution.
more »
« less
- Award ID(s):
- 2021149
- PAR ID:
- 10508217
- Publisher / Repository:
- Linguistic Society of America
- Date Published:
- Journal Name:
- Proceedings of the Annual Meetings on Phonology
- Volume:
- 10
- ISSN:
- 2377-3324
- Subject(s) / Keyword(s):
- Optimality Theory, chain shifts, saltations, underspecification, faithfulness
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Robinson, Emma Claire (Ed.)Many research questions in sensory neuroscience involve determining whether the neural representation of a stimulus property is invariant or specific to a particular stimulus context (e.g., Is object representation invariant to translation? Is the representation of a face feature specific to the context of other face features?). Between these two extremes, representations may also be context-tolerant or context-sensitive. Most neuroimaging studies have used operational tests in which a target property is inferred from a significant test against the null hypothesis of the opposite property. For example, the popular cross-classification test concludes that representations are invariant or tolerant when the null hypothesis of specificity is rejected. A recently developed neurocomputational theory suggests two insights regarding such tests. First, tests against the null of context-specificity, and for the alternative of context-invariance, are prone to false positives due to the way in which the underlying neural representations are transformed into indirect measurements in neuroimaging studies. Second, jointly performing tests against the nulls of invariance and specificity allows one to reach more precise and valid conclusions about the underlying representations, particularly when the null of invariance is tested using the fine-grained information from classifier decision variables rather than only accuracies (i.e., using the decoding separability test). Here, we provide empirical and computational evidence supporting both of these theoretical insights. In our empirical study, we use encoding of orientation and spatial position in primary visual cortex as a case study, as previous research has established that these properties are encoded in a context-sensitive way. Using fMRI decoding, we show that the cross-classification test produces false-positive conclusions of invariance, but that more valid conclusions can be reached by jointly performing tests against the null of invariance. The results of two simulations further support both of these conclusions. We conclude that more valid inferences about invariance or specificity of neural representations can be reached by jointly testing against both hypotheses, and using neurocomputational theory to guide the interpretation of results.more » « less
-
Polymer ionization differs from that for their monomeric counterparts due to intramolecular correlations. Such effects are conventionally described in terms of the site-binding model that accounts for short-range interactions between neighboring sites. With an apparent equilibrium constant for each ionizable group and the nearest-neighbor energy as adjustable parameters, the site-binding method is useful to correlate experimental titration curves when the site–site interactions are insignificant at long ranges. This work aims to describe the electrostatic behavior of weak polyelectrolytes in aqueous solutions on the basis of the intrinsic equilibrium constants of the individual ionizable groups and solution conditions underlying the thermodynamic non-ideality. A molecular thermodynamic model is proposed for the protonation of weak polyelectrolytes by incorporating classical density functional theory into the site-binding model to account for the effects of the local ionic environment on both inter-chain and intra-chain correlations. By an extensive comparison of theoretical predictions with experimental titration curves, we demonstrate that the thermodynamic model is able to quantify the ionization behavior of weak polyelectrolytes over a broad range of molecular architectures and solution conditions.more » « less
-
The pursuit of generalizable representations in the realm of machine learning and computer vision is a dynamic field of research. Typically, current methods aim to secure invariant representations by either harnessing domain expertise or leveraging data from multiple domains. In this paper, we introduce a novel approach that involves acquiring Causal Markov Blanket (CMB) representations to improve prediction performance in the face of distribution shifts. Causal Markov Blanket representations comprise the direct causes and effects of the target variable, rendering them invariant across diverse domains. To elaborate, our approach commences with the introduction of a novel structural causal model (SCM) equipped with latent representations, designed to capture the underlying causal mechanisms governing the data generation process. Subsequently, we propose a CMB representation learning framework that derives representations conforming to the proposed SCM. In comparison to state-of-the-art domain generalization methods, our approach exhibits robustness and adaptability under distribution shiftsmore » « less
-
Perlman, Marcus (Ed.)Longstanding cross-linguistic work on event representations in spoken languages have argued for a robust mapping between an event’s underlying representation and its syntactic encoding, such that–for example–the agent of an event is most frequently mapped to subject position. In the same vein, sign languages have long been claimed to construct signs that visually represent their meaning, i.e., signs that are iconic. Experimental research on linguistic parameters such as plurality and aspect has recently shown some of them to be visually universal in sign, i.e. recognized by non-signers as well as signers, and have identified specific visual cues that achieve this mapping. However, little is known about what makes action representations in sign language iconic, or whether and how the mapping of underlying event representations to syntactic encoding is visually apparent in the form of a verb sign. To this end, we asked what visual cues non-signers may use in evaluating transitivity (i.e., the number of entities involved in an action). To do this, we correlated non-signer judgments about transitivity of verb signs from American Sign Language (ASL) with phonological characteristics of these signs. We found that non-signers did not accurately guess the transitivity of the signs, but that non-signer transitivity judgments can nevertheless be predicted from the signs’ visual characteristics. Further, non-signers cue in on just those features that code event representations across sign languages, despite interpreting them differently. This suggests the existence of visual biases that underlie detection of linguistic categories, such as transitivity, which may uncouple from underlying conceptual representations over time in mature sign languages due to lexicalization processes.more » « less
An official website of the United States government

