Asymptotics of solutions in nA+nB->C reaction Diffusion systems

We analyze the long time behavior of initial value problems that model a process where particles of type A and B diffuse in some substratum and react according to $nA+nB\to C$. The case n=1 has been studied before; it presents nontrivial behavior on the reactive scale only. In this paper we discuss in detail the cases $n>3$, and prove that they show nontrivial behavior on the reactive and the diffusive length scale.Comment: 22 pages, 1 figur

Understanding CME and associated shock in the solar corona by merging multi wavelengths observation

Using multi-wavelength imaging observations, in EUV, white light and radio, and radio spectral data over a large frequency range, we analyzed the triggering and development of a complex eruptive event. This one includes two components, an eruptive jet and a CME which interact during more than 30 min, and can be considered as physically linked. This was an unusual event. The jet is generated above a typical complex magnetic configuration which has been investigated in many former studies related to the build-up of eruptive jets; this configuration includes fan-field lines originating from a corona null point above a parasitic polarity, which is embedded in one polarity region of large Active Region (AR). The initiation and development of the CME, observed first in EUV, does not show usual signatures. In this case, the eruptive jet is the main actor of this event. The CME appears first as a simple loop system which becomes destabilized by magnetic reconnection between the outer part of the jet and the ambient medium. The progression of the CME is closely associated with the occurrence of two successive types II bursts from distinct origin. An important part of this study is the first radio type II burst for which the joint spectral and imaging observations allowed: i) to follow, step by step, the evolution of the spectrum and of the trajectory of the radio burst, in relationship with the CME evolution; ii) to obtain, without introducing an electronic density model, the B-field and the Alfven speed.Comment: 17 pages, 13 figure

Quantum Amplitude Amplification and Estimation

Consider a Boolean function $\chi: X \to \{0,1\}$ that partitions set $X$ between its good and bad elements, where $x$ is good if $\chi(x)=1$ and bad otherwise. Consider also a quantum algorithm $\mathcal A$ such that $A |0\rangle= \sum_{x\in X} \alpha_x |x\rangle$ is a quantum superposition of the elements of $X$, and let $a$ denote the probability that a good element is produced if $A |0\rangle$ is measured. If we repeat the process of running $A$, measuring the output, and using $\chi$ to check the validity of the result, we shall expect to repeat $1/a$ times on the average before a solution is found. *Amplitude amplification* is a process that allows to find a good $x$ after an expected number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$, assuming algorithm $A$ makes no measurements. This is a generalization of Grover's searching algorithm in which $A$ was restricted to producing an equal superposition of all members of $X$ and we had a promise that a single $x$ existed such that $\chi(x)=1$. Our algorithm works whether or not the value of $a$ is known ahead of time. In case the value of $a$ is known, we can find a good $x$ after a number of applications of $A$ and its inverse which is proportional to $1/\sqrt{a}$ even in the worst case. We show that this quadratic speedup can also be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform amplitude estimation, a process that allows to estimate the value of $a$. We apply amplitude estimation to the problem of *approximate counting*, in which we wish to estimate the number of $x\in X$ such that $\chi(x)=1$. We obtain optimal quantum algorithms in a variety of settings.Comment: 32 pages, no figure