Particle trajectories in linearized irrotational shallow water flows

We investigate the particle trajectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water's free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending on the strength of underlying uniform current, we obtain that some particle trajectories are undulating path to the right or to the left, some are looping curves with a drift to the right and others are parabolic curves or curves which have only one loop

Methods and compositions for stimulating T-lymphocytes

Disclosed are methods, compositions, antibodies, and therapeutic kits for use in stimulating cytotoxic T-lymphocytes and generating immune responses against epitopes of protooncogenes. Novel peptides are described which have been shown to stimulate cytotoxic T-lymphocytes, and act as antigens in generation of oncogenic epitope-recognizing antibodies. Methods are disclosed for use in treating various proliferative disorders, and diagnosing HER-2/neu-containing cells; also disclosed are therapeutic kits useful in the treatment of cancer and production of potential anti-cancer vaccines.Board of Regents, University of Texas Syste

Gold nanorods as molecular contrast agents in photoacoustic imaging: the promises and the caveats\ud

Rod-shaped gold nanoparticles exhibit intense and narrow absorption peaks for light in the far-red and near-infrared wavelength regions, owing to the excitation of longitudinal plasmons. Light absorption is followed predominantly by non radiative de-excitation, and the released heat and subsequent temperature rise cause strong photoacoustic (optoacoustic) signals to be produced. This feature combined with the relative inertness of gold, and its favorable surface chemistry, which permits affinity biomolecule coupling, has seen gold nanorods (AuNR) attracting much attention as contrast agents and molecular probes for photoacoustic imaging. In this article we provide an short overview of the current status of the use of AuNR in molecular imaging using photoacoustics. We further examine the state of the art in various chemical, physical and biochemical phenomena that have implications for the future photoacoustic applications of these particles. We cover the route through fine-tuning of AuNR synthetic procedures, toxicity reduction by appropriate coatings, in vitro cellular interactions of AuNRs, attachment of targeting antibodies, in vivo fate of the particles and the effects of certain light interactions with the AuN

On the Cauchy problem for a nonlinearly dispersive wave equation

We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite times. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.Comment: arxiv version is already officia

Equations of the Camassa-Holm Hierarchy

The squared eigenfunctions of the spectral problem associated with the Camassa-Holm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy, and their Hamiltonian structures, can be naturally expressed using the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. Using the GFT, we explicitly describe some members of the CH hierarchy, including integrable deformations for the CH equation. We also show that solutions of some $(1+2)$ - dimensional members of the CH hierarchy can be constructed using results for the inverse scattering transform for the CH equation. We give an example of the peakon solution of one such equation.Comment: 10 page

Coercivity and stability results for an extended Navier-Stokes system

In this article we study a system of equations that is known to {\em extend} Navier-Stokes dynamics in a well-posed manner to velocity fields that are not necessarily divergence-free. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of controlling the nonlinear terms. We address questions of global existence and stability in bounded domains with no-slip boundary conditions. Even in two space dimensions, global existence is open in general, and remains so, primarily due to the lack of a self-contained $L^2$ energy estimate. However, through use of new $H^1$ coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a time-discrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included.Comment: 29 pages, no figure

Generating and Adding Flows on Locally Complete Metric Spaces

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.Comment: 29 pages,6 figure

The 'nanobig rods' class of gold nanorods: optimized dimensions for improved in vivo therapeutic and imaging efficacy

Currently, gold nanorods can be synthesized in a wide range of sizes. However, for intended biological applications gold nanorods with approximate dimensions 50 nm x 15 nm are used. We investigate by computer simulation the effect of particle dimensions on the optical and thermal properties in the context of the specific applications of photoacoustic imaging. In addition we discuss the influence of particle size in overcoming the following biophysical barriers when administrated in vivo: extravasation, avoidance of uptake by organs of the reticuloendothelial system, penetration through the interstitium, binding capability and uptake by the target cells. Although more complex biological influences can be introduced in future analysis, the present work illustrates that larger gold nanorods, designated by us as "nanobig rods", may perform relatively better at meeting the requirements for successful in vivo applications compared to their smaller counterparts which are conventionally used

On the stability of travelling waves with vorticity obtained by minimisation

We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)] to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.Comment: NoDEA Nonlinear Differential Equations and Applications, to appea

A stochastic-Lagrangian particle system for the Navier-Stokes equations

This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space \holderspace{1}{\alpha} which consists of differentiable functions whose first derivative is $\alpha$ H\"older continuous (see Section \ref{sGexist} for the precise definition). Further, we show that as $N \to \infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\frac{1}{N})$ times its original energy as $t \to \infty$. Hence the limit $N \to \infty$ and $T\to \infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \to \infty$ explicitly.Comment: v3: Typo fixes, and a few stylistic changes. 17 pages, 2 figure