The well-known proof of Beurling’s Theorem in the Hardy space $H^2$, which describes all shift-invariant subspaces, rests on calculating the orthogonal projection of the unit constant function onto the subspace in question. Extensions to other Hardy spaces $H^p$ for $0 < p < \infty $ are usually obtained by reduction to the $H^2$ case via inner–outer factorization of $H^p$ functions. In this article, we instead explicitly calculate the metric projection of the unit constant function onto a shift-invariant subspace of the Hardy space $H^p$ when $1<p<\infty $. This problem is equivalent to finding the best approximation in $H^p$ of the conjugate of an inner function. In $H^2$, this approximation is always a constant, but in $H^p$, when $p\neq 2$, this approximation turns out to be zero or a non-constant outer function. Further, we determine the exact distance between the unit constant and any shift-invariant subspace and propose some open problems. Our results use the notion of Birkhoff–James orthogonality and Pythagorean inequalities, along with an associated dual extremal problem, which leads to some interesting inequalities. Further consequences shed light on the lattice of shift-invariant subspaces of $H^p$, as well as the behavior of the zeros of optimal polynomial approximants in $H^p$.

Twenty extra-framework S- and C-bearing species including anions (SO42–, SO32–, S2–, S52–, HS–, CO32–, C2O42– and HCO4–), radical anions (S2•–, S3•–, cis- and trans-S4•– and SO4•–) and neutral molecules (CO2, COS, H2S, cis- and trans-S4 and S6) have been identified in natural tectosilicates (members of the sodalite, cancrinite and scapolite groups) using a novel multimethodic approach based on several spectroscopic methods, single-crystal and powder X-ray diffraction, and electron microprobe and wet chemical analyses. Various polysulfide groups were detected in lazurite, haüyne, vladimirivanovite, sapozhnikovite, slyudyankaite, kyanoxalite, balliranoite, marinellite, tounkite, bystrite, sulfhydrylbystrite and meionite. Experimental data on mutual conversions of S-bearing species and structure modulations of these minerals under different temperatures, reducing and oxidizing environment as well as radiation-induced conversions provide a key for their use as markers of the crystallization or transformation conditions in Nature and during synthesis and modification.

We investigated the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, focusing on cotangent bundles of maximal flag manifolds. Our work provides an explicit description of the canonical symplectic structure on these spaces using connections and curvatures of principal bundles naturally associated with the underlying Lie groups. We extend classical results concerning the exactness of symplectic forms on adjoint orbits, previously known for specific Lie algebras, to arbitrary simple Lie groups. In particular, we identify conditions under which the Kostant–Kirillov–Souriau form on a regular adjoint orbit coincides with the canonical symplectic form of the cotangent bundle, yielding exact symplectic structures. The approach combines differential-geometric techniques with Lie-theoretic constructions, offering a unifying framework that connects the geometry of coadjoint orbits with symplectic structures on homogeneous spaces.

This commentary is part of a series of responses to Michele Goodwin’s 2025 LSA presidential address.

The death of chess Grandmaster and content creator Daniel Naroditsky sparked heated debate on the impact of cyberbullying on his mental health in the last 2 years of his life. Cyberbullying remains a widespread public health problem, with strong associations to mental disorders and significant relevance to psychiatric practice worldwide.

Expected returns on market volatility, which can be obtained from VIX futures prices in closed form using standard models, positively predict subsequent realized volatility returns. Volatility returns are negative on average. Following increases in volatility, expected volatility returns and subsequent realized volatility returns become more negative. Because realized volatility returns are negatively correlated with index returns, expected volatility returns also negatively predict S&P 500 index returns, but these results are less significant. The results are robust to a wide range of variations in the empirical setup and to small-sample biases.

We prove under certain conditions that any stable unfolding of a quasi-homogeneous map-germ with finite singularity type is substantial. We then prove that if an equidimensional map-germ is finitely determined, of corank 1, and either it admits a minimal stable unfolding or it is of multiplicity 3, then it admits a substantial unfolding if and only if it is quasi-homogeneous in some coordinate system. Based on this, we pose the following conjecture: a finitely determined map-germ is quasi-homogeneous in some coordinate system if and only if it admits a substantial unfolding.

Newton famously rejected the use of hypotheses in natural philosophy, in stark contrast to many of his contemporaries, such as Descartes, Huygens, and Leibniz, who employed hypothetical methods. This disagreement is often framed as one concerning the Hypothetico-Deductive (HD) method, but I argue that this is mistaken. The relevant hypothesis-based methods at issue were what I call inference to the best hypothesis and its stronger version, inference to the only plausible hypothesis. These methods were far more nuanced and plausible than HD, and they enjoyed widespread popularity among early modern thinkers, even among prominent experimental philosophers. Newton rejected them, nonetheless.

The ability of multirotor unmanned aerial vehicles (UAVs) to perform accurately in windy environments is crucial for extended use in outdoor applications. To design UAVs to operate in these environments, most studies have focused on static performance metrics such as thrust-to-weight ratio and endurance, without directly considering closed-loop control performance. This work develops a simplified metric that serves as a predictor for achievable disturbance rejection performance, enabling efficient UAV design selection without requiring full-scale nonlinear simulations. A reduced-order model is introduced to capture key aerodynamic and actuation characteristics, allowing for rapid evaluation of UAV configurations. The metric is validated against high-fidelity nonlinear simulations, demonstrating strong correlation with actual control performance. By bridging the gap between UAV structural optimization and closed-loop control behavior, this approach provides a practical tool for integrating disturbance rejection capabilities into UAV design processes. The practical utility of this metric is supported by experimental findings from related wind tunnel studies of fully-actuated UAVs, which demonstrate that actual disturbance rejection performance aligns with the trends predicted by the simplified correlation function.

Research syntheses have demonstrated that pronunciation instruction works, which means that whether instruction is effective is no longer an open question. Instead, contemporary intervention research has shifted to investigating how instruction can be further optimized, asking targeted questions about the instructional features that catalyze learning. In this paper, I examine the concept of instructional optimization, focusing on anticipated effect sizes (gains). I outline a four-pronged empirical approach to provide robust data for designing optimal pronunciation interventions. First, I describe the need for replication studies, which provide insight into the precision and stability of effects across distinct research samples and contexts. Second, I advocate for a systematic approach to study design. In such an approach, which is closely tied to the principles of replication, one or two variables are manipulated at a time, leading to a set of maximally comparable studies that lend insight into the impact of specific variables. Third, I explain the need to situate instruction within a longitudinal perspective to examine how robust and durable instructional gains are. Finally, I turn to adaptive approaches, where the surface format that instruction takes is highly variable and responsive to learner needs while the adaptive decision tree that generates the form is fixed and replicable.

Cardiac hydatidosis accounts for less than 2% of Echinococcus granulosus infections. Despite Syria’s high endemicity, paediatric cardiac involvement remains exceptionally rare and underreported. We report two Syrian children (aged 5.5 and 9 years) with giant interventricular septal hydatid cysts. Case 1 presented with significant hemodynamic obstruction, while Case 2 exhibited malignant ventricular arrhythmias. Both underwent successful cardiopulmonary bypass-assisted cystectomy with capitonnage repair and adjuvant albendazole therapy. These cases underscore (1) the life-threatening nature of advanced paediatric cardiac hydatidosis and (2) the critical role of early surgical intervention in endemic regions. Written informed consent for publication was obtained.

Citrus leprosis is a non-systemic disease caused by citrus leprosis virus (CiLV), which is classified as cytoplasmic (CiLV-C) or nuclear (CiLV-N) based on its replication site within host cells. Mite species in the genus Brevipalpus vector this virus. In Mexico, B. californicus and B. yothersi have been recorded in citrus orchards, with the latter species being most abundant and widely distributed. Despite extensive research, knowledge gaps remain regarding interactions between Brevipalpus and CiLV-C, the predominant virus type. We investigated the vector competence of both species, with a detailed analysis of density-dependent acquisition and transmission for B. yothersi. Virus acquisition was assessed using three densities (5, 10, or 15 adult females) for B. yothersi and a single desity (15 mites) for B. californicus. Virus detection and quantification were performed using a TaqMan probes targeting the viral movement protein gene. Transmission assays were conducted on Phaseolus vulgaris plants using viruliferous B. yothersi at all densities. B. californicus did not acquire the virus and was therefore excluded from transmission experiments. In contrast, B. yothersi successfully acquired the virus at all densities. While the proportion of viruliferous mites did not differ significantly among density treatments, viral load per mite was significantly higher at the lowest density. Virus transmission ocurred at all densities, with no significant differences in viral titres in inoculated plants. There results provide insights into density-related mite-virus interactions affecting CiLV-C transmission.