On the probabilities of local behaviors in abelian field extensions

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime p of K, we determine the probability that p splits into r primes in a random G-extension of K that is unramified at p. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev's density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.Comment: 28 pages, submitte

Gauss composition over an arbitrary base

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the integers. However, such extensions have always included hypotheses on the rings, and the theorems involve only binary quadratic forms satisfying further hypotheses. We give a complete statement of the relationship between binary quadratic forms and modules for quadratic algebras over any base ring, or in fact base scheme. The result includes all binary quadratic forms, and commutes with base change. We give global geometric as well as local explicit descriptions of the relationship between forms and modules.Comment: submitte

Mission to Planet Markle: Problem-Based Learning for Teaching Elementary Students Difficult Content and Practices

Young children can struggle to learn difficult disciplinary content and important skills for practicing science. Problem-based learning (PBL) may be useful for addressing such difficulties, yet evidence to support its usefulness in elementary school-aged children is limited. We considered the role of a PBL unit in improving students’ genetics content understanding and their skills specific to creating arguments with coordinated claims, evidence, and reasoning. First- through fifth-grade students participated in a six-week PBL unit about evolution and genetics. Students worked in mixed age groups and were charged with illustrating a fictitious alien species, called markles, based on a series of facts they collected about factors expected to impact markle adaptation. This work was particularly unique in its assessment of student groups’ illustrated design solutions as arguments. Although students demonstrated weaknesses in coordinating claims and evidence overall, they were able to demonstrate success in gaining difficult genetics content knowledge and in preparing arguments with, at minimum, two components of well-constructed arguments, in most cases, providing a claim supported by reasoning. This work is informative for understanding student abilities, the potential of PBL, and considerations for its use

Programas PEP y Core knowledge: un estudio comparativo acerca del nivel de conocimiento de ciencias naturales en niños y niñas de NT1

Tesis (Educación Parvularia, Licenciado en Educación)Esta investigación tiene por objetivo establecer semejanzas en relación al conocimiento que presentan los niños y niñas de NT1 de los colegios Santiago College y San Joaquín en el área de ciencias naturales, específicamente en el núcleo de seres vivos y su entorno, considerando el uso de los programas PEP y Core Knowledge respectivamente y los contextos socioeconómicos en los que se encuentran cada uno. Para esto se evaluará mediante una prueba de alternativas que se llevará a cabo de forma oral, para posteriormente comparar los resultados arrojados por ésta. De esta forma se podrá ver la importancia de las ciencias naturales para el desarrollo del preescolar. Asimismo, con el fin de tener una base teórica sólida, esta investigación se respalda en diversas fuentes bibliográficas principalmente; “Bases curriculares de la educación parvularia”, “Metodología de la investigación”, “Aprendizajes Iniciales desde las Ciencias”, entre otros

Locally performed postoperative circulating tumour DNA testing performed during routine clinical care to predict recurrence of colorectal cancer

Background: Identifying patients at high risk for colorectal cancer recurrence is essential for improving prognosis. In the postoperative period, circulating tumour DNA (ctDNA) has been demonstrated as a significant prognostic indicator of recurrence. These results have been obtained under the strict rigours of clinical trials, but not validated in a real-world setting using in-house testing. We report the outcomes of locally performed postoperative ctDNA testing conducted during routine clinical care and the association with the recurrence of colorectal cancer. Methods: We recruited 36 consecutive patients with newly diagnosed colorectal cancer between 2018 and 2020. Postoperative plasma samples were collected at the first outpatient review following resection. Tumour-informed ctDNA analysis was performed using droplet digital polymerase chain reaction or targeted next-generation sequencing. Results: At the time of surgery, there were 24 patients (66.7%) with localized cancer, nine (25%) with nodal spread, and three (8.3%) with metastatic disease. The median time from surgery to plasma sample donation was 22 days (IQR 20–28 days). At least one somatic mutation was identified in primary tumour tissue for 28 (77.8%) patients. Postoperative ctDNA was detected in five patients (13.9%). The median duration of follow-up was 32.0 months (IQR 27.2–38.1 months). Two patients (5.56%) developed metastatic recurrence. However, neither had detectable postoperative ctDNA. There were no instances of loco-regional recurrence. Conclusion: Analysis of postoperative ctDNA testing can be performed locally, however this study did not reproduce the adverse association between detectable postoperative ctDNA and the development of colorectal cancer recurrence seen in clinical trials

\u27New Conversations About Integrated Employment\u27 Webinars Shine a Fresh Light On Issues In Our Field

This poster session highlights creative thinking about employment supports. The RRTC on Advancing Employment for Individuals with IDD has launched an interactive and dynamic webinar series with a twist ? to expose participants to new ideas, provoke reactions, and inspire them to think differently about such topics as Employment Professionals as Leaders for Change; the Real Meaning of Informed Choice; and Reframing the Benefits Conversation Around Financial Well-Being