Our Work

We report two small Lean 4 formalisations, built on Mathlib, of classical theorems aboutcircles. The first is Casey’s theorem (1866), the generalisation of Ptolemy’s theorem in which thefo…

We report two small Lean 4 formalisations, built on Mathlib, of classical theorems aboutcircles. The first is Casey’s theorem (1866), the generalisation of Ptolemy’s theorem in which thefour points on a circle Γ are replaced by four circles tangent to Γ and distances are replaced bylengths of common tangents. We prove it in every real inner product space, for circles internallytangent to Γ and, through a signed radius, for any mixture of internally and externally tangentcircles; with all radii zero it specialises to Mathlib’s Ptolemy theorem, which the proof reuses.The second is Fuss’s theorem (1798): if a quadrilateral is inscribed in a circle of radius R andits four side lines are tangent to a circle of radius r whose centre is at distance d̸ = R from thefirst centre, with r > 0, then (R2 − d2)2 = 2r2(R2 + d2). We prove it in a Euclidean plane,together with the converse in the form of Poncelet’s porism for quadrilaterals: under Fuss’srelation every non-stalling four-step chain of the tangent construction (with the hypotheses ofTheorem 9) closes, and if the inner circle lies inside the outer one, every point of the outer circleis a vertex of such a quadrilateral. Both developments are short (about 200 and 870 lines),contain no sorry and use only Lean’s standard axioms. As far as we could determine, neithertheorem had a formal proof before

We study the minimum weight cycle problem (MWC) and the minimum weight triangle problem (MWT) in the quantum query model, for n-vertex graphs with integer edge weights in [M] given by their adjacency …

We study the minimum weight cycle problem (MWC) and the minimum weight triangle problem (MWT) in the quantum query model, for n-vertex graphs with integer edge weights in [M] given by their adjacency matrix. Roditty and Vassilevska Williams showed that the two problems are classically equivalent. We show that an exact minimum weight cycle of an undirected graph can be found with $\tilde{O}(n^{19/12})$ queries, while $\Omega(n^{3/2})$ queries are necessary. The algorithm learns only the $n^{7/6}$ lightest vertex pairs, uses quantum counting to locate weight thresholds above them, applies a strict form of the Moore bound to bound the optimum and the size of each weight class, and searches the classes for the few heavy edges of an optimal cycle with nested quantum walks. A simpler version gives $\tilde{O}(n^{5/3})$ queries, and searching for fewer heavy edges trades queries for approximation, with a 2-approximation in $\tilde{O}(n^{8/5})$ queries and a tight $O(\log n)$-approximation in $\tilde{O}(n^{3/2})$ queries. In contrast, MWT costs $O(n^{9/7}\log M)$ queries, so the classical equivalence becomes a polynomial separation. For cycles of constant length we give explicit exponents, such as $\tilde{O}(n^{125/83})$ for the minimum weight 5-cycle, and we prove that deciding whether a graph has a k-cycle of total weight exactly zero requires $\Omega(n^{2-2/(\lfloor k/2\rfloor+1)})$ queries for every constant $k\ge 4$. Finally, non-adaptive learning graphs cannot beat $n^2$ for cycles of about $\log n$ edges, and we conjecture that exact MWC on directed graphs requires $n^{2-o(1)}$ queries.

Short description (English) "Código de alta" (Discharge Code) is an in-person educational escape room for teaching Enhanced Recovery After Surgery (ERAS) in colorectal surgery, based on the Spa…

Short description (English) "Código de alta" (Discharge Code) is an in-person educational escape room for teaching Enhanced Recovery After Surgery (ERAS) in colorectal surgery, based on the Spanish RICA pathway (2026 update) and ERAS Society guidelines. It is designed for surgical, anaesthesiology and nursing residents, and for final-year medical students. Teams of 4 to 6 participants follow a patient with caecal adenocarcinoma from the preoperative clinic to discharge. In 60 minutes they solve challenges on patient education, multimodal prehabilitation, Patient Blood Management, patient safety, intraoperative care (including laparoscopic box-trainer skills) and postoperative care. Each solved challenge adds a piece to a wall mural that rebuilds the pathway and reveals the final discharge code. The resource includes the clinical case, game mechanics, scripts, printable materials, a facilitator solution sheet, a structured debriefing guide and a Kirkpatrick-based evaluation plan.  Descripción breve (español) «Código de alta» es un escape room educativo presencial para enseñar la vía de Recuperación Intensificada del Adulto (vía RICA, actualización 2026) y las recomendaciones ERAS en cirugía colorrectal. Está dirigido a residentes de Cirugía General, Anestesiología y enfermería quirúrgica, y a estudiantes de 5.º y 6.º de Medicina. Equipos de 4 a 6 participantes acompañan a un paciente con adenocarcinoma de ciego desde la consulta prequirúrgica hasta el alta. Durante 60 minutos resuelven retos sobre información al paciente, prehabilitación multimodal, Patient Blood Management, seguridad del paciente, medidas intraoperatorias (con ejercicios de habilidad laparoscópica en endotrainer) y cuidados postoperatorios. Cada reto superado aporta una pieza a un mural que reconstruye la vía clínica y forma el código de alta final. El documento incluye el caso clínico, la mecánica del juego, los guiones, los materiales imprimibles, la hoja de soluciones para el coordinador, una propuesta de debriefing estructurado y la evaluación según el modelo de Kirkpatrick.  

We formalise the Cayley–Bacharach theorem for plane cubics in Lean 4 with Mathlib: underhypotheses made precise below, if two cubics meet in nine points, every cubic through eight ofthem passes …

We formalise the Cayley–Bacharach theorem for plane cubics in Lean 4 with Mathlib: underhypotheses made precise below, if two cubics meet in nine points, every cubic through eight ofthem passes through the ninth. We prove three versions. The first assumes that the eight pointsimpose independent conditions on cubics; we prove the exact criterion for this, in both directions:no five of the points are collinear and the eight do not lie on a conic. The second assumes that thetwo cubics have no common line or conic component. The third is the textbook form, in whichthe two cubics have exactly nine common zeros. None of the three assumes an algebraicallyclosed field. The first two hold over every field. The third holds over every infinite field, and afinite field K exactly when |K| /∈ {4, 5, 7}: we prove it for every other finite field, and it is falseover F4, F5 and F7, for which we give explicit counterexamples, checked by computer but notformalised. The line lemma behind all three (a cubic vanishing on a line contains it) needs afield with at least three elements, and we formalise a counterexample over F2. As applicationswe derive Pascal’s and Pappus’s theorems from Cayley–Bacharach under their natural incidenceand distinctness hypotheses, for fields with more than ten elements. We also prove that a conicis a line pair if and only if its Gram determinant vanishes, over an algebraically closed field ofcharacteristic different from two. The development has about 3,600 lines and uses no sorry andno axioms beyond Lean’s standard three. The mathematics is classical. We found no earlierformalisation of the theorem in the sources we searched (Section 8).

EvalHAQ is an application for registering patients, completing the HAQ questionnaire and tracking their progress through scores and charts. It generates PDF reports, exports data to Excel and supports…

EvalHAQ is an application for registering patients, completing the HAQ questionnaire and tracking their progress through scores and charts. It generates PDF reports, exports data to Excel and supports Spanish and English. All information is stored locally on the computer.   ___________ EvalHAQ es una aplicación para registrar pacientes, realizar el cuestionario HAQ y analizar su evolución mediante resultados y gráficas. Permite generar informes PDF, exportar datos a Excel y trabajar en español o inglés. Toda la información se almacena localmente en el ordenador.

HCM_1000 V2 is an audited VRPTW dataset for Ho Chi Minh City. The archive contains 56 instances, each with one depot and 1,000 customers; eight directed 1,001 × 1,001 road distance/time matrices…

HCM_1000 V2 is an audited VRPTW dataset for Ho Chi Minh City. The archive contains 56 instances, each with one depot and 1,000 customers; eight directed 1,001 × 1,001 road distance/time matrices; node-order mappings; coordinates and geographic data; routing provenance; construction parameters; and independently validated feasible witnesses for all 56 instances. Feasibility does not imply optimality. Validation: extract the archive and run python code/verify_release.py --root . inside HCM_1000_V2 (Python 3 standard library). See DATA_DICTIONARY.md and PROVENANCE.md for units, assumptions and limitations. The complete solver-comparison experiment pipeline is not included. Historical map images and notes with incomplete rights information are excluded. Component licences: database rights — ODbL 1.0; Python code — MIT; newly authored documentation — CC BY 4.0. LICENSING.md defines the exact scope. OpenStreetMap-derived components require attribution to © OpenStreetMap contributors; upstream rights remain applicable. GitHub release: https://github.com/ltlong-SGU/HCM_1000/releases/tag/v2.0Archive SHA-256: 8af1405664af7c05089bd032c07dc94e3eeabf1b8807bc37e8fde7b44019c705

Replication package for the book "The Monitoring Gap: Selective Accountability and the Representativeness of BRICS Compliance Scores (2009–2026)" (A. A. Merlet Echavarría, 2026; The Gap S…

Replication package for the book "The Monitoring Gap: Selective Accountability and the Representativeness of BRICS Compliance Scores (2009–2026)" (A. A. Merlet Echavarría, 2026; The Gap Series, Vol. VII). The book examines how the BRICS Research Group selects the commitments it monitors and what that selection implies for the published compliance figures. It builds a commitment-level dataset of the 18 BRICS leaders' declarations of 2009–2026 (1,714 candidate commitments identified by a rule-based classifier derived from the Global Governance Program's Compliance Coding Manual, 2020) and links it to the 14 final compliance reports for the summits of 2011–2024. Contents:- download/: Windows scripts that download the 18 leaders' declarations, 5 secondary documents, 22 annex documents and 14 final compliance reports from the BRICS Information Centre (University of Toronto).- manifests/: MD5 hashes and file sizes of every downloaded source file, to verify that the rebuilt corpus is identical.- code/: the full pipeline in Python (text extraction, commitment classifier, extraction of priority commitments and compliance scores, database construction, and every analysis in Chapters 4 to 10: representativeness tests, coverage and permutation tests, conditional logit of selection with wild cluster bootstrap and within-summit randomization inference, inverse-probability weighting, assumption-free bounds, sensitivity analysis and robustness checks). run_all.py executes all steps in order.- data/: the coded commitment-level database (commitments_coded.csv: summit, sentence index, MD5 of the sentence, binding level, eleven compliance catalysts, topic, Africa flag and selection indicator), priority commitments, compliance scores by commitment, the ids of the blind validation sample and the output of a zero-shot NLI model used as an interim convergent check.- expected_output/key_results.txt: the results the pipeline must reproduce. Rights: the declarations and compliance reports are published by the BRICS Information Centre and the BRICS Research Group with all rights reserved and are not redistributed. The download scripts and MD5 manifests rebuild and verify the identical corpus, and rebuild_text.py restores the text of each coded commitment locally, checking it against its MD5. Code and derived data are released under CC BY 4.0. Reproducibility: the pipeline was run end to end from the raw downloaded files; the core files (sentences, priority commitments and commitment database) are rebuilt byte-identical, and all reported results are reproduced. Limitation: the precision of the commitment classifier against a human coder has not been estimated in this version; a blind sample of 80 sentences is prepared for version 2. AI use declaration: code drafting, data processing and verification of bibliographic records were assisted by Claude (Anthropic). All results are produced by the executable scripts included in this package, and all research decisions were made and reviewed by the author.

The development of communicative competence is one of the essential conditions for the successful academic, social, and personal development of students in inclusive educational environments. This iss…

The development of communicative competence is one of the essential conditions for the successful academic, social, and personal development of students in inclusive educational environments. This issue becomes particularly significant for blind and visually impaired students, whose opportunities for accessing visual information and participating in conventional communication practices may be limited. The purpose of this study is to identify and substantiate innovative pedagogical approaches to developing the communicative competence of upper-grade students with visual impairments in inclusive schools. The study employed theoretical analysis, observation, questionnaires, interviews, pedagogical experimentation, and comparative analysis. Particular attention was given to interactive learning, cooperative activities, audio-based educational resources, assistive technologies, peer-mediated learning, role-playing, and problem-based communication tasks. The findings indicate that a systematic combination of accessible digital technologies, collaborative learning, structured communication activities, and individualized pedagogical support can contribute to the development of students’ verbal interaction, active listening, self-expression, social participation, and confidence in communication.

AURORA pipeline data

Glioblastoma (GBM) is an aggressive brain tumor with poor survival. While immune checkpoint inhibitors (ICIs) show promise in many cancers, their efficacy in GBM is limited. Epigenetic regulation, suc…

Glioblastoma (GBM) is an aggressive brain tumor with poor survival. While immune checkpoint inhibitors (ICIs) show promise in many cancers, their efficacy in GBM is limited. Epigenetic regulation, such as DNA methylation, may influence immune checkpoint gene expression and contribute to therapy resistance. This study investigated the association between DNA methylation and expression of immune checkpoint genes in GBM, identified genes potentially contributing to resistance against ICIs, and evaluated the potential of combining ICIs with demethylating agents. Data were collected from PubMed and TCGA, and 32 immune checkpoint genes were intersected across datasets. Pearson correlation analysis between methylation and expression was performed in R, followed by pathway enrichment analysis using Enrichr and protein–protein interaction network analysis using STRING. Twelve genes showed the strongest negative correlations between DNA methylation and gene expression. These genes were enriched in immune-related pathways, including B cell receptor signaling, T cell receptor signaling, and negative regulation of lymphocyte proliferation, cytokine production and interferon production. The protein–protein interaction network highlighted LILRB1 and LILRB4 as potential immune checkpoint biomarkers in GBM. Overall, DNA methylation inversely affects immune checkpoint gene expression in glioblastoma, and a subset of strongly correlated genes represents potential targets for further research, including combination therapy with demethylating agents.