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Abstract The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation $-{\mathrm{\Lambda }}^{\sigma }$, where $\widehat{{\mathrm{\Lambda }}^{\sigma }}=|k{|}^{\sigma }$, both for the problem on the circle $x\in [-\pi ,\pi ]$and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\sigma \text{⩾}1$and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when $a=1/2$, for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory. 

Broadening interest in computer science is a major research goal of the 21st century. Many initiatives use traditional “hooks” to attract students, such as video games and robotics. Unfortunately, this tends to attract only those already interested in computer science. One alternative domain gaining momen- tum in computer science education research is music, which is showing inter- esting results with participants that have previous music knowledge. This paper presents a case study of teaching computer programming with music, in Brazil, to students with limited formal music experience. Through data collected in sur- veys, focus groups, and researchers’ observations, we show that in this context students can still learn and thrive as musical programmers. 

Broadening interest in computer science is a major research goal of the 21st century. Many initiatives use traditional “hooks” to attract students, such as video games and robotics. Unfortunately, this tends to attract only those already interested in computer science. One alternative domain gaining momentum in computer science education research is music, which is showing interesting results with participants that have previous music knowledge. This paper presents a case study of teaching computer programming with music, in Brazil, to students with limited formal music experience. Through data collected in surveys, focus groups, and researchers’ observations, we show that in this context students can still learn and thrive as musical programmers. 

Characterizing exoplanets’ spectra is a crucial step in understanding the chemical and physical processes shaping their atmospheres and constraining their formation and evolutionary history. The X-SHYNE library is a homogeneous sample of 43 medium-resolution (R_λ~ 8000) infrared (0.3–2.5 μm) spectra of young (<500 Myr), low-mass (<20 M_Jup), and cold (T_eff~600–2000 K) isolated brown dwarfs and wide-separation companions observed with the VLT/X-Shooter instrument. To characterize our targets, we performed a global comparative analysis. We first applied a semiempirical approach. By refining their age and bolometric luminosity, we derived key atmospheric and physical properties, such as T_eff, mass, surface gravity (g), and radius, using the evolutionary modelCOND03. These results were then compared with the results from a synthetic analysis based on three self-consistent atmospheric models: the cloudy modelsExo-REMandSonora Diamondback, and the cloudless modelATMO. To compare our spectra with these grids we used the Bayesian inference codeForMoSA. We found similar L_bolestimates between both approaches, but an underestimated T_efffrom the cloudy models, likely due to a lack of absorbers that could dominate the J and H bands of early L. We also observed a discrepancy in the log(g) estimates, which are dispersed between 3.5 and 5.5 dex for mid-L objects. We interpret this as a bias caused by a range of rotational velocities leading to cloud migration toward equatorial latitudes, combined with a variety of viewing angles that result in different observed atmospheric properties (cloud column densities, atmospheric pressures, etc.). This interpretation is supported by the correlation of the color anomaly Δ(J–K) of each object with log(g) and the parameter f_sedthat drives the sedimentation of the clouds. Finally, while providing robust estimates of [M/H] and C/O for individual objects remains challenging, the X-SHYNE library globally suggests solar values that are consistent with a formation via stellar formation mechanisms. This study highlights the strength of homogeneous datasets in performing comparative analyses, reducing the impact of systematics, and ensuring robust conclusions while avoiding overinterpretation. 

Numerically computed with high accuracy are periodic travelling waves at the free surface of a two-dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are the following. (i) There is a boundary layer where the angle rises sharply from $$0^\circ$$ at the crest to a local maximum, which converges to $$30.3787\ldots ^\circ$$ , independently of the vorticity, as the amplitude increases towards that of the extreme wave, which displays a corner at the crest with a $$30^\circ$$ angle. (ii) There is an outer region where the angle descends to $$0^\circ$$ at the trough for negative vorticity, while it rises to a maximum, greater than $$30^\circ$$ , and then falls sharply to $$0^\circ$$ at the trough for large positive vorticity. (iii) There is a transition region where the angle oscillates about $$30^\circ$$ , resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form.