We give an enumeration of all positive definite primitive $ \mathbb{Z} $ -lattices in dimension $n\geq 3$ whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith–Minkowski–Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson.

We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson’s classifications in other dimensions. A list of all single-class primitive $ \mathbb{Z} $ -lattices has been compiled and incorporated into the Catalogue of Lattices.

For a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger’s theorem relates the critical value of the $\mathrm{L} $ -function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger’s recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our ‘simplified Waldspurger’ by giving several examples.

We describe an algorithm for enumerating the set of level one systems of Hecke eigenvalues arising from modular forms (mod  $p$ ).

Supplementary materials are available with this article.

We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck’s dessins d’enfants. In the particular case of the index 36 subgroups, the corresponding Calabi–Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over ${ \mathbb{P} }^{1} $ as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.

Self-identification with peer crowds (jocks, popular kids, druggies, etc.) has an important influence on adolescent substance use behavior. However, little is known about the impact of the shared nature of crowd identification on different stages of adolescent drinking behavior, or the way crowd identification interacts with participation in school-sponsored sports activities. This study examines drinking influences from (1) peers with shared crowd identities, and (2) peers who jointly participate in organized sports at their school (activity members). This study introduces a new network analytic approach that can disentangle the effects of crowd identification and sports participation on individual behavior. Using survey data from adolescents in five high schools in a predominantly Hispanic/Latino district (N = 1,707), this paper examines the association between social influences and each stage of drinking behavior (intention to drink, lifetime, past-month, and binge drinking) by conducting an ordinal regression analysis. The results show that both shared identities and joint participation were associated with all stages of drinking, controlling for friends' influence. Additionally, shared identification overlapped with joint participation was associated with more frequent drinking. Related policy implications are discussed.

Entailment recognition is a primary generic task in natural language inference, whose focus is to detect whether the meaning of one expression can be inferred from the meaning of the other. Accordingly, many NLP applications would benefit from high coverage knowledgebases of paraphrases and entailment rules. To this end, learning such knowledgebases from the Web is especially appealing due to its huge size as well as its highly heterogeneous content, allowing for a more scalable rule extraction of various domains. However, the scalability of state-of-the-art entailment rule acquisition approaches from the Web is still limited. We present a fully unsupervised learning algorithm for Web-based extraction of entailment relations. We focus on increased scalability and generality with respect to prior work, with the potential of a large-scale Web-based knowledgebase. Our algorithm takes as its input a lexical–syntactic template and searches the Web for syntactic templates that participate in an entailment relation with the input template. Experiments show promising results, achieving performance similar to a state-of-the-art unsupervised algorithm, operating over an offline corpus, but with the benefit of learning rules for different domains with no additional effort.

This paper presents a variable neighborhood search (VNS) algorithm that is specially designed for the blockmodeling of two-mode binary network matrices in accordance with structural equivalence. Computational results for 768 synthetic test networks revealed that the VNS heuristic outperformed a relocation heuristic (RH) and a tabu search (TS) method for the same problem. Next, the three heuristics were applied to two-mode network data pertaining to the votes of member countries on resolutions in the United Nations General Assembly. A comparative analysis revealed that the VNS heuristic often provided slightly better criterion function values than RH and TS, and that these small differences in criterion function values could sometimes be associated with substantial differences in the actual partitions obtained. Overall, the results suggest that the VNS heuristic is a promising approach for blockmodeling of two-mode binary networks. Recommendations for extensions to stochastic blockmodeling applications are provided.

Biologists depend on visual representations, and their use of diagrams has drawn the attention of philosophers, historians, and sociologists interested in understanding how these images are involved in biological reasoning. These studies, however, proceed from identification of diagrams on the basis of their spare visual appearance, and do not draw on a foundational theory of the nature of diagrams as representations. This approach has limited the extent to which we understand how these diagrams are involved in biological reasoning. In this paper, I characterize three different kinds of figures among those previously identified as diagrams. The features that make these figures distinctive as representational types, furthermore, illuminate the ways in which they are involved in biological reasoning.

The concept of manipulative abduction is devoted to capture the role of action in many interesting cognitive situations: action provides otherwise unavailable information that enables the agent to solve problems by starting and performing a suitable abductive process of generation or selection of hypotheses. We observe that many external things, usually inert from an epistemological point of view, can be transformed into epistemic mediators. I will present some details derived from the history of the discovery of the non-Euclidean geometries that illustrate the relationships between strategies for anomaly resolution and visual thinking. Geometrical diagrams are external representations that play both a mirror role (to externalize rough mental models) and an unveiling role (as gateways to imaginary entities). I describe them as epistemic mediators able to perform various explanatory, non-explanatory, and instrumental abductive tasks (discovery of new properties or new propositions/hypotheses, provision of suitable sequences of models as able to convincingly verifying theorems, etc.). I am also convinced that they can be exploited and studied in everyday non-mathematical applications also to the aim of promoting new trends in artificial intelligence modeling of various aspects of hypothetical reasoning: finding routes, road signs, buildings maps, for example, in connection with various zooming effects of spatial reasoning. I also think that the cognitive activities of optical, mirror, and unveiling diagrams can be studied in other areas of manipulative and model-based reasoning, such as the ones involving creative, analogical, and spatial inferences, both in science and everyday situations so that this can extend the epistemological, computational, and the psychological theory.

We use data on a real, large-scale social network of 27 million individuals interacting daily, together with the day-by-day adoption of a new mobile service product, to inform, build, and analyze data-driven simulations of the effectiveness of seeding (network targeting) strategies under different social conditions. Three main results emerge from our simulations. First, failure to consider homophily creates significant overestimation of the effectiveness of seeding strategies, casting doubt on conclusions drawn by simulation studies that do not model homophily. Second, seeding is constrained by the small fraction of potential influencers that exist in the network. We find that seeding more than 0.2% of the population is wasteful because the gain from their adoption is lower than the gain from their natural adoption (without seeding). Third, seeding is more effective in the presence of greater social influence. Stronger peer influence creates a greater than additive effect when combined with seeding. Our findings call into question some conventional wisdom about these strategies and suggest that their overall effectiveness may be overestimated.