On the existence of four or more curved foldings with common creases and crease patterns

Consider an oriented curve $$\Gamma $$ Γ in a domain D in the plane $${\varvec{R}}^2$$ R 2 . Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space $${\varvec{R}}^3$$ R 3 . This can be expressed as the image of an “origami map” $$\Phi :D\rightarrow {\varvec{R}}^3$$ Φ : D → R 3 such that $$\Gamma $$ Γ is the singular set of $$\Phi $$ Φ , the word “origami” coming from the Japanese term for paper folding. We call the singular set image $$C:=\Phi (\Gamma )$$ C : = Φ ( Γ ) the crease of $$\Phi $$ Φ and the singular set $$\Gamma $$ Γ the crease pattern of $$\Phi $$ Φ . We are interested in the number of origami maps whose creases and crease patterns are C and $$\Gamma $$ Γ , respectively. Two such possibilities have been known. In the authors’ previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possible values for the number N of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if C is a non-closed simple arc, then $$N=4$$ N = 4 if and only if both $$\Gamma $$ Γ and C do not admit any symmetries. On the other hand, when C is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.

Synergistic effects of carbon-encapsulated cobalt/tricobalt tetroxide nanocapsules on hydrogenation of 4-nitrophenol

In this paper, we fabricated carbon-encapsulated cobalt/tricobalt tetroxide complex nanomaterials (Co/Co3O4@C) by a simple arc-discharged method and controlled annealing in air. Controlled annealing can solve the problem of poor hydrophilicity of materials prepared by arc-discharged method, and what’s more, a large number of Co/Co3O4 heterointerfaces can be constructed. Carbon-encapsulated Co/Co3O4 exhibited excellent catalytic performance in the catalytic hydrogenation of 4-nitrophenol (4-NP), even better than that of the reported precious metal catalysts in the literature. Excellent catalytic performance not only contributed to improved hydrophilicity, but also contributed to the synergistic effects of Co and Co3O4 on the catalytic 4-NP hydrogenation. The preparation process is appropriate for synthesizing various defective-rich metals or alloy@carbon electrocatalysts.

Seeking Freedom in Multiple Contexts

Many individuals have re-evaluated their aspirations, sense of self, and possible futures as they have moved from one social and cultural context to another. Eighteenth and nineteenth-century slave narratives commonly depicted a simple arc from darkness and suffering to Christian salvation. The life trajectory of one Sudanese woman named Halima led from her original home in Dar Fur’s hinterland and across the Sahara to Cairo’s slave market. There, in 1817, the French doctor Charles Dussap purchased her, making her his slave-wife. Multiple primary documents reveal Halima’s sense of self and aspirations for freedom throughout her life trajectory. By combining her autoethnography of her original culture, a detailed description of her conversion, her husband’s account of their early relationship, the observations of CMS missionaries, the writings of Ismayl Urbain (her lover), and the Saint-Simonian archive, her iterative construction of self, multiple imagined futures, and interactions with Sudanese, Egyptians, and Europeans will be examined and evaluated in six contexts.