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1. Residually finite dimensional algebras and polynomial almost identities

   https://doi.org/10.1142/S0219498822500384  

   Larsen, Michael; Shalev, Aner (February 2022, Journal of Algebra and Its Applications)

   Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon, 

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2. Primitivity index bounds in free groups, and the second Chebyshev function

   https://doi.org/10.1142/S021819672350008X  

   Kapovich, Ilya; Simon, Zachary (February 2023, International Journal of Algebra and Computation)

   Motivated by results about “untangling” closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, [Formula: see text] and [Formula: see text], where [Formula: see text], and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence [Formula: see text] as [Formula: see text]. Answering a question from [17], we prove that this sequence is unbounded and that for [Formula: see text], we have [Formula: see text]. By contrast, we show that for all [Formula: see text], one has [Formula: see text]. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs. 

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3. Quantum Galois groups of subfactors

   https://doi.org/10.1142/S0129167X22500136  

   Bhattacharjee, Suvrajit; Chirvasitu, Alexandru; Goswami, Debashish (February 2022, International Journal of Mathematics)

   For a finite-index [Formula: see text] subfactor [Formula: see text], we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on [Formula: see text] in a trace-preserving fashion and fixing [Formula: see text] pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse. 

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4. ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations

   https://doi.org/10.1287/moor.2020.1078  

   Chen, Yi; Dong, Jing; Ni, Hao (May 2021, Mathematics of Operations Research)

   null (Ed.)

   Consider a fractional Brownian motion (fBM) [Formula: see text] with Hurst index [Formula: see text]. We construct a probability space supporting both B H and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one for any user-specified error bound [Formula: see text]. When [Formula: see text], we further enhance our error guarantee to the α-Hölder norm for any [Formula: see text]. This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations [Formula: see text]. Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one. Our algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently. 

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5. DNA fragmentation in a steady shear flow

   https://doi.org/10.1063/5.0109361  

   Qiao, Yiming; Ma, Zixue; Onyango, Clive; Cheng, Xiang; Dorfman, Kevin D. (September 2022, Biomicrofluidics)

   We have determined the susceptibility of T4 DNA (166 kilobase pairs, kbp) to fragmentation under steady shear in a cone-and-plate rheometer. After shearing for at least 30 min at a shear rate of [Formula: see text], corresponding to a Reynolds number of [Formula: see text] and a Weissenberg number of [Formula: see text], [Formula: see text]% of the sample is broken into a polydisperse mixture with a number-averaged molecular weight of [Formula: see text] kbp and a polydispersity index of [Formula: see text], as measured by pulsed-field gel electrophoresis (with a 95% confidence interval). The molecular weight distributions observed here from a shear flow are similar to those produced by a (dominantly extensional) sink flow of DNA and are qualitatively different than the midpoint scission observed in simple extensional flow. Given the inability of shear flow to produce a sharp coil–stretch transition, the data presented here support a model where polymers can be fragmented in flow without complete extension. These results further indicate that DNA fragmentation by shear is unlikely to be a significant issue in microfluidic devices, and anomalous molecular weight observations in experiments are due to DNA processing prior to observation in the device. 

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