The maximum number of cycles in a triangular-grid billiards system with a given perimeter

Given a (simple) grid polygon $P$ in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside of $P$. We study the relationship between the perimeter $\operatorname{perim}(P)$ of $P$ and the number of different trajectories $\operatorname{cyc}(P)$ that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality $\operatorname{cyc}(P) \leq (\operatorname{perim}(P) + 2)/4$ and characterize the equality cases.Comment: 21 pages, 21 figure

Evacuating "O''- and "Y''-shaped houses on fire: the connectivity of friends-and-strangers graphs on complete multipartite graphs

For simple graphs $X$ and $Y$ on $n$ vertices, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $\sigma: V(X) \to V(Y)$, where two bijections $\sigma$ and $\sigma'$ are adjacent if and only if they agree on all but two adjacent vertices $a, b \in V(X)$ such that $\sigma(a), \sigma(b) \in V(Y)$ are adjacent in $Y$. Resolving a conjecture of Wang, Lu, and Chen, we completely characterize the connectedness of $\mathsf{FS}(X, Y)$ when $Y$ is a complete bipartite graph. We further extend this result to when $Y$ is a complete multipartite graph. We also determine when $\mathsf{FS}(X, Y)$ has exactly two connected components where $X$ is bipartite and $Y$ is a complete bipartite graph.Comment: 22 pages, 23 figure

Classifying Isometries

An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry. We then classify all isometries of the plane equipped with the $\ell_p$ metric for any $p \geq 1$ and $p = \infty$.Comment: 12 pages, 7 figure

Generalized inverses and their relations with clean decompositions

An element a in a ring R is called clean if it is the sum of an idempotent e and a unit u. Such a clean decomposition a = e + u is said to be strongly clean if eu = ue and special clean if aR eR = (0). In this paper, we prove that a is Drazin invertible if and only if there exists an idempotent e and a unit u such that an = e + u is both a strongly clean decomposition and a special clean decomposition, for some positive integer n. Also, the existence of the Moore-Penrose and group inverses is related to the existence of certain - clean decompositions.This research is supported by the Natural Science Foundation of Anhui Province (No. 1808085QA16), the Fundamental Research Funds for the Central Universities (No. JZ2018HGTB0233), China Postdoctoral Science Foundation (No. 2018M632385) and the Portuguese Funds through FCT- ‘Fundação para a Ciência e a Tecnologia’, within the project UID-MAT-00013/2013

DoctorGLM: Fine-tuning your Chinese Doctor is not a Herculean Task

The recent progress of large language models (LLMs), including ChatGPT and GPT-4, in comprehending and responding to human instructions has been remarkable. Nevertheless, these models typically perform better in English and have not been explicitly trained for the medical domain, resulting in suboptimal precision in diagnoses, drug recommendations, and other medical advice. Additionally, training and deploying a dialogue model is still believed to be impossible for hospitals, hindering the promotion of LLMs. To tackle these challenges, we have collected databases of medical dialogues in Chinese with ChatGPT's help and adopted several techniques to train an easy-deploy LLM. Remarkably, we were able to fine-tune the ChatGLM-6B on a single A100 80G in 13 hours, which means having a healthcare-purpose LLM can be very affordable. DoctorGLM is currently an early-stage engineering attempt and contain various mistakes. We are sharing it with the broader community to invite feedback and suggestions to improve its healthcare-focused capabilities: https://github.com/xionghonglin/DoctorGLM

Whole-genome transcription and DNA methylation analysis of peripheral blood mononuclear cells identified aberrant gene regulation pathways in systemic lupus erythematosus

Hypermethylated CpG sites in PBMC of SLE patients comparing normal controls by genome-wide DNA methylation analysis. Three group comparisons, SLE LN+ vs. NC (S3-1), SLE LN− vs. NC (S3-2), and SLE (LN− and LN+) vs. NC (S3-3), were performed to identify hypermethylated DNA CpG sites by fold change (FC) > 1.2 and q value < 5% within each group. The 1813 common hypermethylated sites identified in each of the three group comparisons were then selected and listed in S3-4. (XLSX 814 kb

Transmission of H7N9 influenza virus in mice by different infective routes.

BackgroundOn 19 February 2013, the first patient infected with a novel influenza A H7N9 virus from an avian source showed symptoms of sickness. More than 349 laboratory-confirmed cases and 109 deaths have been reported in mainland China since then. Laboratory-confirmed, human-to-human H7N9 virus transmission has not been documented between individuals having close contact; however, this transmission route could not be excluded for three families. To control the spread of the avian influenza H7N9 virus, we must better understand its pathogenesis, transmissibility, and transmission routes in mammals. Studies have shown that this particular virus is transmitted by aerosols among ferrets.MethodsTo study potential transmission routes in animals with direct or close contact to other animals, we investigated these factors in a murine model.ResultsViable H7N9 avian influenza virus was detected in the upper and lower respiratory tracts, intestine, and brain of model mice. The virus was transmissible between mice in close contact, with a higher concentration of virus found in pharyngeal and ocular secretions, and feces. All these biological materials were contagious for naïve mice.ConclusionsOur results suggest that the possible transmission routes for the H7N9 influenza virus were through mucosal secretions and feces