The Critical Exponent is Computable for Automatic Sequences

The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341

A new code for Fourier-Legendre analysis of large datasets: first results and a comparison with ring-diagram analysis

Fourier-Legendre decomposition (FLD) of solar Doppler imaging data is a promising method to estimate the sub-surface solar meridional flow. FLD is sensible to low-degree oscillation modes and thus has the potential to probe the deep meridional flow. We present a newly developed code to be used for large scale FLD analysis of helioseismic data as provided by the Global Oscillation Network Group (GONG), the Michelson Doppler Imager (MDI) instrument, and the upcoming Helioseismic and Magnetic Imager (HMI) instrument. First results obtained with the new code are qualitatively comparable to those obtained from ring-diagram analyis of the same time series.Comment: 4 pages, 2 figures, 4th HELAS International Conference "Seismological Challenges for Stellar Structure", 1-5 February 2010, Arrecife, Lanzarote (Canary Islands

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow up solutions for the wave map problem from 2+1 dimensions into the 2-sphere. These solutions are the sum of a dynamically rescaled ground-state harmonic map plus a radiation term. The local energy of the latter tends to zero as time approaches blow up time. This is accomplished by first "renormalizing" the rescaled ground state harmonic map profile by solving an elliptic equation, followed by a perturbative analysis

Executing Underspecified OCL Operation Contracts with a SAT Solver

Executing formal operation contracts is an important technique for requirements validation and rapid prototyping. Current approaches require additional guidance from the user or exhibit poor performance for underspecified contracts that describe the operation results non-constructively. We present an efficient and fully automatic approach to executing OCL operation contracts which uses a satisfiability (SAT) solver. The operation contract is translated to an arithmetic formula with bounded quantifiers and later to a satisfiability problem. Based on the system state in which the operation is called and the arguments to the operation, an off-the-shelf SAT solver computes a new state that satisfies the postconditions of the operation. An effort is made to keep the changes to the system state as small as possible. We present a tool for generating Java method bodies for operations specified with OCL. The efficiency of our method is confirmed by a comparison with existing approaches

Nondispersive solutions to the L2-critical half-wave equation

We consider the focusing $L^2$-critical half-wave equation in one space dimension $$i \partial_t u = D u - |u|^2 u,$$ where $D$ denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold $M_* > 0$ such that all $H^{1/2}$ solutions with $\| u \|_{L^2} < M_*$ extend globally in time, while solutions with $\| u \|_{L^2} \geq M_*$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass $\| u_0 \|_{L^2} = M_*$. More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy $E_0 >0$ and the linear momentum $P_0 \in \R$. In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for $L^2$-critical nonlinear PDE with nonlocal dispersion.Comment: 51 page