Finite-time Singularity Formation for Strong Solutions to the 3D3D Euler Equations, I

In this paper and the companion paper [EJE2], we establish finite-time singularity formation for finite-energy strong solutions to the axi-symmetric $3D$ Euler equations in the domain $\{(x,y,z)\in\mathbb{R}^3:z^2\leq c(x^2+y^2)\}$ for some $c>0$. In the spirit of our previous works, [EJSI] and [EJB], we do this by first studying scale-invariant solutions which satisfy a one dimensional PDE system and proving that they may become singular in finite time for properly chosen initial data. We then prove local well-posedness for the $3D$ Euler equations in a natural regularity class which includes scale-invariant solutions. While these solutions have uniformly bounded vorticity from time zero until right before the blow-up time, they do not have finite energy. To remedy this, we cut off the scale-invariant data to ensure finite energy and prove that the corresponding local solution must also become singular in finite time. This paper focuses only on the analysis of the scale-invariant solutions themselves and the proof that they can become singular in finite time. The local well-posedness theorem and the cut-off argument are very close to those in [JSI] and [EJB] and are left for the companion paper [EJE2]. It is important to remark that while the fluid domain is the exterior of a cone, we prove global regularity for the axi-symmetric $3D$ Euler equations without swirl in the exact same regularity classes and in the same domain. It is quite possible that the methods we use can be adapted to establish finite-time singularity formation for $C^\infty$ finite-energy solutions to the $3D$ Euler equations on $\mathbb{R}^3_+$.Comment: There was a sign error in the version of the axi-symmetric Euler equations that we have adopted. While Theorems A and D remain valid as stated, it is unclear whether finite-time singularity formation results hold for the systems with correct sign

Crummer/Suntrust Portfolio: Analysis and Recommendations [2008]

Unfavorable economic conditions have prompted the investment team to meet a short-term goal: preserve the value of the portfolio by reducing overall risk. A top down analysis of the current portfolio resulted in recommendations to change the portfolio by eliminating funds, investing in long term government bonds, TIPS and large cap value type securities. Allocations of securities was performed by either overweighting or underweighting a particular sector, depending on its historical and expected return during a recession. Individual securities were analyzed by our sector analystsand recommendations of hold, buy or sell were made. The outcome.... a robust portfolio, that will preserve its value during the recession by reducing its overall risk by 25%

Circadian Network Interactions with Jasmonate Signaling and Defense

Plants experience specific stresses at particular, but predictable, times of the day. The circadian clock is a molecular oscillator that increases plant survival by timing internal processes to optimally match these environmental challenges. Clock regulation of jasmonic acid (JA) action is important for effective defenses against fungal pathogens and generalist herbivores in multiple plant species. Endogenous JA levels are rhythmic and under clock control with peak JA abundance during the day, a time when plants are more likely to experience certain types of biotic stresses. The expression of many JA biosynthesis, signaling, and response genes is transcriptionally controlled by the clock and timed through direct connections with core clock proteins. For example, the promoter of Arabidopsis transcription factor MYC2, a master regulator for JA signaling, is directly bound by the clock evening complex (EC) to negatively affect JA processes, including leaf senescence, at the end of the day. Also, tobacco ZEITLUPE, a circadian photoreceptor, binds directly to JAZ proteins and stimulates their degradation with resulting effects on JA root-based defenses. Collectively, a model where JA processes are embedded within the circadian network at multiple levels is emerging, and these connections to the circadian network suggest multiple avenues for future research

Gibberellin driven growth in <i>elf3</i> mutants requires PIF4 and PIF5

<div><p>The regulatory connections between the circadian clock and hormone signaling are essential to understand, as these two regulatory processes work together to time growth processes relative to predictable environmental events. Gibberellins (GAs) are phytohormones that control many growth processes throughout all stages of the plant life cycle, including germination and flowering. An increasing number of examples demonstrate that the circadian clock directly influences GA biosynthesis and signaling. EARLY FLOWERING 3 (ELF3) participates in a tripartite transcriptional complex known as the Evening Complex (EC). In this capacity, ELF3 is fundamental to core circadian clock activity, as well as time-of-day specific regulation of genes directly responsible for growth control, namely the <i>PHYTOCHROME INTERACTING FACTOR 4</i> (<i>PIF4</i>) and <i>PIF5</i> genes. Here we show that the GA biosynthesis inhibitor paclobutrazol substantially reduces the long hypocotyl and petiole phenotypes of Arabidopsis <i>elf3</i> mutants. In addition, loss of ELF3 activity causes upregulation of the key GA biosynthesis genes <i>GA20ox1</i> and <i>GA20ox2</i>. Moreover, <i>GA20ox1</i> and <i>GA20ox2</i> expression depends strongly on the redundant activities of <i>PIF4</i> and <i>PIF5</i>. These findings indicate that the defining growth phenotypes of <i>elf3</i> mutants arise from altered GA biosynthesis due to misregulation of <i>PIF4</i> and <i>PIF5</i>. These observations agree with recent work linking increased GA production with the elongated growth phenotypes of the barley <i>elf3</i> mutant. Thus, the role of the EC in regulation of GA biosynthesis and signaling in eudicots is shared with monocots and, therefore, is a highly conserved mechanism for growth control.</p></div