We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity $p \geq 5$, which is known to be globally well-posed in the Sobolev space $H^\sigma(\mathbb{T})$, for every $\sigma \geq 1$, thanks to the conservation and finiteness of the energy. For regularities σ < 1, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures µs, with covariance operator $(1-\Delta)^s$, for s in a range $(s_p,\frac{3}{2}]$. We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures µs, with additional Lq-bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These Lq-bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain’s invariant argument [7] and to arecent work by Forlano-Tolomeo in [18].

This article explores biophilic (nature-centred) instrument design and its intersection with architecture and music. While the connection between these disciplines is often discussed figuratively, they are less often combined in practice. The Biophilic Instrument Pavilion (BIP), a site-responsive sound and light installation, serves as a model for such a collaboration using biophilic design as a unifying principle. This multidisciplinary project demonstrates spatialisation in ecological, sonic, visual and social contexts, offering insights into environmental instrument practices and collaborative creative processes.

This article examines women’s work in rural areas of the Republic of Venice between the seventeenth and eighteenth centuries, with a focus on the district of Padua. By applying the verb-oriented method to judicial sources, it reconstructs a detailed and nuanced picture of female labour, highlighting the extent and continuity of women’s economic activities. The findings show that women were engaged across all sectors of the economy, including sectors far beyond care and housework. They played an active role in agriculture, trade, and manufacturing. Moreover, their work was fully integrated into the household economy, structured around seasonal labour demands, and performed throughout the year. A comparison with previous studies suggests that women’s work in the Venetian countryside aligned closely with broader European patterns. These findings underscore the fundamental contribution of female labour to household survival, challenging long-standing assumptions about women’s economic roles in Italy and the Mediterranean.

We prove that every irreducible component of the coarse Kollár-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi–Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of Hacking–Keel–Yu. The proof combines tools from log smooth deformation theory, the minimal model program, punctured log Gromov–Witten theory, and mirror symmetry.

This guide provides a philosophical framework and practical advice for gathering, analyzing, and reporting a particular type of qualitative data. These data are obtained from including an open-text box following the key quantitative question in survey-style studies with the request to ‘Please explain your response’. While many studies currently collect such data, they often either fail to report or analyze it, or they conduct unstructured analyses with limited detail, often mistakenly referring to it as ‘thematic analysis’. Content analysis provides a well-established framework for analyzing such data, and the simplicity of the data form allows for a highly pragmatic and flexible approach. The guide integrates the concept of reflexivity from qualitative research to navigate the large number of researcher degrees of freedom involved in the process, particularly in working with the second coder. It begins by arguing for the value of this data, before outlining the guide’s philosophy, offering advice on maximizing the validity of your data, and addressing the common concern of confabulation. It then provides advice on developing a coding scheme, recruiting and collaborating with a second coder, and writing your report, considering the potential role of large language models at these various stages. Additionally, it provides a checklist for reviewers to evaluate the quality of a given analysis. Throughout the guide, a running example is used to demonstrate the implementation of the provided advice, accompanied by extensive example materials in the online repository, which can be used to practice the method.

The paper discusses the stochastic dynamics of the vortex shedding process in the presence of external harmonic excitation and coloured multiplicative noise. The situation is encountered in a turbulent practical combustor experiencing combustion instability. Acoustic feedback and turbulent flow are imitated by the harmonic and stochastic excitations, respectively. The Ornstein–Uhlenbeck process is used to generate the noise. A low-order model for vortex shedding is used. The Fokker–Planck framework is used to obtain the evolution of the probability density function of the shedding time period. Stochastic lock-in and resonance characteristics are studied for various parameters associated with the harmonic (amplitude, frequency) and noise (amplitude, correlation time, multiplicative noise factor) excitations. We observed that: (i) the stochastic lock-in (s-lock-in) boundary strongly depends on the noise correlation time; (ii) the parameter sites for s-lock-in can be approximately identified from the noise-induced shedding statistics; and (iii) stochastic resonance is significant for some intermediate correlation times. The effects of the above-mentioned observations are discussed in the context of combustion instability.

It has often been considered that the representations of the gods in Ancient Central Mexico were purely symbolic and that we should not look for the presence of glottograms, i.e. signs that encode linguistic units pronounced in the Nahuatl language. This article intends to demonstrate that we should reject the image/writing dichotomy in this context. In order to understand the identity of the Nahua gods, it is necessary to combine symbolic deciphering with a reading of the names embedded in their bodies and ornaments. This article takes the example of several representations of gods in codices of the Aztec tradition. It shows that this embedded script used the main scriptural techniques known in Nahuatl writing: logograms, phonograms, and indicators. In this way, the identity of the god, and therefore its ritual effectiveness, was expressed simultaneously visually and phonically.

A deep-learning-based closure model to address energy loss in low-dimensional surrogate models based on proper-orthogonal-decomposition (POD) modes is introduced. Using a transformer-encoder block with an easy-attention mechanism, the model predicts the spatial probability density function of fluctuations not captured by the truncated POD modes. The methodology is demonstrated on the wake of the Windsor body at yaw angles of $\delta = [2.5^\circ ,5^\circ ,7.5^\circ ,10^\circ ,12.5^\circ ]$, with $\delta = 7.5^\circ$ as a test case, and in a realistic urban environment at wind directions of $\delta = [-45^\circ ,-22.5^\circ ,0^\circ ,22.5^\circ ,45^\circ ]$, with $\delta = 0^\circ$ as a test case. Key coherent modes are identified by clustering them based on dominant frequency dynamics using Hotelling’s $T^2$ on the spectral properties of temporal coefficients. These coherent modes account for nearly $60 \,\%$ and $75 \,\%$ of the total energy for the Windsor body and the urban environment, respectively. For each case, a common POD basis is created by concatenating coherent modes from training angles and orthonormalising the set without losing information. Transformers with different size on the attention layer, (64, 128 and 256), are trained to model the missing fluctuations in the Windsor body case. Larger attention sizes always improve predictions for the training set, but the transformer with an attention layer of size 256 slightly overshoots the fluctuation predictions in the Windsor body test set because they have lower intensity than in the training cases. A single transformer with an attention size of 256 is trained for the urban flow. In both cases, adding the predicted fluctuations close the energy gap between the reconstruction and the original flow field, improving predictions for energy, root-mean-square velocity fluctuations and instantaneous flow fields. For instance, in the Windsor body case, the deepest architecture reduces the mean energy error from $37 \,\%$ to $12 \,\%$ and decreases the Kullback–Leibler divergence of velocity distributions from ${\mathcal{D}}_{\mathcal{KL}}=0.2$ to below ${\mathcal{D}}_{\mathcal{KL}}=0.026$.

In the last few years, Hindu nationalism’s effort to shape the Hindu identity of the nation has intensified. Apart from its move to assert cultural homogenisation over the diverse landscape, this ideology produces a newer understanding of spaces in the land. When it is read as a part of the broader Hindutva movement, the use of violence, bureaucratic overreach, or judicial intervention to rewrite the sacred topography of the land unmasks the territorial goal of Hindu Rashtra. The territorial manifestation of this ideology takes a strident effort inside the country to encroach and reclaim the spaces inhabited by the “other” as Hindu spaces in the name of the nation. This immediately establishes a clear and precise correlation between the spaces and the nature of the spaces. This territorialisation of the spaces indicates the spatial rearrangement of the public spaces to marginalise minorities, invisibilise Muslims, and push them into the “private” space.

Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then completely characterized by the number of elastic collisions. The rules of mathematical billiards may be simple, but the possible behaviours of billiard trajectories are endless. In fact, several fundamental theory questions in mathematics can be recast as billiards problems. A billiard trajectory is called a periodic orbit if the number of distinct collisions in the trajectory is finite. We show that periodic orbits on such billiard tables cannot have an odd number of distinct collisions. We classify all possible equivalence classes of periodic orbits on square and rectangular tables. We also present a connection between the number of different equivalence classes and Euler’s totient function, which for any positive integer N, counts how many positive integers smaller than N share no common divisor with N other than $1$. We explore how to construct periodic orbits with a prescribed (even) number of distinct collisions and investigate properties of inadmissible (singular) trajectories, which are trajectories that eventually terminate at a vertex (a table corner).

There is an increasing global focus on gender diversity and equality in the workplace, particularly regarding women in leadership roles. Our study explores this focus in the wine industry in Australia, examining women's representation in CEO, winemaker, viticulturist, and marketing roles. By using results from a previous Australian study, we find that women have significantly increased their presence in all roles but one (marketing role) when comparing 2007–2013 with 2021–2023. Our study also confirms that women are more likely to be in winemaking and viticulturist roles, conditional upon a woman being in the CEO role. However, women in winemaking and viticulturist roles still lag behind women in leadership roles across other industries in Australia. We offer conclusions and directions for future research.

Should voting rights be tied to citizenship? Over 20 million noncitizens pay taxes, own businesses and homes, send their children to schools, and make countless economic, social, and cultural contributions every day. Yet they cannot vote to select politicians who make policy that affects their daily lives. Today, noncitizens currently vote legally in local elections in 22 cities and towns in Maryland, Vermont, California, and Washington, DC. These practices have their roots in another little-known fact: noncitizens voted in 40 states at some point in time from the Founding until 1926. Noncitizens voted not only in local elections but also in state and federal elections, and they could hold office such as alderman. “Alien suffrage” was seen as a means to facilitate immigrant incorporation and citizenship, which it did in practice. This article examines the politics and practices of immigrant voting in the US, chronicling the rise and fall—and reemergence—of immigrant voting rights. It explores arguments for and against noncitizen voting, reviews evidence about its impact on policy and American political development, and considers its implications for immigration policy and democratic practice. Debate about immigrant voting rights can be viewed as a microcosm of broader debate about immigration, citizenship, and democracy reflected in scholarship and political conflict embroiling the nation, which holds valuable lessons for scholars and policy makers today. I argue, in a country where “no taxation without representation” was a rallying cry for revolution, such a proposition might not be so outlandish upon further scrutiny.