1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

Biological shape and visual science (part I)

Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic the- ory are based on three important pillars: (1) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotive-like terms; (2) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov dis- tribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms derived from the quadratic nonlinearities in the gyrocenter Hamiltonian; and (3) an exact energy conservationlaw for the gyrokineticVlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on the rigorous applications of Lagrangian and Hamiltonian Lie-transform perturba- tion methods used in the variationalderivationof nonlineargyrokineticVlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations, which describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry, are also discussed.

/pdf/foundations-of-nonlinear-gyrokinetic-theory-3l78lnlpai.pdf

Foundations of nonlinear gyrokinetic theory

We present a fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid. The arbitrary Lagrangian–Eulerian description is utilized for the fluid subdomain and the Lagrangian description is utilized for the solid subdomain. Particular attention is paid to the derivation of various forms of the conservation equations; the conservation properties of the semi-discrete and fully discretized systems; a unified presentation of the generalized-α time integration method for fluid-structure interaction; and the derivation of the tangent matrix, including the calculation of shape derivatives. A NURBS-based isogeometric analysis methodology is used for the spatial discretization and three numerical examples are presented which demonstrate the good behavior of the methodology.

Isogeometric fluid-structure interaction: theory, algorithms, and computations

This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.

/pdf/geometric-control-of-mechanical-systems-59lvmkw96d.pdf

Geometric control of mechanical systems

Differential Forms, with Applications to the Physical Sciences.

Differential Forms with Applications to the Physical Sciences

(1973). Differentiation Under the Integral Sign. The American Mathematical Monthly: Vol. 80, No. 6, pp. 615-627.

Differentiation Under the Integral Sign

On Spaces of Linear Transformations with Bounded Rank

W. E. Roth gave necessary and sufficient conditions for the solvability of each of the matrix equations $AX - XB = C$ and $AX - YB = C$. His proofs were based on canonical forms. In this note we give invariant proofs of Roth’s theorems.

Harley Flanders

Papers

Differential Forms, with Applications to the Physical Sciences.

Differential Forms with Applications to the Physical Sciences

Differentiation Under the Integral Sign

On Spaces of Linear Transformations with Bounded Rank

On the matrix equations $AX - XB = C$ and $AX - YB = C$