Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type

We study the Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion and with monotonically increasing initial data using the Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero dispersion limit, is obtained using the steepest descent method for oscillatory Riemann-Hilbert problems. The asymptotic solution is completely described by a scalar function \g that satisfies a scalar RH problem and a set of algebraic equations constrained by algebraic inequalities. The scalar function \g is equivalent to the solution of the Lax-Levermore maximization problem. The solution of the set of algebraic equations satisfies the Whitham equations. We show that the scalar function \g and the Lax-Levermore maximizer can be expressed as the solution of a linear overdetermined system of equations of Euler-Poisson-Darboux type. We also show that the set of algebraic equations and algebraic inequalities can be expressed in terms of the solution of a different set of linear overdetermined systems of equations of Euler-Poisson-Darboux type. Furthermore we show that the set of algebraic equations is equivalent to the classical solution of the Whitham equations expressed by the hodograph transformation.Comment: 32 pages, 1 figure, latex2

Extremal GG-invariant eigenvalues of the Laplacian of GG-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere $S^2$ endowed with $S^1$-invariant metrics, we consider the subsequence $\lambda_k^G$ of the spectrum of a Riemannian manifold $M$ which corresponds to metrics and functions invariant under the action of a compact Lie group $G$. If $G$ has dimension at least 1, we show that the functional $\lambda_k^G$ admits no extremal metric under volume-preserving $G$-invariant deformations. If, moreover, $M$ has dimension at least three, then the functional $\lambda_k^G$ is unbounded when restricted to any conformal class of $G$-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on $S^n$; however, if we also require the metric to be induced by an embedding of $S^n$ in $\mathbb{R}^{n+1}$, we get an optimal upper bound on $\lambda_k^G$.Comment: To appear in Mathematische Zeitschrif

1.25 Gbit/s ITU-T G.984.2 burst-mode transimpedance amplifier without reset pins

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Effects of centrifugal stress on cell disruption and glycerol leakage from Dunaliella salina

Dunaliella salina accumulates large amounts of intracellular glycerol in response to the increases in salt concentration, thus is a potential source for producing fuel grade glycerol as an alternative to biodiesel-derived crude glycerol. D. salina lacks a cell wall; therefore the mode of harvesting Dunaliella cells is critical to avoid cell disruption caused by extreme engineering conditions. This study explored cell disruption and glycerol leakage of D. salina under various centrifugal stresses during cell harvesting. Results show a centrifugal g-force lower than 5000 g caused little cell disruption, while a g-force higher than 9000 g led to ~40 % loss of the intact cells and glycerol yields from the recovered algal pellets. Theoretical calculations of the centrifugal stresses that could rupture Dunaliella cells were in agreement with the experimental results, indicating optimisation of centrifugation conditions is important for recovering intact cells of from D. salina enriched in glycerol

Greatest least eigenvalue of the Laplacian on the Klein bottle

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda\_1 (\mathbb{K}, g) A (\mathbb{K}, g)\le 12 \pi E(2\sqrt 2/3),$$ where $\lambda\_1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of the Laplacian and the area of $(\mathbb{K},g)$, respectively, and $E$ is the complete elliptic integral of the second kind. Moreover, the equality is uniquely achieved, up to dilatations, by the metric $$g\_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} (du^2 + {dv^2\over 1+8\cos ^2v}),$$ with $0\le u,v <\pi$. The proof of this theorem leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.Comment: 17 page

Ultra-high-sensitivity two-dimensional bend sensor

A multicore fibre Fabry-Perot-based strain sensor interrogated with tandem interferometry for bend measurement is described. Curvature in two dimensions is obtained by measuring the difference in strain between three co-located low finesse Fabry-Perot interferometers formed in each core of the fibre by pairs of Bragg gratings. This sensor provides a responsivity enhancement of up to 30 times that of a previously reported fibre Bragg grating based sensor. Strain resolutions of 0.6 n epsilon/Hz(1/2) above 1 Hz are demonstrated, which corresponds to a curvature resolution of similar to 0.012 km(-1)/Hz(1/2)