Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

Autonomous micro- and nanomotors should, in principle, deliver materials in a site-directed fashion, powering the assembly of dynamic, nonequilibrium superstructures. Here we demonstrate that catalytic Pt−Au nanomotors can transport a prototypical cargo: polystyrene microspheres. In addition, motors with Ni segments can overcome both Brownian orientational fluctuations and biased rotation of the rod−sphere doublet to enable persistent steerable uniaxial motion in an external magnetic field. Assuming a cargo-independent motive force, the speeds are inversely proportional to the Stokes resistance, which we compute using a completed double-layer boundary integral equation. In addition, we demonstrate motors transporting cargo via chemotaxis toward a H2O2 fuel source.

Catalytic Motors for Transport of Colloidal Cargo

The swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.

Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations

The swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. The validity of the far-field description is probed for a selection of model swimmers of varying geometry and propulsive activity by comparison with full numerical simulations. The reduced model is then used to deliver simple but accurate predictions of hydrodynamically generated wall attraction and pitching dynamics, and may help to explain a number of experimental results.

Numerical simulations are extensively used to investigate the motion of suspended particles in a fluid and their influence on the dynamics of the overall flow. Contexts range from the rheology of concentrated suspensions in a viscous fluid to the dynamics of particle-laden turbulent flows. This review summarizes several current approaches to the numerical simulation of rigid particles suspended in a flow, pointing out both common features and differences, along with their primary range of application. The focus is on non-Brownian systems for which thermal fluctuations do not play a role, whereas interparticle forces may result in particle self-assembly. Applications may include the motion of a few isolated particles with complex shape or the collective dynamics of many suspended particles.

Simulation Methods for Particulate Flows and Concentrated Suspensions

The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear Fredholm integral equations of the second kind for a ...

Second kind integral equation formulation of Stokes' flows past a particle of arbitary shape

Introduction Continuum Mathematical Models Physically Motivated Mathematical Concepts and Theorems General 3-D Use of Flux Vector Densities Illustrative Examples of PDEs A Comment on the Numerical Treatment of the grad Operator Notes on Numerical Analysis Computational Errors Order of Accuracy Norms and Condition Numbers Linear Systems of Equations Solution of Nonlinear Equations Mimetic Differential Operators Castillo-Grone Method for 1-D Uniform Staggered Grids Higher-Dimensional CGM 2-D Staggerings 3-D Staggerings Gradient Compositions Nullity Tests Higher-Order Operators Formulation of Nonlinear and Time-Dependent Problems Object-Oriented Programming and C++ From Structured to Object-Oriented Programming Fundamental Concepts in Object-Oriented Programming Object-Oriented Modeling and UML Inheritance and Polymorphism Mimetic Methods Toolkit (MTK) MTK Usage Philosophy Study of a Diffusive-Reactive Process Using the MTK Collaborative Development of the MTK: Flavors and Concerns Downloading the MTK Nonuniform Structured Meshes Divergence Operator Gradient Operator Case Studies Porous Media Flow and Reservoir Simulation Modeling Carbon Dioxide Geologic Sequestration Maxwell's Equations Wave Propagation Geophysical Flow Appendix A: Heuristic Deduction of the Extended Form of Gauss' Divergence Theorem Appendix B: Tensor Concept: An Intuitive Approach Appendix C Total Force Due to Pressure Gradients Appendix D: Heuristic Deduction of Stokes' Formula Appendix E: Curl in a Rotating Incompressible Inviscid Liquid Appendix F: Curl in Poiseuille's Flow Appendix G: Green's Identities Appendix H: Fluid Volumetric Time-Tate of Change Appendix I: General Formulation of the Flux Concept Appendix J: Fourth-Order Castillo-Grone Divergence Operators References Index Sample Problems appear at the end of each chapter.

Mimetic Discretization Methods

Integral equation formulation for the creeping flow of an incompressible viscous fluid between two arbitrarily closed surfaces, and a possible mathematical model for the brain fluid dynamics

Guillermo Miranda

Papers

Second kind integral equation formulation of Stokes' flows past a particle of arbitary shape

Mimetic Discretization Methods

Integral equation formulation for the creeping flow of an incompressible viscous fluid between two arbitrarily closed surfaces, and a possible mathematical model for the brain fluid dynamics

Integral equation solution of oseen’s flow with a free surface

Green's Identities