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Full-Text Articles in Physical Sciences and Mathematics
Comparing The Strength Of Diagonally Non-Recursive Functions In The Absence Of Σ02 Induction, Francois G. Dorais, Jeffry L. Hirst, Paul Shafer
Comparing The Strength Of Diagonally Non-Recursive Functions In The Absence Of Σ02 Induction, Francois G. Dorais, Jeffry L. Hirst, Paul Shafer
Dartmouth Scholarship
We prove that the statement "there is a k such that for every f there is a k-bounded diagonally non-recursive function relative to f" does not imply weak K\"onig's lemma over RCA0+BΣ02. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally non-recursive function computes a 2-bounded diagonally non-recursive function may fail in the absence of IΣ02.
A Fast Algorithm For Simulating Multiphase Flows Through Periodic Geometries Of Arbitrary Shape, Gary R. Marple, Alex Barnett, Adrianna Gillman, Shravan Veerapaneni
A Fast Algorithm For Simulating Multiphase Flows Through Periodic Geometries Of Arbitrary Shape, Gary R. Marple, Alex Barnett, Adrianna Gillman, Shravan Veerapaneni
Dartmouth Scholarship
This paper presents a new boundary integral equation (BIE) method for simulating particulate and mul- tiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system—multiple vesicles suspended in a periodic channel of arbitrary shape—to describe the numerical method and test its performance. Rather than relying on the periodic Green’s function as classical BIE methods do, the method combines the free-space Green’s function with a small auxiliary basis, and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms, and handle a …
Robust And Efficient Solution Of The Drum Problem Via Nyström Approximation Of The Fredholm Determinant, Lin Zhao, Alex Barnett
Robust And Efficient Solution Of The Drum Problem Via Nyström Approximation Of The Fredholm Determinant, Lin Zhao, Alex Barnett
Dartmouth Scholarship
The “drum problem''---finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition---has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary integral equations are appealing for large-scale problems, yet certain difficulties have limited their use. We introduce the following two ideas to remedy this: (1) We solve the resulting nonlinear eigenvalue problem using Boyd's method for analytic root-finding applied to the Fredholm determinant, and we show that this is many times faster than the usual iterative minimization of a singular value. (2) We fix the problem of spurious exterior resonances …
Robust Fast Direct Integral Equation Solver For Quasi-Periodic Scattering Problems With A Large Number Of Layers, Min Hyung Cho, Alex H. Barnett
Robust Fast Direct Integral Equation Solver For Quasi-Periodic Scattering Problems With A Large Number Of Layers, Min Hyung Cho, Alex H. Barnett
Dartmouth Scholarship
We present a new boundary integral formulation for time-harmonic wave diffraction from two-dimensional structures with many layers of arbitrary periodic shape, such as multilayer dielectric gratings in TM polarization. Our scheme is robust at all scattering parameters, unlike the conventional quasi-periodic Green’s function method which fails whenever any of the layers approaches a Wood anomaly. We achieve this by a decomposition into near- and far-field contributions. The former uses the free-space Green’s function in a second-kind integral equation on one period of the material interfaces and their immediate left and right neighbors; the latter uses proxy point sources and small …