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We solve the Einstein constraint equations for a first-order causal viscous relativistic hydrodynamic theory in the case of a conformal fluid. For such a theory, a direct application of the conformal method does not lead to a decoupling of the equations, even for constant-mean curvature initial data. We combine the conformal method applied to a background perfect fluid theory with a perturbative argument in order to obtain the result.

We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of operators in geometric analysis. The proofs rely on a nonstandard technique using rescaling estimates and apply to operators having coefficients with low regularity. For each class of function space for an operator's coefficients, we exhibit a natural associated range of function spaces of the same type for the domain of the operator and we provide regularity inference along with interior estimates. Additionally, we present a unified set of multiplication results for the function spaces we consider., Comment: 73 pages, 3 figures

The creation of relevance assessments by human assessors (often nowadays crowdworkers) is a vital step when building IR test collections. Prior works have investigated assessor quality & behaviour, though into the impact of a document's presentation modality on assessor efficiency and effectiveness. Given the rise of voice-based interfaces, we investigate whether it is feasible for assessors to judge the relevance of text documents via a voice-based interface. We ran a user study (n = 49) on a crowdsourcing platform where participants judged the relevance of short and long documents sampled from the TREC Deep Learning corpus-presented to them either in the text or voice modality. We found that: (i) participants are equally accurate in their judgements across both the text and voice modality; (ii) with increased document length it takes participants significantly longer (for documents of length > 120 words it takes almost twice as much time) to make relevance judgements in the voice condition; and (iii) the ability of assessors to ignore stimuli that are not relevant (i.e., inhibition) impacts the assessment quality in the voice modality-assessors with higher inhibition are significantly more accurate than those with lower inhibition. Our results indicate that we can reliably leverage the voice modality as a means to effectively collect relevance labels from crowdworkers., Comment: Accepted at SIGIR 2023

The Search Engine Results Page (SERP) has evolved significantly over the last two decades, moving away from the simple ten blue links paradigm to considerably more complex presentations that contain results from multiple verticals and granularities of textual information. Prior works have investigated how user interactions on the SERP are influenced by the presence or absence of heterogeneous content (e.g., images, videos, or news content), the layout of the SERP (list vs. grid layout), and task complexity. In this paper, we reproduce the user studies conducted in prior works-specifically those of Arguello et al. [4] and Siu and Chaparro [29]-to explore to what extent the findings from research conducted five to ten years ago still hold today as the average web user has become accustomed to SERPs with ever-increasing presentational complexity. To this end, we designed and ran a user study with four different SERP interfaces: (i) a heterogeneous grid; (ii) a heterogeneous list; (iii) a simple grid; and (iv) a simple list. We collected the interactions of 41 study participants over 12 search tasks for our analyses. We observed that SERP types and task complexity affect user interactions with search results. We also find evidence to support most (6 out of 8) observations from [4 , 29] indicating that user interactions with different interfaces and to solve tasks of different complexity have remained mostly similar over time., Comment: Figure 2 in the SIGIR proceedings version has a mistake---the legends of Remember and Navigation are flipped. This submission has the correct version of the plot

We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity, and introduce several technical tools for studying PDEs on such manifolds. Our results employ two novel families of function spaces suitable for metrics potentially having a large amount of interior differentiability, but whose Sobolev regularity implies only a H\"older continuous conformal structure at infinity. We establish Fredholm theorems for elliptic operators arising from metrics in these families. To demonstrate the utility of our methods, we solve the Yamabe problem in this category. As a special case, we show that the asymptotically hyperbolic Yamabe problem is solvable so long as the metric admits a $W^{1,p}$ conformal compactification, with $p$ greater than the dimension of the manifold.

Complex search tasks - such as those from the Search as Learning (SAL) domain - often result in users developing an information need composed of several aspects. However, current models of searcher behaviour assume that individuals have an atomic need, regardless of the task. While these models generally work well for simpler informational needs, we argue that searcher models need to be developed further to allow for the decomposition of a complex search task into multiple aspects. As no searcher model yet exists that considers both aspects and the SAL domain, we propose, by augmenting the Complex Searcher Model (CSM), the Subtopic Aware Complex Searcher Model (SACSM) - modelling aspects as subtopics to the user's need. We then instantiate several agents (i.e., simulated users), with different subtopic selection strategies, which can be considered as different prototypical learning strategies (e.g., should I deeply examine one subtopic at a time, or shallowly cover several subtopics?). Finally, we report on the first large-scale simulated analysis of user behaviours in the SAL domain. Results demonstrate that the SACSM, under certain conditions, simulates user behaviours accurately., Comment: 15 pages, 5 figures. Part of the proceedings of BCS-IRSG ECIR 2022 in Stavanger, Norway

Search engines are considered the primary tool to assist and empower learners in finding information relevant to their learning goals-be it learning something new, improving their existing skills, or just fulfilling a curiosity. While several approaches for improving search engines for the learning scenario have been proposed, instructional scaffolding has not been studied in the context of search as learning, despite being shown to be effective for improving learning in both digital and traditional learning contexts. When scaffolding is employed, instructors provide learners with support throughout their autonomous learning process. We hypothesize that the usage of scaffolding techniques within a search system can be an effective way to help learners achieve their learning objectives whilst searching. As such, this paper investigates the incorporation of scaffolding into a search system employing three different strategies (as well as a control condition): (I) AQE_{SC}, the automatic expansion of user queries with relevant subtopics; (ii) CURATED_{SC}, the presenting of a manually curated static list of relevant subtopics on the search engine result page; and (iii) FEEDBACK_{SC}, which projects real-time feedback about a user's exploration of the topic space on top of the CURATED_{SC} visualization. To investigate the effectiveness of these approaches with respect to human learning, we conduct a user study (N=126) where participants were tasked with searching and learning about topics such as `genetically modified organisms'. We find that (I) the introduction of the proposed scaffolding methods does not significantly improve learning gains. However, (ii) it does significantly impact search behavior. Furthermore, (iii) immediate feedback of the participants' learning leads to undesirable user behavior, with participants focusing on the feedback gauges instead of learning., Comment: Published At CHIIR 2021, won the best student paper

We present a uniform (and unambiguous) procedure for scaling the matter fields in implementing the conformal method to parameterize and construct solutions of Einstein constraint equations with coupled matter sources. The approach is based on a phase space representation of the space-time matter fields after a careful $n+1$ decomposition into spatial fields $B$ and conjugate momenta $\Pi_B$, which are specified directly and are conformally invariant quantities. We show that if the Einstein-matter field theory is specified by a Lagrangian which is diffeomorphism invariant and involves no dependence on derivatives of the space-time metric in the matter portion of the Lagrangian, then fixing $B$ and $\Pi_B$ results in conformal constraint equations that, for constant-mean curvature initial data, semi-decouple just as they do for the vacuum Einstein conformal constraint equations. We prove this result by establishing a structural property of the Einstein momentum constraint that is independent of the conformal method: For an Einstein-matter field theory which satisfies the conditions just stated, if $B$ and $\Pi_B$ satisfy the matter Euler-Lagrange equations, then (in suitable form) the right-hand side of the momentum constraint on each spatial slice depends only on $B$ and $\Pi_B$ and is independent of the space-time metric. We discuss the details of our construction in the special cases of the following models: Einstein-Maxwell-charged scalar field, Einstein-Proca, Einstein-perfect fluid, and Einstein-Maxwell-charged dust. In these examples we find that our technique gives a theoretical basis for scaling rules, such as those for electromagnetism, that have worked pragmatically in the past, but also generates new equations with advantageous features for perfect fluids that allow direct specification of total rest mass and total charge in any spatial region., Comment: Reordered sections to help with exposition. Other minor changes

The drift method, introduced by the second author, provides a new formulation of the Einstein constraint equations, either in vacuum or with matter fields. The natural of the geometry underlying this method compensates for its slightly greater analytic complexity over, say, the conformal or conformal thin sandwich methods. We review this theory here and apply it to the study of solutions of the constraint equations with non-constant mean curvature. We show that this method reproduces previously known existence results obtained by other methods, and does better in one important regard. Namely, it can be applied even when the underlying metric admits conformal Killing (but not true Killing) vector fields. We also prove that the absence of true Killing fields holds generically., Comment: 32 pages; added small section on other asymptotic geometries

The conformal formulation of the Einstein constraint equations has been studied intensively since the modern version of the conformal method was first pub- lished in the early 1970s. Proofs of existence and uniqueness of solutions were limited to the constant mean curvature (CMC) case through the early 90s, with analogous results for the near-CMC case beginning to appear thereafter. In the last decade, there has been some limited progress towards understanding the properties of the conformal method for far-from-CMC solutions as well. Although it was initially conceivable that that these far-from-CMC results would lead to a solution theory for the non-CMC case that would mirror the good properties of the CMC and near-CMC cases, examples of bifurcations and of nonexistence of solutions have been since discovered. Nevertheless, the general properties of the conformal method for far-from-CMC data remain unknown. In this article we apply analytic and numerical continuation techniques to the study of the con- formal method, in an attempt to give some insight into what the solution behavior is in the far-from-CMC case in various scenarios., Comment: 22 pages, 14 figures; expanded and revised

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also show that the Yamabe class of an asymptotically Euclidean manifold is the same as the Yamabe class of its conformal compactification., Comment: 30 pages

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant mean curvature (CMC) solutions of the constraint equations, it is unknown how well it applies in the non-CMC setting, and there have been indications that it encounters difficulties there. We are therefore motivated to investigate alternative generalizations of the CMC conformal method. Introducing a densitized lapse into the ADM Lagrangian, we find that solutions of the momentum constraint can be described in terms of three parameters. The first is conformal momentum as it appears in the standard conformal method. The second is volumetric momentum, which appears as an explicit parameter in the CMC conformal method, but not in the non-CMC formulation. We have called the third parameter drift momentum, and it is the conjugate momentum to infinitesimal motions in superspace that preserve conformal class and volume form up to independent diffeomorphisms. This decomposition of solutions of the momentum constraint leads to extensions of the CMC conformal method where conformal and volumetric momenta both appear as parameters. There is more than one way to treat drift momentum, in part because of an interesting duality that emerges, and we identify three candidates for incorporating drift into a variation of the conformal method., Comment: 53 pages

The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn our attention to non-CMC slices of the smaller family of flat Kasner spacetimes. In this restricted setting we obtain a full description of the construction of certain $U^{n-1}$ symmetric slices, even in the far-from-CMC regime. Among the conformal data sets generating these slices we find that most data sets construct a single flat Kasner spacetime, but that there are also far-from-CMC data sets that construct one-parameter families of slices. Although these non-CMC families are analogues of well-known CMC one-parameter families, they differ in important ways. Most significantly, unlike the CMC case, the condition signaling the appearance of these non-CMC families is not naturally detected from the conformal data set itself. In light of this difficulty, we propose modifications of the conformal method that involve a conformally transforming mean curvature., Comment: 34 pages, 5 figures

The conformal method developed in the 1970s and the more recent Lagrangian and Hamiltonian conformal thin-sandwich methods are techniques for finding solutions of the Einstein constraint equations. We show that they are manifestations of a single conformal method: there is a straightforward way to convert back and forth between the parameters for these methods so that the corresponding solutions of the Einstein constraint equations agree. The unifying idea is the need to clearly distinguish tangent and cotangent vectors to the space of conformal classes on a manifold, and we introduce a vocabulary for working with these objects without reference to a particular representative background metric. As a consequence of these conceptual advantages, we demonstrate how to strengthen previous near-CMC existence and non-existence theorems for the original conformal method to include metrics with scalar curvatures that change sign., Comment: 34 pages; minor misprints corrected, and a reference added

We consider the alternating method of Kozlov and Maz'ya for solving the Cauchy problem for elliptic boundary-value problems. Considering the case of the Laplacian, we show that this method can be recast as a form of Landweber iteration. In addition to conceptual advantages, this observation leads to some practical improvements. We show how to accelerate Kozlov-Maz'ya iteration using the conjugate gradient algorithm, and we show how to modify the method to obtain a more practical stopping criterion., Comment: 28 pages, 4 figures

We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by taking the quotient of certain symmetric data on conformally flat tori. Specializing the model problem to a three-parameter family of conformal data we observe a number of new phenomena for the conformal and CTS methods. Within this family, we obtain a general existence theorem so long as the mean curvature does not change sign. When the mean curvature changes sign, we find that for certain data solutions exist if and only if the transverse-traceless tensor is sufficiently small. When such solutions exist, there are generically more than one. Moreover, the theory for mean curvatures changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations., Comment: 40 pages, 4 figures

We give a sufficient condition, with no restrictions on the mean curvature, under which the conformal method can be used to generate solutions of the vacuum Einstein constraint equations on compact manifolds. The condition requires a so-called global supersolution but does not require a global subsolution. As a consequence, we construct a class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, extending a recent result of Holst, Nagy, and Tsogtgerel [HNT07] which constructed similar solutions in the presence of matter. We give a second proof of this result showing that vacuum solutions can be obtained as a limit of [HNT07] non-vacuum solutions. Our principal existence theorem is of independent interest in the near-CMC case, where it simplifies previously known hypotheses required for existence., Comment: 21 pages