Solving DSGE Models with a Nonlinear Moving Average

We introduce a nonlinear infinite moving average as an alternative to the standard state-space policy function for solving nonlinear DSGE models. Perturbation of the nonlinear moving average policy function provides a direct mapping from a history of innovations to endogenous variables, decomposes the contributions from individual orders of uncertainty and nonlinearity, and enables familiar impulse response analysis in nonlinear settings. When the linear approximation is saddle stable and free of unit roots, higher order terms are likewise saddle stable and first order corrections for uncertainty are zero. We derive the third order approximation explicitly and examine the accuracy of the method using Euler equation tests.Perturbation, nonlinear impulse response, DSGE, solution methods

Existence and Uniqueness of Perturbation Solutions to DSGE Models

We prove that standard regularity and saddle stability assumptions for linear approximations are sufficient to guarantee the existence of a unique solution for all undetermined coefficients of nonlinear perturbations of arbitrary order to discrete time DSGE models. We derive the perturbation using a matrix calculus that preserves linear algebraic structures to arbitrary orders of derivatives, enabling the direct application of theorems from matrix analysis to prove our main result. As a consequence, we provide insight into several invertibility assumptions from linear solution methods, prove that the local solution is independent of terms first order in the perturbation parameter, and relax the assumptions needed for the local existence theorem of perturbation solutions.Perturbation, matrix calculus, DSGE, solution methods, Bézout theorem; Sylvester equations

Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field

Chirality is one of the most fundamental properties of many physical, chemical and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the BZ reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media.Comment: 21 pages with 9 figures; update references; to appear in J. Chem. Phy

P2: Image Analysis and Quantification of 3D Cancer Cell Migration

Metastatic tumors are known for their ability to migrate toward circulatory apparatus and detach from the primary tumor. Generally, metastasis is quantified in vitro using migration assays that are normally measured in two dimensions (2D). Threedimensional (3D) migration assays can better mimic cancers by providing similar microenvironments to those observed in vivo. Imaging 3D cell cultures requires multiple 2D images stacked along a Z-axis; however, imaged cells would be in-focus at varied z-positions at different time points due to the characteristics of cell migration. Our goal in this study was to analyze in-focus cell images and quantify cell migration in 3D in high throughput. Briefly, Hep3B human hepatoma cell line in alginate was printed on top of a layer of chemoattractants in a microwell chip and cultured over time to model hepatocellular carcinoma. Acquired cell images were analyzed using a Fast Fourier Transform (FFT) to create a histogram of pixel brightness variation within an image. We selected a specific frequency range that would correspond to a sharp change in pixel brightness, a spheroid\u27s edge, while the rest was subtracted to delete out-of-focus cells. In-focus cell images were recreated by reverse FFT, and ImageJ macros have been used to calculate the brightness of each corrected image in our 3D culture. By correlating pixel brightness to cell number, it allowed us to calculate the average position of all the cells in our 3D culture, based on brightness and z-position of the cell image. By measuring the change in average position over time, we created a quantifiable method to measure cell migration in 3D.https://engagedscholarship.csuohio.edu/u_poster_2017/1054/thumbnail.jp

P2: Image Analysis and Quantification of 3D Cancer Cell Migration

Metastatic tumors are known for their ability to migrate toward circulatory apparatus and detach from the primary tumor. Generally, metastasis is quantified in vitro using migration assays that are normally measured in two dimensions (2D). Threedimensional (3D) migration assays can better mimic cancers by providing similar microenvironments to those observed in vivo. Imaging 3D cell cultures requires multiple 2D images stacked along a Z-axis; however, imaged cells would be in-focus at varied z-positions at different time points due to the characteristics of cell migration. Our goal in this study was to analyze in-focus cell images and quantify cell migration in 3D in high throughput. Briefly, Hep3B human hepatoma cell line in alginate was printed on top of a layer of chemoattractants in a microwell chip and cultured over time to model hepatocellular carcinoma. Acquired cell images were analyzed using a Fast Fourier Transform (FFT) to create a histogram of pixel brightness variation within an image. We selected a specific frequency range that would correspond to a sharp change in pixel brightness, a spheroid\u27s edge, while the rest was subtracted to delete out-of-focus cells. In-focus cell images were recreated by reverse FFT, and ImageJ macros have been used to calculate the brightness of each corrected image in our 3D culture. By correlating pixel brightness to cell number, it allowed us to calculate the average position of all the cells in our 3D culture, based on brightness and z-position of the cell image. By measuring the change in average position over time, we created a quantifiable method to measure cell migration in 3D.https://engagedscholarship.csuohio.edu/u_poster_2017/1054/thumbnail.jp

Freestanding dielectric nanohole array metasurface for mid-infrared wavelength applications

We designed and simulated freestanding dielectric optical metasurfaces based on arrays of etched nanoholes in a silicon membrane. We showed $2\pi$ phase control and high forward transmission at mid-infrared wavelengths by tuning the dimensions of the holes. We also identified the mechanisms responsible for high forward scattering efficiency and showed that these conditions are connected with the well-known Kerker conditions already proposed for isolated scatterers. A beam deflector was designed and optimized through sequential particle swarm and gradient descent optimization to maximize transmission efficiency and reduce unwanted grating orders. Such freestanding silicon nanohole array metasurfaces are promising for the realization of silicon based mid-infrared optical elements