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In this study, we develop a hybrid model to represent membranes of biological cells and use the distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) formulation for simulating the fluid/cell interactions. The hybrid model representing the cellular structure consists of a continuum representation of the lipid bilayer, from which the bending force is(More)
Alveolar macrophages play a large role in the innate immune response of the lung. However, when these highly immune-regulatory cells are unable to eradicate pathogens, the adaptive immune system, which includes activated macrophages and lymphocytes, particularly T cells, is called upon to control the pathogens. This collection of immune cells surrounds,(More)
Atherosclerosis, the leading death in the United State, is a disease in which a plaque builds up inside the arteries. As the plaque continues to grow, the shear force of the blood flow through the decreasing cross section of the lumen increases. This force may eventually cause rupture of the plaque, resulting in the formation of thrombus, and possibly heart(More)
Bertini real is a command line program for numerically decomposing the real portion of a oneor two-dimensional complex irreducible algebraic set in any reasonable number of variables. Using numerical homotopy continuation to solve a series of polynomial systems via regeneration from a witness set, a set of real vertices is computed, along with connection(More)
A Matlab implementation, multiplicity, of a numerical algorithm for computing the multiplicity structure of a nonlinear system at an isolated zero is presented. The software incorporates a newly developed equation-by-equation strategy that significantly improves the efficiency of the closedness subspace algorithm and substantially reduces the storage(More)
Homotopy continuation is an efficient tool for solving polynomial systems. Its efficiency relies on utilizing adaptive stepsize and adaptive precision path tracking, and endgames. In this article, we apply homotopy continuation to solve steady state problems of hyperbolic conservation laws. A third-order accurate finite difference weighted essentially(More)
The growth of tumors can be modeled as a free boundary problem involving partial differential equations. We consider one such model and compute steady-state solutions for this model. These solutions include radially symmetric solutions where the free boundary is a sphere and nonradially symmetric solutions. Linear and nonlinear stability for these solutions(More)
We consider the Bethe equations for the isotropic spin-1/2 Heisenberg quantum spin chain with periodic boundary conditions. We formulate a conjecture for the number of solutions with pairwise distinct roots of these equations, in terms of numbers of so-called singular (or exceptional) solutions. Using homotopy continuation methods, we find all such(More)
We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ and the cell-to-cell(More)
We consider a free boundary problem for a system of partial differential equations, which arise in a model of cell cycle. For the quasi steady state system, it depends on a positive parameter β, which describes the signals from the microenvironment. Upon discretizing this model, we obtain a family of polynomial systems parameterized by β. We numerically(More)