The Euler numbers are the coefficients En in the expansion

(1)

Since sec x is an even function and tan x an odd one, this decouples into

(2)

and

(3)

where all three series converge if and E2n + 1 are sometimes called the secant numbers (or secant coefficients) and tangent numbers (or tangent coefficients) respectively.

We document a strong, negative relation between the curvature of stock price paths (i.e., price-path convexity) and future short-horizon returns at both the aggregate and firm levels. This relation obtains regardless of the cumulative return during the convexity estimation period. At the aggregate level, convexity is a better predictor of future returns than many commonly used predictors. At the firm level, this effect is not explained by known return predictors, microstructure frictions, or illiquidity. Using survey-based expectations of short-horizon returns, we show that the negative relation between convexity and future returns is driven in part by overextrapolation of past returns.

The article argues for an emendation in Plin. HN 9.126. Modern editors are accustomed to print the text cum testa uiuas, adopting J. Hardouin’s conjecture for cum terra uitis, the reading transmitted in most manuscripts. Nevertheless, the overlooked manuscript reading contritis conchis allows us to deduce a palaeographically neater solution contritis if conchis is considered a gloss which entered the text.

The crystal structure of delamanid has been solved and refined using synchrotron X-ray powder diffraction data and optimized using density functional theory techniques. Solution and refinement of the structure presented significant difficulties, and the result should be considered proposed or approximate. Delamanid crystallizes in the space group P212121 (#19) with a = 67.3701(18), b = 12.86400(9), c = 5.65187(12) Å, V = 4,898.19(14) Å3, and Z = 8 at 295 K. There are two independent delamanid molecules, with different conformations, which are essentially identical in energy. The crystal structure consists of layers of delamanid molecules perpendicular to the a-axis. The imidazooxazole ring systems stack along the b-axis, and the trifluoromethyl groups make up the boundaries of the corrugated layers. There are no classical hydrogen bonds in the crystal structure. Eight C–H···O and one C–H···N hydrogen bonds contribute to the lattice energy. The powder pattern has been submitted to the International Centre for Diffraction Data for inclusion in the Powder Diffraction File™ (PDF®).

In this article, the author examines the influence of Immanuel Kant’s philosophical ideas on Hans Kelsen’s early theory of international law. He situates Kelsen’s work within the post-World War I context, where Kant’s vision of perpetual peace significantly impacted the creation of international organizations. The article delves into Kelsen’s seminal work “Das Problem der Souveränität und die Theorie des Völkerrechts,” exploring how Kelsen’s pure theory of law parallels and diverges from Kant’s concepts. While Kelsen’s ideas were shaped by Kantian philosophy, particularly in promoting a lawful international order, Kelsen transcended Kant by developing a more rigorous, epistemologically grounded legal theory. The author argues that Kelsen’s adaptation of Kantian principles reflects both a continuation and transformation of Kant’s vision, tailored to the political and cultural challenges of early 20th-century Europe.

The crystal structure of iprodione has been solved and refined using synchrotron X-ray powder diffraction data and optimized using density functional theory techniques. Iprodione crystallizes in the space group P21/c (#14) with a = 15.6469(3), b = 22.8436(3), c = 8.67226(10) Å, β = 94.1303(7)°, V = 3,091.70(9) Å3, and Z = 8 at 298 K. The crystal structure contains clusters of four iprodione molecules. The only two classical N–H···O hydrogen bonds in the structure are both intramolecular. The powder pattern has been submitted to the International Centre for Diffraction Data for inclusion in the Powder Diffraction File™ (PDF®).

The application by states of economic principles in education has not produced good results in access to education in low-income and less-developed countries. This prompted UNESCO to designate countries with substantial problems of access to education and illiteracy as the E-9 countries, which include Nigeria. Nigeria’s status as an E-9 country indicates the existence of considerable problems in education, and where necessary, statistical evidence will be used to elucidate Nigeria’s E-9 status. This article argues that the nature of the laws and policy mechanisms that control education in Nigeria suggests that the country seems to be responding to the contemporaneous demands of global programmes of action in education that are predicated on economic principles and driven by the tides of globalization instead of to the requirements of international human rights law.

This scoping review aims to offer a panoptic overview of the research on grit and L2 grit in second and foreign language learning. To this end, a “hybrid search strategy” (Wohlin et al., 2022) was implemented. Out of 1,111 records identified across 15 databases and 78 found applying the backward/forward snowballing technique, 233 empirical studies published between 2013 and 2025 were finally included. With a focus on study and scale quality, the results present (1) a zoom-in/zoom-out description of the research landscape, considering 30 bibliometric and methodological variables, and (2) an in-depth comparative analysis of the psychometric instruments used to measure both grit and L2 grit, examining 45 variables arranged into four categories: (a) scale design and administration, (b) means and standard deviations, (c) reliability of scales and subscales, (d) content, construct, and predictive validity. The review concludes with a discussion of relevant findings and evidence-based suggestions for future and quality-enhanced research.

The investigation of truncated theta series was popularized by Andrews and Merca. In this article, we establish an explicit expression with nonnegative coefficients for the bivariate truncated Jacobi triple product series:

\begin{equation*}\frac{1}{(z,q/z,q;q)_\infty}\sum_{n=0}^{m}(-1)^n(1-z^{n+1})(1-(q/z)^{n+1})q^{\binom{n+1}{2}},\end{equation*}

which can be regarded as a companion to Wang and Yee’s truncation of the triple product identity. As applications, our result confirms a conjecture of Li, Lin, and Wang and implies a family of linear inequalities for a bi-parametric partition function. We also work on another truncated triple product series arising from the work of Xia, Yee, and Zhao and derive similar nonnegativity results and linear inequalities.

We study weighted Sobolev inequalities on open convex cones endowed with α-homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal rearrangement-invariant function spaces for these weighted Sobolev inequalities. Both optimal target and optimal domain spaces are characterized. Abstract results are accompanied by general yet concrete examples of optimal function spaces. For these examples, the class of so-called Lorentz–Karamata spaces, which contains in particular Lebesgue spaces, Lorentz spaces, and some Orlicz spaces, is used.