An asymptotic relationship between coupling methods for stochastically modeled population processes

This paper is concerned with elucidating a relationship between two common coupling methods for the continuous time Markov chain models utilized in the cell biology literature. The couplings considered here are primarily used in a computational framework by providing reductions in variance for different Monte Carlo estimators, thereby allowing for significantly more accurate results for a fixed amount of computational time. Common applications of the couplings include the estimation of parametric sensitivities via finite difference methods and the estimation of expectations via multi-level Monte Carlo algorithms. While a number of coupling strategies have been proposed for the models considered here, and a number of articles have experimentally compared the different strategies, to date there has been no mathematical analysis describing the connections between them. Such analyses are critical in order to determine the best use for each. In the current paper, we show a connection between the common reaction path (CRP) method and the split coupling (SC) method, which is termed coupled finite differences (CFD) in the parametric sensitivities literature. In particular, we show that the two couplings are both limits of a third coupling strategy we call the "local-CRP" coupling, with the split coupling method arising as a key parameter goes to infinity, and the common reaction path coupling arising as the same parameter goes to zero. The analysis helps explain why the split coupling method often provides a lower variance than does the common reaction path method, a fact previously shown experimentally.Comment: Edited Section 4.

A Decimation-in-Frequency Fast-Fourier Transform for the Symmetric Group

A Discrete Fourier Transform (DFT) changes the basis of a group algebra from the standard basis to a Fourier basis. An efficient application of a DFT is called a Fast Fourier Transform (FFT). This research pertains to a particular type of FFT called Decimation in Frequency (DIF). An efficient DIF has been established for commutative algebra; however, a successful analogue for non-commutative algebra has not been derived. However, we currently have a promising DIF algorithm for CSn called Orrison-DIF (ODIF). In this paper, I will formally introduce the ODIF and establish a bound on the operation count of the algorithm

Extremely broad hysteresis in the magnetization process of α-Dy2S3 single crystal induced by high field cooling

α-Dy2S3 possesses orthorhombic crystal structure having two crystallograpically inequivalent Dy sites. Magnetization process of α-Dy2S3 single crystal after cooling in the high magnetic field of 18 T has been investigated. The magnetization under the field of 18 T along the α-axis on the cooling process from 150 K shows step-like rises at 70 and 40 K and reaches about 9 μB per one Dy3+ at 1.5 K. This value, which corresponds to 90 % of full saturation moment of Dy3+, is much larger than 6 μB obtained at the same conditions after cooling in no magnetic field (zero-field cooling; ZFC). After cooling to 1.5 K, the magnetization while decreasing field shows abrupt drops at 3.0 and 1.7 T, and then comes to 0 μB at 0 T. Subsequently, while increasing field, the magnetization demonstrates a similar curve to that obtained after ZFC without step-like rise below 13.1 T. At μ0H = 13.1 T, the magnetization rises suddenly and agrees with the curve for the decreasing process. This irreversible magnetization process yields extremely broad hysteresis having width of μ0ΔH = 11.4 T. Broader hysteresis and narrower one are also observed at 4.2 and 10 K, respectively