We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0 < q < 1. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1.Our algorithm and the proof build on the recent results of Dinur, Kaufman, Mass and Oppenheim [KM17; DK17; KO18] who show that high dimensional walks on simplicial complexes mix rapidly if the corresponding localized random walks on 1-skeleton of links of all complexes are strong spectral expanders. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if X is a pure simplicial complex with positive weights on its maximal faces, we can associate with X a multiaffine homogeneous polynomial p X such that the eigenvalues of the localized random walks on X correspond to the eigenvalues of the Hessian of derivatives of p X .

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We say a probability distribution µ is spectrally independent if an associated correlation matrix has a bounded largest eigenvalue for the distribution and all of its conditional distributions. We prove that if µ is spectrally independent, then the corresponding high dimensional simplicial complex is a local spectral expander. Using a line of recent works on mixing time of high dimensional walks on simplicial complexes [KM17; DK17; KO18; AL19], this implies that the corresponding Glauber dynamics mixes rapidly and generates (approximate) samples from µ.As an application, we show that natural Glauber dynamics mixes rapidly (in polynomial time) to generate a random independent set from the hardcore model up to the uniqueness threshold. This improves the quasi-polynomial running time of Weitz's deterministic correlation decay algorithm [Wei06] for estimating the hardcore partition function, also answering a long-standing open problem of mixing time of Glauber dynamics [LV97; LV99; DG00; Vig01; Eft+16].

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Objectives The aim of study was to investigate the anticancer activities of Ivermectin (IVM) and the possible mechanisms in cells level via cell proliferation inhibition, apoptosis and migration inhibition in model cancer cell HeLa. Materials and methods The MTT assay was used to study the inhibitory effect of IVM on the proliferation of Hela cells, and the cell cycle was analysed by flow cytometry. The neutral comet assay was used to study the DNA damage. The presence of apoptosis was confirmed by DAPI nuclear staining and flow cytometry. Changes in mitochondrial membrane potential and reactive oxygen species (ROS) levels were determined using Rhodamine 123 staining and DCFH‐DA staining. Western blot analysis for apoptosis‐related proteins was carried out. We use scratch test to analyse the antimigration potential of IVM . Results Ivermectin can inhibit the viability of HeLa cells significantly. In addition, treatment with IVM resulted in cell cycle arrest at the G1/S phase which partly account for the suppressed proliferation. Typical apoptosis morphological changes were shown in IVM treatment cells including DNA fragmentation and chromatin condensation. At the same time, the results of flow cytometry analysis showed that the number of apoptotic cells increased significantly with the increase of IVM concentration. Moreover, we observed that the mitochondrial membrane potential collapses and the ratio of Bax/Bcl‐2 in the cytoplasm increases, which induces cytochrome c release from the mitochondria to the cytoplasm, activates caspase‐9/‐3 and finally induces apoptosis. We also found that IVM can significantly increase intracellular ROS content. At the same time, we determined that IVM can significantly inhibit the migration of HeLa cells. Conclusions Our experimental results show that IVM might be a new potential anticancer drug for therapy of human cancer.

We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. ( 2020) and shows O(n log n) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity λ, we establish O(n log n) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree ∆ when λ < λ c (∆) where λ c (∆) is the critical point for the uniqueness/non-uniqueness phase transition on the ∆-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n log n) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(n log n) mixing for q-colorings of triangle-free graphs of maximum degree ∆ when the number of colors satisfies q > α∆ where α ≈ 1.763, and O(m log n) mixing for generating random matchings of any graph with bounded degree and m edges.Our approach is based on two steps. First, we show that the approximate tensorization of entropy (i.e., factorizing entropy into single vertices), which is a key step for establishing the modified log-Sobolev inequality in many previous works, can be deduced from entropy factorization into blocks of fixed linear size. Second, we adapt the local-to-global scheme of Alev and Lau (2020) to establish such block factorization of entropy in a more general setting of pure weighted simplicial complexes satisfying local spectral expansion; this also substantially generalizes the result of Cryan et al. (2019).