Logarithmic Sobolev inequality for the invariant measure of the periodic Korteweg--de Vries equation.

The periodic KdV equation arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure \nu on balls in the phase space such that the Cauchy problem for KdV is well posed on the support of \nu, and \nu is invariant under the KdV flow. This paper shows that \nu satisfies a logarithmic Sobolev inequality. The stationary points of the Hamiltonian on spheres are found in terms of elliptic functions, and they are shown to be linearly stable. The paper also presents logarithmic Sobolev inequalities for the modified period KdV equation and the cubic nonlinear Schrodinger equation for small values of the number operator N

GIS in the cloud: implementing a web map service on Google App Engine

Many producers of geographic information are now disseminating their data using open web service protocols, notably those published by the Open Geospatial Consortium. There are many challenges inherent in running robust and reliable services at reasonable cost. Cloud computing provides a new kind of scalable infrastructure that could address many of these challenges. In this study we implement a Web Map Service for raster imagery within the Google App Engine environment. We discuss the challenges of developing GIS applications within this framework and the performance characteristics of the implementation. Results show that the application scales well to multiple simultaneous users and performance will be adequate for many applications, although concerns remain over issues such as latency spikes. We discuss the feasibility of implementing services within the free usage quotas of Google App Engine and the possibility of extending the approaches in this paper to other GIS applications

Operational calculus and integral transforms for groups with finite propagation speed

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an operational calculus for integral transforms and functions of $A$ using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on $E$. Examples include the Mellin transform and the Mehler--Fock transform. The paper uses functional calculus for the cosine family $\cos( t\sqrt {\Delta})$ which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm--Liouville hypergroups with Laplace representation as well as analogues to the Kunze--Stein phenomenon in the hypergroup convolution setting.Comment: arXiv admin note: substantial text overlap with arXiv:1304.5868. Substantial revision to version

Concentration of measure on product spaces with applications to Markov processes.

For a stochastic process with state space some Polish space, this paper gives sufficient conditions on the initial and conditional distributions for the joint law to satisfy Gaussian concentration and transportation inequalities. In the case of Euclidean space, there are sufficient conditions for the joint law to satisfy a logarithmic Sobolev inequality. In several cases, the constants obtained are of optimal growth with respect to the number of random variables, or are independent of this number. These results extend results known for mutually independent random variables and weakly dependent random variabels under Dobrushkin--Shlosman type conditions. The paper also contains applications to Markov processes including the ARMA process

Discrete Tracy--Widom operators.

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper considers discrete Tracy--Widom operators, and gives sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equatio and the Fourier transform of Mathieu's equation