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

Constructing sonified haptic line graphs for the blind student: first steps

Line graphs stand as an established information visualisation and analysis technique taught at various levels of difficulty according to standard Mathematics curricula. It has been argued that blind individuals cannot use line graphs as a visualisation and analytic tool because they currently primarily exist in the visual medium. The research described in this paper aims at making line graphs accessible to blind students through auditory and haptic media. We describe (1) our design space for representing line graphs, (2) the technology we use to develop our prototypes and (3) the insights from our preliminary work

First-principles phase diagram calculations for the HfC–TiC, ZrC–TiC, and HfC–ZrC solid solutions

We report first-principles phase diagram calculations for the binary systems HfC–TiC, TiC–ZrC, and HfC–ZrC. Formation energies for superstructures of various bulk compositions were computed with a plane-wave pseudopotential method. They in turn were used as a basis for fitting cluster expansion Hamiltonians, both with and without approximations for excess vibrational free energies. Significant miscibility gaps are predicted for the systems TiC–ZrC and HfC–TiC, with consolute temperatures in excess of 2000 K. The HfC–ZrC system is predicted to be completely miscibile down to 185 K. Reductions in consolute temperature due to excess vibrational free energy are estimated to be ~7%, ~20%, and ~0%, for HfC–TiC, TiC–ZrC, and HfC–ZrC, respectively. Predicted miscibility gaps are symmetric for HfC–ZrC, almost symmetric for HfC–TiC and asymmetric for TiC–ZrC

Canadian Beaufort Sea 2000: The Environmental and Social Setting

The Beaufort Sea Conference 2000 brought together a diverse group of scientists and residents of the Canadian Beaufort Sea region to review the current state of the region's renewable resources and to discuss the future management of those resources. In this paper, we briefly describe the physical environment, the social context, and the resource management processes of the Canadian Beaufort Sea region. The Canadian Beaufort Sea land area extends from the Alaska-Canada border east to Amundsen Gulf and includes the northwest of Victoria Island and Banks Island. The area is defined by its geology, landforms, sources of freshwater, ice and snow cover, and climate. The social context of the Canadian Beaufort Sea region has been set by prehistoric Inuit and Gwich'in, European influence, more recent land-claim agreements, and current management regimes for the renewable resources of the Beaufort Sea.La Conférence de l'an 2000 sur la mer de Beaufort a attiré un groupe hétérogène de scientifiques et de résidents de la région de la mer de Beaufort en vue d'examiner le statut actuel des ressources renouvelables de cette zone et de discuter de leur gestion future. Dans cet article, on décrit brièvement l'environnement physique, le contexte social et les processus de gestion des ressources de la zone canadienne de la mer de Beaufort. La superficie terrestre de la mer de Beaufort au Canada s'étend de la frontière entre ce pays et l'Alaska jusqu'au golfe Amundsen à l'est, et elle englobe le nord-ouest de l'île Victoria et de l'île Banks. Cette zone est définie par sa géologie, son relief, ses sources d'eau douce, son couvert glaciel et nival ainsi que son climat. Le contexte social de la région de la mer de Beaufort canadienne a été établi par les Inuits et Gwich'in préhistoriques, l'influence européenne, les récentes ententes territoriales ainsi que les régimes actuels de gestion des ressources renouvelables de la mer de Beaufort

Dispersion representations and anomalous singularities of the triangle diagram

We discuss dispersion representations for the triangle diagram $F(p_1^2,p_2^2,q^2)$, the single dispersion representation in $q^2$ and the double dispersion representation in $p_1^2$ and $p_2^2$, with special emphasis on the appearance of the anomalous singularities and the anomalous cuts in these representations. For the double dispersion representation in $p_1^2$ and $p_2^2$, the appearance of the anomalous cut in the region $q^2>0$ is demonstrated, and a new derivation of the anomalous double spectral density is given. We point out that the double spectral representation is particularly suitable for applications in the region of $p_1^2$ and/or $p_2^2$ above the two-particle thresholds. The dispersion representations for the triangle diagram in the nonrelativistic limit are studied and compared with the triangle diagram of the nonrelativistic field theory.Comment: 10 pages, revtex, added a reference, version to be published in Phys. Rev.

Computable analysis of linear rearrangement optimization

Optimization problems over rearrangement classes arise in various areas such as mathematics, fluid mechanics, biology, and finance. When the generator of the rearrangement class is two-valued, they reduce to shape optimization and free boundary problems which can exhibit intriguing symmetry breaking phenomena. A robust framework is required for computable analysis of these problems. In this paper, as a first step towards such a robust framework, we provide oracle Turing machines that compute the distribution function, decreasing rearrangement, and linear rearrangement optimizers, with respect to functions that are continuous and have no significant flat zones. This assumption on the reference function is necessary, as otherwise, the aforementioned operations may not be computable. We prove that the results can be computed to within any degree of accuracy, conforming to the framework of Type-II Theory of Effectivity

On the stability of travelling waves with vorticity obtained by minimization

We modify the approach of Burton and Toland Comm. Pure Appl. Math. LXIV. 975-1007 (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 minimization of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimization appears in Comm. Pure Appl. Math. LXIV. 975-1007 2011 after a first maximization). 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. Our proofs depend on the assumption that the surface offers some resistance to stretching and bendin