We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the $\ell $-adic local systems on affine spaces in characteristic $p> 0$ whose trace functions are these exponential sums. The exponential sums here are much more general than we previously were able to consider. As a byproduct, we determine the number of irreducible components of maximal dimension in certain intersections of Fermat surfaces. We also show that in any family of such local systems, say parameterized by an affine space S, there is a dense open set of S over which the geometric monodromy group of the corresponding local system is a fixed known group.

The studies of Bonaventura Cavalieri’s indivisibles by Giusti, Andersen, Mancosu and others provide a comprehensive picture of Cavalieri’s mathematics, as well as of the mathematical objections to it as formulated by Paul Guldin and other critics. Issues that have been studied in less detail concern the theological underpinnings of the contemporary debate over indivisibles, its historical roots, the geopolitical situation at the time, and its relation to the ultimate suppression of Cavalieri’s religious order. We analyze sources from the seventeenth through twenty-first centuries to investigate this relationship.

Global challenges such as climate change demand transnational responses, including from legal clinics. Building on earlier community legal clinic and international human rights clinic models, transnational legal clinics combine the objectives of legal clinics with the framework of transnational law to work across domestic and international planes. This article focuses on a Canadian–Peruvian legal clinic collaboration to research and draft an amicus curiae brief for landmark climate litigation in Peru. While the global north–south axis of collaboration raises structural challenges, adopting a transnational approach unites participants around the principle of solidarity and decentres assumptions about expertise. A transnational approach also contributes to the progressive development of law, in this case by offering insights into remedies in climate litigation. Overall, we argue that transnational legal clinic collaboration can spur participants’ reflective learning and make substantive contributions to the growing number of climate cases.

Machine learning has exhibited substantial success in the field of natural language processing (NLP). For example, large language models have empirically proven to be capable of producing text of high complexity and cohesion. However, at the same time, they are prone to inaccuracies and hallucinations. As these systems are increasingly integrated into real-world applications, ensuring their safety and reliability becomes a primary concern. There are safety critical contexts where such models must be robust to variability or attack and give guarantees over their output. Computer vision had pioneered the use of formal verification of neural networks for such scenarios and developed common verification standards and pipelines, leveraging precise formal reasoning about geometric properties of data manifolds. In contrast, NLP verification methods have only recently appeared in the literature. While presenting sophisticated algorithms in their own right, these papers have not yet crystallised into a common methodology. They are often light on the pragmatical issues of NLP verification, and the area remains fragmented. In this paper, we attempt to distil and evaluate general components of an NLP verification pipeline that emerges from the progress in the field to date. Our contributions are twofold. First, we propose a general methodology to analyse the effect of the embedding gap – a problem that refers to the discrepancy between verification of geometric subspaces, and the semantic meaning of sentences which the geometric subspaces are supposed to represent. We propose a number of practical NLP methods that can help to quantify the effects of the embedding gap. Second, we give a general method for training and verification of neural networks that leverages a more precise geometric estimation of semantic similarity of sentences in the embedding space and helps to overcome the effects of the embedding gap in practice.

Increasing numbers of Americans are affected by serious mental illness and severe substance use disorders. While funding has increased for the treatment of these conditions in recent years, increases in service needs have outstripped resources. Further, too often those living with these conditions are incarcerated, held for inordinate periods without treatment in emergency departments, and/or relegated to the streets as part of the burgeoning numbers of homeless in the United States. These conditions require innovative approaches to care that should include integrated medical care and community resources to decrease isolation and to improve the response to crises as they occur. There are numerous opportunities already in place that, used appropriately, can improve outcomes for some of our most vulnerable people and will improve community living for all. This perspective describes available resources that can better address the mental health and substance use crisis facing the American people.

The international order that the United States has for decades led and maintained is undergoing dramatic change. In this Essay, we explain that international law during this period was constituted with, and dependent on, U.S. power; that the two became (in odd-couple fashion) entwined together; and that, as the international order changes, the international legal system, its content and its architecture, will also inevitably change.

The impact of required durations of therapy for antibiotic orders at the time of order entry has not been reported. Requiring ordering clinicians to enter stop dates at the time of antibiotic order entry decreased DOT/1000 patient days for orders with empiric indication from 154 to 119 (–34.9 (–55.7 to –14)).

For the purpose of reaching a decision about potentially defective administrative action into which he is conducting an investigation, the Ombudsman may wish to compel people to engage in various sorts of testimonial activity—the furnishing of information in writing, the production of documents, the answering of questions orally. This article examines the Ombudsman’s powers in that regard, pointing to matters which may give rise to difficulties in the exercise of such powers and suggesting a number of changes to the relevant provisions. Some of the matters discussed are relevant to the information-gathering powers of other Commonwealth agencies, for example, the Taxation Commissioner and the Trade Practices Commission. Not discussed in the article is the question of excuses which can be made to avoid complying with a valid request once made, a subject which deserves its own treatment separately.

An index is proposed to measure the extent of agreement of the data of a sociometric test with another test made at an earlier time or on another test criterion. The index is used to define an index of concordance between the two tests. It is shown how the index may be used for either individuals or groups. Tests of the hypothesis that agreement is random are given for all cases and applied to an example.

The variance of the number of mutual dyads in a sociometric situation where each member of a group chooses independently and at random is derived for unrestricted numbers of choices per group member, as well as for a fixed number of choices. The distribution of the number of mutuals is considered.

A new criterion for rotation to an oblique simple structure is proposed. The results obtained are similar to that obtained by Cattell and Muerle's maxplane criterion. Since the proposed criterion is smooth it is possible to locate the local maxima using simple gradient techniques. The results of the application of the Functionplane criterion to three sets of data are given. In each case a better fit to the subjective solution was obtained using the functionplane criterion than was reported for by Hakstian for the oblimax, promax, maxplane, or the Harris-Kaiser methods.

To explain changes in the sociometric configuration of a group through time, a problem arises of the extent to which such changes may be viewed as the aggregation of part-processes occurring at the level of two-person choice structures. A possible model is a Markov chain in which three possible states are mutual choice, one-way choice, and indifference, one realization for each pair of choosing individuals in the group. Choice data for an eighth-grade classroom are fitted to this model and are used to answer questions of constancy of transition probabilities, order of the chain, and sex differences.