On discrete analogues of potential vorticity for variational shallow water systems

We outline how discrete analogues of the conservation of potential vorticity may be achieved in Finite Element numerical schemes for a variational system which has the particle relabelling symmetry, typically shallow water equations. We show that the discrete analogue of the conservation law for potential vorticity converges to the smooth law for potential vorticity, and moreover, for a strong solution, is the weak version of the potential vorticity law. This result rests on recent results by the author with T. Pryer concerning discrete analogues of conservation laws in Finite Element variational problems, together with an observation by P. Hydon concerning how the conservation of potential vorticity in smooth systems arises as a consequence of the linear momenta. The purpose of this paper is to provide all the necessary information for the implementation of the schemes and the necessary numerical tests. A brief tutorial on Noether's theorem is included to demonstrate the origin of the laws and to demonstrate that the numerical method follows the same basic principle, which is that the law follows directly from the Lie group invariance of the Lagrangian.Comment: 12 pages, 1 figur

Moving Frames and Noether’s Conservation Laws – the General Case

In recent works [1, 2], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables

Using Lie group integrators to solve two and higher dimensional variational problems with symmetry

The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. In the one dimensional case, Noether's laws give first integrals of the Euler–Lagrange equations. In higher dimensional problems, the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of $SU(2)$. We then consider variational problems for evolving curves which are invariant under the projective action of $SL(2)$ and finally the standard affine action of $SE(2)$

What Are the Public Health Effects of Direct-to-Consumer Drug Advertising?

Background to the debate: Only two industrialized countries, the United States and New Zealand, allow direct-to-consumer advertising (DTCA) of prescription medicines, although New Zealand is planning a ban [ 1]. The challenge for these governments is ensuring that DTCA is more beneficial than harmful. Proponents of DTCA argue that it helps to inform the public about available treatments and stimulates appropriate use of drugs for high-priority illnesses (such as statin use in people with ischemic heart disease). Critics argue that the information in the adverts is often biased and misleading, and that DTCA raises prescribing costs without net evidence of health benefits

Porcine Bladder Urothelial, Myofibroblast, and Detrusor Muscle Cells: Characterization and ATP Release

ATP is released from the bladder mucosa in response to stretch, but the cell types responsible are unclear. Our aim was to isolate and characterize individual populations of urothelial, myofibroblast, and detrusor muscle cells in culture, and to examine agonist-stimulated ATP release. Using female pig bladders, urothelial cells were isolated from bladder mucosa following trypsin-digestion of the luminal surface. The underlying myofibroblast layer was dissected, minced, digested, and cultured until confluent (10–14 days). A similar protocol was used for muscle cells. Cultures were used for immunocytochemical staining and/or ATP release investigations. In urothelial cultures, immunoreactivity was present for the cytokeratin marker AE1/AE3 but not the contractile protein α-smooth muscle actin (α-SMA) or the cytoskeletal filament vimentin. Neither myofibroblast nor muscle cell cultures stained for AE1/AE3. Myofibroblast cultures partially stained for α-SMA, whereas muscle cultures were 100% stained. Both myofibroblast and muscle stained for vimentin, however, they were morphologically distinct. Ultrastructural studies verified that the suburothelial layer of pig bladder contained abundant myofibroblasts, characterized by high densities of rough endoplasmic reticulum. Baseline ATP release was higher in urothelial and myofibroblast cultures, compared with muscle. ATP release was significantly stimulated by stretch in all three cell populations. Only urothelial cells released ATP in response to acid, and only muscle cells were stimulated by capsaicin. Tachykinins had no effect on ATP release. In conclusion, we have established a method for culture of three cell populations from porcine bladder, a well-known human bladder model, and shown that these are distinct morphologically, immunologically, and pharmacologically