Synchronization hypothesis in the Winfree model

We consider $N$ oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength $\kappa$ and the spectrum width $\gamma$ of the frequencies of each oscillator. In the uncoupled regime, $\kappa=0$, each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. We say that $N$ oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in $(\gamma,\kappa)$ of synchronization contains coupling values that are both weak and strong. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We give a numerical counter-example which shows that this hypothesis is necessary for the existence of synchronization

Target Site Recognition by a Diversity-Generating Retroelement

Diversity-generating retroelements (DGRs) are in vivo sequence diversification machines that are widely distributed in bacterial, phage, and plasmid genomes. They function to introduce vast amounts of targeted diversity into protein-encoding DNA sequences via mutagenic homing. Adenine residues are converted to random nucleotides in a retrotransposition process from a donor template repeat (TR) to a recipient variable repeat (VR). Using the Bordetella bacteriophage BPP-1 element as a prototype, we have characterized requirements for DGR target site function. Although sequences upstream of VR are dispensable, a 24 bp sequence immediately downstream of VR, which contains short inverted repeats, is required for efficient retrohoming. The inverted repeats form a hairpin or cruciform structure and mutational analysis demonstrated that, while the structure of the stem is important, its sequence can vary. In contrast, the loop has a sequence-dependent function. Structure-specific nuclease digestion confirmed the existence of a DNA hairpin/cruciform, and marker coconversion assays demonstrated that it influences the efficiency, but not the site of cDNA integration. Comparisons with other phage DGRs suggested that similar structures are a conserved feature of target sequences. Using a kanamycin resistance determinant as a reporter, we found that transplantation of the IMH and hairpin/cruciform-forming region was sufficient to target the DGR diversification machinery to a heterologous gene. In addition to furthering our understanding of DGR retrohoming, our results suggest that DGRs may provide unique tools for directed protein evolution via in vivo DNA diversification

Non symmetry of the function ζ(s)s\frac{\zeta(s)}{s}

We prove that $| \frac{\zeta(\overline{s})}{\overline{s}}-\frac{\zeta(1-s)}{1-s}|\neq 0$ for every $\Re(s)\in(0,\frac{1}{2})$ and $\Im(s)\in \mathbb{R}^*$, where $\zeta$ is the Riemann Zeta function. At the end of the paper, we give a discussion about the Riemann hypothesi

Notes et exercices du cours d'Équations Différentielles

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Synchronization in Winfree model with N oscillators

We consider $N$ oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength $\kappa$ and the spectrum width $\gamma$ of the frequencies of each oscillator. In the uncoupled regime, $\kappa=0$, each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. We say that $N$ oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in $(\gamma,\kappa)$ of synchronization contains coupling values that are both weak and strong. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We give a numerical counter-example which shows that this hypothesis is necessary for the existence of synchronization

Invariant cone and synchronization state stability of the mean field models

In this article we prove the stability of mean field systems as the Winfree model in the synchronized state. The model is governed by the coupling strength parameter κ and the natural frequency of each oscillator. The stability is proved independently of the number of os-cillators and the distribution of the natural frequencies. In order to prove the main result, we introduce the positive invariant cone and we start by studying the linearized system. The method can be applied to others mean field models as the Kuramoto model

MarineFisheries Advisory (2006-02-10; 600-Lb. Spiny Dogfish Possession Limit Set For 2006 Fishing Year)

In this article we prove the stability of mean field systems as the Winfree model in the synchronized state. The model is governed by the coupling strength parameter $\kappa$ and the natural frequency of each oscillator. The stability is proved independently of the number of os-cillators and the distribution of the natural frequencies. In order to prove the main result, we introduce the positive invariant cone and we start by studying the linearized system. The method can be applied to others mean field models as the Kuramoto model

Reduced dimension and Rotation vector formula of ordinary differential equation

The leader trajectory function defined in this article is an approximate solution of a differential equation. It is defined by some independent one-dimensional differential equations. The generalized main result of this article asserts that if the leader trajectory exists then it is at finite distance from the solution of the system. The application of the generalized main result is to control the trajectory of the periodic systems. We prove that for any periodic system and any initial condition there exists a leader trajectory which is a linear function of the time variable. In other words, we find an exact Rotation vector formula which is the relation between the rotation vector and the initial condition. In addition, we present a necessary and sufficient condition for the existence of a locally constant rotation vector under perturbation of the system, known by the Arnold tongue