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In L 2 ( R d ; C n ) , we consider a semigroup e - t A ε , t ⩾ 0 , generated by a matrix elliptic second-order differential operator A ε ⩾ 0 . Coefficients of A ε are periodic, depend on x / ε , and oscillate rapidly as ε → 0 . Approximations for e - t A ε were obtained by Suslina (Funktsional Analiz i ego Prilozhen 38(4):86-90, 2004) and Suslina (Math Model Nat Phenom 5(4):390-447, 2010) via the spectral method and by Zhikov and Pastukhova (Russ J Math Phys 13(2):224-237, 2006) via the shift method. In the present note, we give another short proof based on the contour integral representation for the semigroup and approximations for the resolvent with two-parametric error estimates obtained by Suslina (2015)., (© The Author(s) 2020.)

In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces. Some other results in this article are convergence of the exploration process of the loop ensemble (or the branch of the exploration tree) to SLE ( κ , κ - 6 ) , κ = 16 / 3 , and convergence of a generalization of this process for 4 marked points to SLE [ κ , Z ] , κ = 16 / 3 , where Z refers to a partition function. The latter SLE process is a process that can't be written as a SLE ( κ , ρ 1 , ρ 2 , … ) process, which are the most commonly considered generalizations of SLEs., (© The Author(s) 2019.)

The cycle graph introduced by Bafna and Pevzner is an important tool for evaluating the distance between two genomes, that is, the minimal number of rearrangements needed to transform one genome into another. We interpret this distance in topological terms and relate it to the random matrix theory. Namely, the number of genomes at a given 2-break distance from a fixed one (the Hultman number) is represented by a coefficient in the genus expansion of a matrix integral over the space of complex matrices with the Gaussian measure. We study generating functions for the Hultman numbers and prove that the two-break distance distribution is asymptotically normal.

As an RNA virus, hepatitis C virus (HCV) is able to rapidly acquire drug resistance, and for this reason the design of effective anti-HCV drugs is a real challenge. The HCV subgenomic replicon-containing cells are widely used for experimental studies of the HCV genome replication mechanisms, for drug testing in vitro and in studies of HCV drug resistance. The NS3/4A protease is essential for virus replication and, therefore, it is one of the most attractive targets for developing specific antiviral agents against HCV. We have developed a stochastic model of subgenomic HCV replicon replication, in which the emergence and selection of drug resistant mutant viral RNAs in replicon cells is taken into account. Incorporation into the model of key NS3 protease mutations leading to resistance to BILN-2061 (A156T, D168V, R155Q), VX-950 (A156S, A156T, T54A) and SCH 503034 (A156T, A156S, T54A) inhibitors allows us to describe the long term dynamics of the viral RNA suppression for various inhibitor concentrations. We theoretically showed that the observable difference between the viral RNA kinetics for different inhibitor concentrations can be explained by differences in the replication rate and inhibitor sensitivity of the mutant RNAs. The pre-existing mutants of the NS3 protease contribute more significantly to appearance of new resistant mutants during treatment with inhibitors than wild-type replicon. The model can be used to interpret the results of anti-HCV drug testing on replicon systems, as well as to estimate the efficacy of potential drugs and predict optimal schemes of their usage.