Quasi-continuous symmetries of non-Lie type

We introduce a smooth mapping of some discrete space-time symmetries into quasi-continuous ones. Such transformations are related with q-deformations of the dilations of the Euclidean space and with the non-commutative space. We work out two examples of Hamiltonian invariance under such symmetries. The Schrodinger equation for a free particle is investigated in such a non-commutative plane and a connection with anyonic statistics is found. PACS: 03.65.Fd, 11.30.E

Hydrodynamics

The mathematical description of the states of a fluid is based on the study of three fields defined on the domain occupied by the fluid: the velocity field V, the density ρ, and the pressure field P. These three “unknowns” are determined by integrating other five scalar equations, namely the mass conservation (continuity equation), the three components of the equation of momentum balance (Euler or Navier–Stokes), and the energy balance. This last equation needs in addition information about the thermodynamics of the fluid, so it may need to be supplied with some equation of state. In addition to these five equations, we request regularity, asymptotic and, if it is the case, boundary conditions, to provide a unique solution. When we study the dynamics of the fluid confined in a compact domain with free boundaries, the system is slightly more complicated, and we have to add the kinematical equation of the free surface, as well as equations of momentum balance at the surface. If we take into account the nonlinear terms in the dynamical equations, and in the associated curved geometry, some interesting solutions occur. Special nonlinear effects related to fluids on compact domains with free surface could be Gibbs–Marangoni effect, dividing the flow in cells (Bènard effect), couplings between different modes, collective effects, separation of flow in layer (boundary layer, turbulence), standing traveling surface waves, etc. In this chapter, we introduce some elements of general hydrodynamics which we will use later on in the book, boundary conditions especially at free surfaces, surface pressure theory, and representation theorems

Kinematics of Fluids

The goal of this chapter is to discuss the general frame of hydrodynamics, like particle trajectories (path lines), streamlines, streak lines, free surfaces, and fluid surfaces, and to compare their behavior in the Eulerian and Lagrangian frames. The following sections and chapters proceed on the assumption that the fluid is practically continuous and homogeneous in structure. Of course, the concept of continuum is an abstraction that does not take into account the molecular and nuclear structure of matter. In that, we assume that the properties of the fluid do not change if we consider smaller and smaller amounts of matter [1]. May be the wisest point of view while we remain at the level of general laws of fluid dynamics (or fluid mechanics) is to keep the physical scales rather vague [2]. This aspect is in direct relation with the fact that these laws can be made dimensionless in a large variety of situations

Book Review: Visual Motion of Curves and Surfaces

This is Dr. Ludu\u27s review of the book Visual Motion of Curves and Surfaces by Roberto Cipolla and Peter Giblin. Published by Cambridge University Press in 2009. ISBN: 978-0-521-63251-5

Differential Geometry of Moving Surfaces and Its Relation to Solitons

In this article we present an introduction in the geometrical theory of motion of curves and surfaces in R3, and its relations with the nonlinear integrable systems. The working frame is the Cartan\u27s theory of moving frames together with Cartan connection. The formalism for the motion of curves is constructed in the Serret-Frenet frames as elements of the bundle of adapted frames. The motion of surfaces is investigated in the Gauss-Weingarten frame. We present the relations between types of motions and nonlinear equations and their soliton solutions

Ice Spiral Patterns on the Ocean Surface

We investigate a new two-dimensional compressible Navier-Stokes hydrodynamic model design to explain and study large scale ice swirls formation at the surface of the ocean. The linearized model generates a basis of Bessel solutions from where various types of spiral patterns can be generated and their evolution and stability in time analyzed. By restricting the nonlinear system of equations to its quadratic terms we obtain swirl solutions emphasizing logarithmic spiral geometry. The resulting solutions are analyzed and validated using three mathematical approaches: one predicting the formation of patterns as Townes solitary modes, another approach mapping the nonlinear system into a sine-Gordon equation, and a third approach uses a series expansion. Pure radial, azimuthal and spiral modes are obtained from the fully nonlinear equations. Combinations of multiple-spiral solutions are also obtained, matching the experimental observations. The nonlinear stability of the spiral patterns is analyzed by Arnold's convexity method, and the Hamiltonian of the solutions is plotted versus some order parameters showing the existence of geometric phase transitions.Comment: 45 pages, 11 figure

2N-Dimensional Canonical Systems and Applications

We study the 2N-dimensional canonical systems and discuss some properties of its fundamental solution. We then discuss the Floquet theory of periodic canonical systems and observe the asymptotic behavior of its solution. Some important physical applications of the systems are also discussed: linear stability of periodic Hamiltonian systems, position-dependent effective mass, pseudo-periodic nonlinear water waves, and Dirac systems

Topology and Geometry of Mixing of Fluids

We present an enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. We describe the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. Further studies are needed for non-compact manifolds, or compact manifolds with Dirichlet BC. The optimal shape of the fixed obstacles in the mixing device could be obtained through such a research

A Mathematical Model for Transport and Growth of Microbes in Unsaturated Porous Soil

In this work, we develop a mathematical model for transport and growth of microbes by natural (rain) water infiltration and flow through unsaturated porous soil along the vertical direction under gravity and capillarity by coupling a system of advection diffusion equations (for concentration of microbes and their growth-limiting substrate) with the Richards equation. (e model takes into consideration several major physical, chemical, and biological mechanisms. (e resulting coupled system of PDEs together with their boundary conditions is highly nonlinear and complicated to solve analytically. We present both a partial analytic approach towards solving the nonlinear system and finding the main type of dynamics of microbes, and a full-scale numerical simulation. Following the auxiliary equation method for nonlinear reaction-diffusion equations, we obtain a closed form traveling wave solution for the Richards equation. Using the propagating front solution for the pressure head, we reduce the transport equation to an ODE along the moving frame and obtain an analytic solution for the history of bacteria concentration for a specific test case. To solve the system numerically, we employ upwind finite volume method for the transport equations and stabilized explicit Runge–Kutta–Legendre super-time-stepping scheme for the Richards equation. Finally, some numerical simulation results of an infiltration experiment are presented, providing a validation and backup to the analytic partial solutions for the transport and growth of bacteria in the soil, stressing the occurrence of front moving solitons in the nonlinear dynamics