Random walks generated by Ewens distribution on the symmetric group

This paper studies Markov chains on the symmetric group $S_n$ where the transition probabilities are given by Ewens distribution with parameter $\theta>0$. The eigenvalues are identified to be content polynomials of partitions divided by n$th$ rising factorial of $\theta.$ The mixing time analysis is carried out for two cases: If $\theta$ is a constant integer, the chain is fast enough that the mixing time does not depend on $n.$ If $\theta=n$, the chain exhibits a total variation cutoff at $\frac{\log n}{\log 2}$ steps.Comment: 24 pages, 2 figure

A rare cause of peripheral facial paralysis in childhood in our country: lyme disease

Lyme disease is a zoonosis caused by Spirochetes called Borrelia burgdorferi, involving several areas, such as the skin, heart and central nervous system. In this case report, we present a 10-year-old male who had complaints of fever, extensive oral aphthae, perioral dried sores, rash, blurred vision and peripheral facial paralysis, and was diagnosed with Lyme disease. In this report, we want to emphasize that Lyme disease should be kept in mind for differential diagnosis in patients with fever and peripheral facial paralysis